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On a class of residuated semilattice monoids
Fuzzy Sets and Systems 138 (2003) 149 – 176 www.elsevier.com/locate/fss
On a class of residuated semilattice monoids P. Flondor∗ , M. Sularia Department of Mathematics II, Polytechnic University of Bucharest, Splaiul Independentei St. 313, 77206 Bucharest, Romania Received 22 November 2001; received in revised form 18 July 2002; accepted 5 August 2002
Abstract In this paper, we consider the structure of residuated join semilattice monoid and we study its 3rst properties. The class of residuated join semilattice monoids is a variety of algebras. A representation of this structure as residuated mappings is given. Then we present the notion of normal 3lter together with a characterization theorem of homomorphic images based on this notion. Di6erent classes of residuated join semilattice monoids are introduced. c 2002 Elsevier B.V. All rights reserved. Keywords: Fuzzy logic; Real membership space; Multiple-valued logic; Residuated semilattice monoid; Residuated mapping; Quantale
1. Introduction Residuated monoids [1,2] are proved to be important structures of multiple-valued logic. The real membership space of fuzzy sets introduced by Zadeh is the real unit interval endowed with its usual linear order relation “6” and the three structures of commutative residuated monoid corresponding, respectively, to the binary operations of meet “∧”, bounded sum “⊕” and algebraic product “·” [26,27]. The precedent structures are particular cases of Heyting algebra, MV-algebra and BL-algebra. Starting from this fact di6erent ways to put together these structures have been considered with the aim to improve the mathematical foundation of the notion of fuzzy set [15,8,17,24,21].
∗
Corresponding author. E-mail addresses: [email protected] (P. Flondor), Mircea [email protected] (M. Sularia).
c 2002 Elsevier B.V. All rights reserved. 0165-0114/03/$ - see front matter PII: S 0 1 6 5 - 0 1 1 4 ( 0 2 ) 0 0 3 9 0 - 1
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Following HHajek [15] a BL-algebra is a residuated lattice (L; ∧; ∨; ∗; →; 0; 1) such that (i) x ∧ y = x ∗ (x → y), (ii) (x → y) ∨ (y → x) = 1, for all x; y ∈ L. It is to say that in [15] the notion of BL-algebra is considered basic for fuzzy logic (in the narrow sense). After the extension of the notion of MV-algebra [4,5] to the noncommutative case by Georgescu and Iorgulescu [11], a noncommutative extension of BL-algebras was introduced [12,16]. The new fact emerging from this extension (noncommutativity apart) is that condition (i) is not to be expected in the general case (this is due to the continuity of T-norms and a theorem of Aczel [9]). Another way to put this is that “∧” is not determined by “*” and “→” (in other terms “∧” is somewhat independent of “*” and “→”). From the logical point of view, we interpret “*” as (a kind of) conjunction (may be commutative) and “∨” as disjunction (“∧” is another conjunction). It seems natural to consider a “minimal” structure with only one conjunction “*” (may be noncommutative), disjunction “∨” (of semilattice type) and implication “→” (two of them in the noncommutative case) as a possible starting point for a theory of not necessarily commutative fuzzy logic. We call such a structure a WBL-algebra (we do not consider (ii) as necessary for the beginning). One can consider a WBL-algebra as the “compact” part of a quantale [23] (see below). This connection with the theory of quantales is viewed as a “completion”. As for some concrete “models” of logic to be developed over such structures, we can consider the logic of actions; after all applying actions is a noncommutative process. That is why we consider the composition of functions as the basic model for the noncommutative conjunction. The idea to represent such structures as residuated mappings comes from the possible applications in the theory of actions and causality.
2. Residuated semilattice monoids We recall that an integral ordered monoid [1,10] is an ordered monoid (L; ·; 6; 1) such that its unity element 1 is a greatest element, i.e. L is a set, · is a binary operation on L called multiplication, 6 is an order relation on L and 1 ∈ L is a constant such that the following conditions hold, for every element a ∈ L: 1. (L; ·; 1) is a monoid with the unity element 1; 2. the left translation a and the right translation a on L given by a (x) = a · x and a (x) = x · a are isotone mappings, i.e. for all x; y ∈ L, if x6y then a · x6a · y and x · a6y · a; 3. 1 = max L, i.e. (∀x ∈ L)x61. Denition 2.1. Let K() be the class of all algebras of type = (2; 2; 2; 2; 0). A residuated join semilattice monoid is a system L = (L; ∨; ·; →; →; ˜ 1) ∈ K(), such that the following conditions hold: (i) (L; ∨) is a join semilattice, where ∨ is the binary supremum on L;
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(ii) (L; ·; 6; 1) is an integral ordered monoid, where 6 is the unique order relation on L associated with ∨ and de3ned by x6y i6 x ∨ y = y; (iii) for all a; b ∈ L, the element a → b ∈ L is the right residual of b by a, i.e. a → b = max{x ∈ L=a · x 6 b}; (iv) for all a; b ∈ L, the element a→b ˜ ∈ L is the left residual of b by a, i.e. a→b ˜ = max{x ∈ L=x · a 6 b}: Lemma 2.2. Let L = (L; ∨; ·; →; →; ˜ 1) be a residuated join semilattice monoid. Then the following conditions hold, for all a; b; x; y ∈ L: (i) (ii) (iii) (iv) (v) (vi) (vii)
x · y6x and x · y6y; x · (x → y)6y and (x → ˜ y) · x6y; x6a → b i8 a · x6b; x6a→b ˜ i8 x · a6b; x6y i8 x → y = 1 i8 x → ˜ y = 1; a · (x ∨ y) = (a · x) ∨ (a · y); (x ∨ y) · a = (x · a) ∨ (y · a).
Proof. Let a; b; x; y ∈ L. Conditions 2.2(i)–(v) are the consequences of De3nition 2.1 which are easy to prove. We show that Eq. 2.2(vi) holds. From x6x ∨ y and y6x ∨ y it follows that a · x 6 a · (x ∨ y)
and
a · y 6 a · (x ∨ y):
This implies that (a · x) ∨ (a · y)6a · (x ∨ y). Suppose now that z ∈ L veri3es a · x6z and a · y6z. Using 2.2(iii) this implies that x6a → z and y6a → z, thus x ∨ y6a → z. Using again 2.2(iii), from the precedent relation it follows that a · (x ∨ y)6z. Thus, Eq. 2.2(vi) is veri3ed. In a similar way, one shows that Eq. 2.2(vii) also holds. Lemma 2.3. Let L = (L; ∨; ·; →; →; ˜ 1) ∈ K(). Then the following conditions are equivalent: (i) L is a residuated join semilattice monoid; (ii) L satis:es the following equations, for all x; y; z ∈ L: x ∨ x = x;
(1)
x ∨ y = y ∨ x;
(2)
(x ∨ y) ∨ z = x ∨ (y ∨ z);
(3)
x ∨ 1 = 1;
(4)
(x · y) · z = x · (y · z);
(5)
x · 1 = x;
(6)
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1 · x = x;
(7)
z · (x ∨ y) = (z · x) ∨ (z · y);
(8)
(x ∨ y) · z = (x · z) ∨ (y · z);
(9)
(x · (x → y)) ∨ y = y;
(10)
((x→y) ˜ · x) ∨ y = y;
(11)
x → (x ∨ y) = 1;
(12)
x→(x ˜ ∨ y) = 1;
(13)
x → (y → z) = (y · x) → z;
(14)
x→(y ˜ →z) ˜ = (x · y)→z: ˜
(15)
Proof. (i) ⇒ (ii): Suppose that L veri3es condition 2.3(i). We show that all Eqs. (1)–(15) are valid in L. Eqs. (1)–(9) follow from De3nition 2.1(i) and (ii) and Lemma 2.2(vi) and (vii). Eqs. (10)–(13) follow from Lemma 2.2(ii) and (v). De3nition 2.1(ii) and Lemma 2.2(ii) imply (y · x) · (x → (y → z)) = y · (x · (x → (y → z))) 6 y · (y → z) 6 z: Using Lemma 2.2(iii) this implies x → (y → z) 6 (y · x) → z: Using again De3nition 2.1(ii) and Lemma 2.2(ii) and (iii) one obtains that y · (x · ((y · x) → z)) = (y · x) · ((y · x) → z) 6 z; which imply (y · x) → z 6 x → (y → z): Therefore, Eq. (14) holds. One can also verify that Eq. (15) holds. This completes the proof of condition 2.3(ii). (ii) ⇒ (i): Suppose that condition 2.3(ii) holds. We verify that all conditions 2.1(i)–(iv) are valid in L. Using Eqs. (1)–(9) one obtains that 2.1(i) and (ii) are satis3ed. We show that condition 2.1(iii) holds. Suppose that x6a → b. Using Eq. (10) one obtains that a·x6a·(a → b)6b, therefore a·x6b. Now suppose that a · x6b, i.e. b = (a · x) ∨ b. Using Eqs. (12) and (14) this implies x → (a → b) = (a · x) → b = (a · x) → ((a · x) ∨ b) = 1: The precedent relations imply x = x · 1 = x · (x → (a → b))6a → b. Thus condition 2.1(iii) holds. In a similar way, one obtains that condition 2.1(iv) also holds. This completes the proof of 2.3(i). Denition 2.4. Let R() be the class of all residuated join semilattice monoids and L; M ∈ R(). (i) P() is the algebraic category associated with the class R().
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(ii) Hom[L; M] is the set of all morphisms from L to M in the category P(); (L) is the set of all subalgebras of L; H(L) is the set of all homomorphic images of L and Is(L) is the set of all M which are isomorphic with L. (iii) i∈I Li is the direct product of a family (Li )i∈I of algebras from K() and j : i∈I Li → Lj is the canonical projection of rank j, for all j ∈ I . Consequence 2.5. The class R() is a variety of algebras from K(), i.e. the following conditions hold: (i)
(L) ∪ H(L) ∪ Is(L) ⊆ R(), for all L ∈ R(). (ii) i∈I Li ∈ R(), for any family (Li )i∈I of algebras from R(). Proof. From Lemma 2.3 it follows that R() is an equational class of algebras from K(). Therefore, both conditions 2.5(i) and (ii) hold. Example 2.6. (a) Let L = [0; 1] be the real unit interval and a; b ∈ L such that 0¡a¡b¡1. For all x; y ∈ L, de3ne x ∨ y = max{x; y}; x ∧ y = min{x; y}; x·y =
0 if x 6 a and y 6 b; x ∧ y if a ¡ x or b ¡ y;
1 if x 6 y; x → y = b if y ¡ x 6 a; y if y ¡ x and a ¡ x; if x 6 y; 1 x→y ˜ = a ∨ y if y ¡ x 6 b; y if y ¡ x and b ¡ x: Then L[0; 1] = ([0; 1]; ∨; ·; →; →; ˜ 1) ∈ R() (see [9]). (b) The negative cone (G − ; +; 6; 0) of any lattice group (G; +; ∧; ∨; 0) has a structure of residuated join semilattice monoid, L[G − ] = (G − ; ∨; +; →; →; ˜ 0) ∈ R(), where G − = {x ∈ G=x60} such that x → y = ((−x) + y) ∧ 0
and
x→y ˜ = (y + (−x)) ∧ 0; for all x; y ∈ G − :
(c) Let L be a set of points from the real plane R2 de3ned by L = {(x; x) ∈ R2 =0 6 x ¡ 1} ∪ {a; b; c; 1};
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where a = (1; 2); b = (2; 1); c = (2; 2); 1 = (3; 3). De3ne four binary operations ∨; ·; → and → ˜ on L such that for all (x; y); (x ; y ); u; v ∈ L, (x; y) ∨ (x ; y ) = (max{x; x }; max{y; y }) u · 1 = u = 1 · u;
u · v = v · u = (0; 0); if u = 1 and v = 1; 1; if u ∨ v = v; u → v = u→v ˜ = v; if u = 1; c; if u = 1 and u ∨ v = v: Then (L; ∨; ·; →; →; ˜ 1) ∈ R() and the underlying poset (L; 6; 1) is not a lattice: the set of all lower bounds of the two elements set {a; b} is de3ned by [0; 1) = {(x; x) ∈ R2 =06x¡1} and [0; 1) is a subset of L without a greatest element. Now we introduce speci3c residuated mappings on a join semilattice. Denition 2.7. Let f : L → L be an endomorphism of a join semilattice (L; ∨), i.e. f(x ∨ y) = f(x) ∨ f(y), for all x; y ∈ L. We call f a residuated endomorphism if there exists a function g : L → L such that the following conditions hold, for all x; y ∈ L: (i) x6y implies g(x)6g(y); (ii) f(g(x))6x6g(f(x)). Let Res(L; ∨) the set of all residuated endomorphisms of (L; ∨). If f ∈ Res(L; ∨) then the function g satisfying 2.7(i) and (ii) is unique. We denote this unique function by f+ and call it the residual of f [2]. If f ∈ Res(L; ∨) then the following conditions hold, for all ∈ L: • f+ () = max{x ∈ L=f(x)6}; • f() = min{x ∈ L=6f+ (x)}. Example 2.8. Let L = (L; ∨; ·; →; →; ˜ 1) ∈ R(). The left translation a and the right translation a given by a (x) = a · x and a (x) = x · a are residuated endomorphisms of (L; ∨) such that a+ = ra and a+ = la , where ra is the right residual and la is the left residual by a ∈ L given by ra (x) = a → x and la (x) = a→x. ˜ The following lemma shows that a residuated join semilattice monoid can be associated with some families of residuated endomorphisms (see De3nition 2.7). Then a representation theorem for algebras from R() will be proved. Lemma 2.9. Let T = (L; ∨; 1); ; be an upper bounded join semilattice (L; ∨; 1) together with two functions and from L to Res(L; ∨) such that the following conditions hold, for all a; b ∈ L: (C1) (1)(a) = a = (a)(1) and (1)(a) = a = (a)(1);
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(C2) a6b i8 (a)(x)6(b)(x), for all x ∈ L i8 (a)(x)6(b)(x), for all x ∈ L; (C3) (a) ◦ (b) = (b) ◦ (a), where ◦ is the usual composition of functions. There exists a structure of residuated join semilattice monoid L[T] = (L[T]; ∨; ·; →; →; ˜ 1) associated with T such that L[T] = L and the following relations hold: (i) (x)(y) = x · y = (y)(x); (ii) x → y = + (x)(y); (iii) x → ˜ y = + (x)(y). Proof. De3ne three binary operations ·; → and → ˜ on L by x · y = (x)(y);
x → y = + (x)(y)
and
x→y ˜ = + (x)(y)
for all x; y ∈ L, where + (x) : L → L is the residual of (x) ∈ Res(L; ∨) and + (x) : L → L is the residual of (x) ∈ Res(L; ∨). Using (C1) and (C3) it follows that the following relations holds, for all x; y; z ∈ A: x · y = (x)(y) = (x)((y)(1)) = ((x) ◦ (y))(1) = ((y) ◦ (x))(1) = (y)((x)(1)) = (y)(x); x · 1 = (x)(1) =x = (1)(x) = 1 · x; x · (y · z) = (x)(y · z) = (x)((z)(y)) = ((x) ◦ (z))(y) = ((z) ◦ (x))(y) = (z)((x)(y)) = (z)(x · y) = (x · y) · z: Suppose now that x6y and a ∈ L. Using (i) and (C2) it follows that x · a = (x)(a) 6 (y)(a) = y · a
and
a · x = (x)(a) 6 (y)(a) = a · y:
From the precedent properties it follows that both conditions (i) and (ii) from De3nition 2.1 hold. Let a; x; b ∈ L. From De3nition 2.7 one obtains that a · (a → b) = (a)(+ (a)(b)) 6 b; (a→b) ˜ · a = (a)(+ (a)(b)) 6 b; a · x 6 b implies (a)(x) 6 b; thus x 6 + (a)(b) = a → b; x · a 6 b implies (a)(x) 6 b; thus x 6 + (a)(b) = a→b: ˜ This completes the proof of Lemma 2.9.
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Theorem 2.10 (Representation of a residuated join semilattice monoid). Let L=(L; ∨; ·; →; →; ˜ 1) be a residuated join semilattice monoid. Then there exists a system T = (L; ∨; 1); ; such that conditions (C1)–(C3) from Lemma 2.9 hold and L = L[T]. Proof. From Example 2.8 it follows that one can de3ne a function : L → Res(L; ∨) by (a) = a and a function : L → Res(L; ∨) by (a) = a . Then the system T = (L; ∨; 1); ; satis3es all conditions (C1)–(C3) from Lemma 2.9 and L = L[T]. Consequence 2.11 (Cayley-type representation): Let L ∈ R() and 1L : L → L be the identity function on L. Then there exists an isomorphism from L onto a residuated join semilattice monoid (L) of residuated endomorphisms such that its underlying monoid is a submonoid of (Res(L; ∨); ◦; 1L ). Proof. There exists an order relation 6 on Res(L; ∨) such that for all f; g ∈ Res(L; ∨), f 6 g i6 f(x) 6 g(x)
for all x ∈ L
(16)
and the supremum of the two element subset {f; g} exists such that sup{f; g} = f ∨ g ∈ Res(L; ∨);
(17)
where f ∨ g : L → L is de3ned by (f ∨ g)(x) = f(x) ∨ g(x)
for all x ∈ L:
(18)
The residual of f ∨ g is the function (f ∨ g)+ : L → L de3ned by (f ∨ g)+ (x) = f+ (x) ∨ g+ (x)
for all x ∈ L:
(19)
Let Res(1L ] be the subset of Res(L; ∨) de3ned by Res(1L ] = {f ∈ Res(L; ∨)=f 6 1L }:
(20)
(Res(1L ]; ∨) is a join semilattice;
(21)
(Res(1L ]; ◦; 6; 1L ) is an integral ordered monoid:
(22)
Then
From Theorem 2.10 it follows that there exists a system T = (L; ∨; 1); ; such that conditions (C1)–(C3) from Lemma 2.9 hold and L = L[T]. Let (L) be the image set of the function : L → Res(L; ∨). Then the following conditions hold for all x; y ∈ L: (L) = {(a)=a ∈ L} = {f ∈ Res(1L ]=f(x) = f(1) · x for all x ∈ L} ⊆ Res(1L ];
(23)
(x ∨ y) = (x) ∨ (y);
(24)
(x · y) = (x) ◦ (y);
(25)
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(x → y) = max{f ∈ (L)=(x) ◦ f 6 (y)};
(26)
(x→y) ˜ = max{f ∈ (L)=f ◦ (x) 6 (y)}:
(27)
From 2.9 (C1) it follows that the function : L → Res(L; ∨) is injective: a = b implies (a)(1) = a = b = (b)(1); thus (a) = (b): From relations (21)–(27) it follows that there exists a structure of a residuated join semilattice monoid (L) = ((L); ∨; ◦; ⇒; ⇒; ˜ 1L ) ∈ R(); with the underlying poset ((L); 6) such that for all x; y ∈ L, (x) ⇒ (y) = (x → y); (x) ⇒ ˜ (y) = (x→y): ˜ The restriction of the function to its image set (L) de3ned by the correspondence L x → (x) ∈ (L) is an isomorphism from L onto (L) in the class R(). This completes the proof of Consequence 2.11. Consequence 2.12. Let (L; ∨; 1) be an upper bounded join semilattice and Res(1L ] be the set of all residuated endomorphisms f ∈ Res(L; ∨) such that f61L . Then the following conditions are equivalent: (i) There exists a structure L = (L; ∨; ·; →; →; ˜ 1) ∈ R() with the underlying lattice (L; ∨; 1). (ii) There exists a function : L → Res(1L ] such that the following conditions hold, for all x; y ∈ L: (x ∨ y) = (x) ∨ (y);
(28)
(1)(x) = x = (x)(1);
(29)
((x)(y)) = (x) ◦ (y):
(30)
If one of conditions (i) and (ii) holds then there exists a structure of residuated join semilattice monoid (L) with the underlying set (L) ⊆ Res(1L ] such that the restriction of the function to its image set (L) is an isomorphism from L onto (L). Proof. (i) ⇒ (ii): Suppose that (i) holds. From Theorem 2.10 it follows that there exists a system T = (L; ∨; 1); ; such that conditions (C1)–(C3) from Lemma 2.9 hold and L = L[T]. From (C1) it follows that (29) holds. From Lemmas 2.9(i), 2.2(i) and 2.3(10) it follows that for all x; y; z ∈ L, the following relations hold: (x)(y) = x · y 6 y = 1L (y);
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(x ∨ y)(z) = (x ∨ y) · z = (x · z) ∨ (y · z) = (x)(z) ∨ (y)(z) = ((x) ∨ (y))(z); ((x)(y))(z) = (x · y)(z) = (x · y) · z = x · (y · z) = (x)(y · z) = (x)((y)(z)) = ((x) ◦ (y))(z): Therefore, condition (ii) holds. (ii) ⇒ (i): Suppose that (ii) holds. De3ne a function : L → Res(1L ] by (x)(y) = (y)(x)
for all x; y ∈ L:
(31)
We verify that the system T = (L; ∨; 1); ; satis3es all conditions (C1)–(C3) from Lemma 2.9, which implies that condition (i) holds. Let a; b ∈ L. Then (1)(a) = (a)(1) = a = (1)(a) = (a)(1); i.e. condition 2.9(C1) holds. From (28) and (29) it follows that a 6 b i6 (a)(x) 6 (b)(x)
for all x ∈ L:
(32)
Let x ∈ L. Then using relation (31) one obtains (a ∨ b)(x) = (x)(a ∨ b) = (x)(a) ∨ (x)(b) = (a)(x) ∨ (b)(x) = ((a) ∨ (b))(x): Thus (a ∨ b) = (a) ∨ (b), which implies a 6 b i6 (a)(x) 6 (b)(x)
for all x ∈ L:
(33)
From (32) and (33) it follows that condition 2.9(C2) holds. Using relations (30) and (31) one obtains ((a) ◦ (b))(x) = (a)((b)(x)) = (a)((x)((b)) = ((a) ◦ (x))(b) = ((a)(x))(b) = ((b) ◦ (a))(x); for all a; b; x ∈ L. This implies that condition 2.9(C3) holds. Thus, equivalence (i) ⇔ (ii) is proved. The proof of the second part of consequence 2.12 is included in the proof of Consequence 2.11. Comment. Let R∗ () be the class of all (L) ∈ R() previously de3ned in the proof of consequence 2.11 for every L ∈ R(). Using the precedent results, one can prove that there exists an equivalence of categories from P() to the algebraic category P∗ () associated with the class R∗ (). Let = (2; 2; 2; 2; 2; 0). For every L = (L; ∨; ·; →; →; ˜ 1) ∈ R(), let L = (L; ∨; ∗l ; →; ∗r ; →; ˜ 1) be an algebra of type , where ∗l and ∗r are binary operations on L de3ned by x ∗l y = x · y = y ∗r x
for all x; y ∈ L:
We remark also that there exists an isomorphism of categories from P() to the algebraic category P( ) associated with the class R( ) of all algebras L , for any L ∈ R(). We have HomP() [L; M] = HomP( ) [L ; M ]
for all L; M ∈ P():
We shall use in the sequel the following duality principle: let e be an equation having the set of symbols of operations included in {∨; ∗l ; →; 1}; let eo be the equation obtained from e by replacing
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each occurrence of ∗l in e by ∗r and each occurrence of → in e by →; ˜ if e holds in R( ) then eo holds in R( ). 3. Homomorphic images In this section, we present a characterization theorem for homomorphic images of L in the algebraic category P() introduced in De3nition 2.4. We will use the usual notions of congruence relation and semilattice 3lter. Denition 3.1. Let L = (L; ∨; ·; →; →; ˜ 1) be a residuated join semilattice monoid. A normal :lter on L is a subset F of L with the property that there exists a congruence relation R of L such that F = {x ∈ L=xR1}: Lemma 3.2. Let R be a congruence relation of L and FR = {x ∈ L=xR1} be the normal :lter associated with R. For all x; y ∈ L, de:ne er (x; y) = (x → y) · (y → x) and el (x; y) = (x → ˜ y) · (y→x). ˜ Then the following conditions hold: (i) xRy i8 er (x; y) ∈ FR ; (ii) xRy i8 el (x; y) ∈ FR ; (iii) For any normal :lter F on L there exists an unique congruence relation R such that F = {x ∈ L=xR1}: Proof. (i) Let x; y ∈ L. Suppose that xRy. From yRy; yRx and xRx it follows that x → yRy → y
and
y → xRx → x;
but 1 = y → y = x → x, thus er (x; y)R1, i.e. er (x; y) ∈ FR . Suppose now that er (x; y) ∈ FR , i.e. (x → y) · (y → x)R1: From x → yRx → y; 1 = (x → y)·(y → x) → (x → y); 1 → (x → y) = x → y, one obtains that x → yR1, but xRx, thus x · (x → y)Rx. We have also yRy. Then (x · (x → y)) ∨ yRx ∨ y, but using Eq. (10) [Lemma 2.3(ii)] we have (x · (x → y)) ∨ y = y, thus yRx ∨ y. Also we have xRx ∨ y, therefore xRy. (ii) The equivalence (ii) follows in a similar way with (i). (iii) Let F be a normal 3lter and R; R be congruence relations such that F = {x ∈ L=xR1} = {x ∈ L=xR 1}: From (i) it follows that xRy i6 er (x; y) ∈ FR = F = FR i6 xR y. Therefore, R = R . The following theorem presents an algebraic characterization of normal 3lters. Theorem 3.3. Let F ⊆ L. The following conditions are equivalent: (i) F is a normal :lter of L.
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(ii) F satis:es: (FN1) F is a :lter of the semilattice (L; ∨); (FN2) F is a submonoid of (L; ·); (FN3) er (a · x; x · a) ∈ F and el (a · x; x · a) ∈ F, for all a ∈ F and x ∈ L. (iii) F satis:es: (FN1∗ ) 1 ∈ F; (FN2∗ ) x ∈ F and x → y ∈ F implies y ∈ F; (FN3∗ ) x → y ∈ F i8 x → ˜ y ∈ F. Proof. (i) ⇒ (ii): Suppose that (i) holds, i.e. F = {x ∈ L=xR1}, for some congruence relation R. It is clear that F satis3es (FN1) and (FN2). We veri3es now the 3rst part of (FN3). Let a ∈ F and x ∈ L. It follows that aR1; a · xR1 · x
and
x · aRx · 1
but x · 1 = x = 1 · x and 1 = x → x, thus (a · x) → (x · a)R1
and
(x · a) → (a · x)R1:
This implies that er (a · x; x · a) ∈ F. The relation el (a · x; x · a) ∈ F from the second part of (FN3) follows in a similar way by a dual argument. Thus (ii) holds. (ii) ⇒ (iii): Suppose that (ii) holds. We have F = ∅. Let x0 ∈ F. From x0 61 and (FN1) it follows that (FN1∗ ) holds. Let x; y ∈ L such that x ∈ F and x → y ∈ F. From (FN2) it follows that x · (x → y) ∈ F, but x · (x → y)6y, thus using (FN1) one obtains that y ∈ F. Therefore, condition (FN2∗ ) holds. One obtains in a similar way the following relation, for all x; y ∈ L: (1) x ∈ F and x → ˜ y ∈ F implies y ∈ F. We verify now that (FN3∗ ) also holds, i.e. for all x; y ∈ L: (2) x → y ∈ F implies x → ˜ y ∈ F. (3) x → ˜ y ∈ F implies x → y ∈ F. For (2) let x; y ∈ L such that a = x → y ∈ F. From (FN3) and (FN1∗ ) it follows that (4) u = (a · x)→(x ˜ · a) ∈ F. (5) v = (x · a)→y ˜ = 1 ∈ F. From Eqs. (15) and (11) [Lemma 2.3(ii)] one obtains that (6) v→(u ˜ →((a ˜ · x)→y)) ˜ = 1 ∈ F. Using (1), relations (4)–(6) imply (a · x)→y ˜ ∈ F, but (a · x)→y ˜ = a→(x ˜ → ˜ y), thus (7) a→(x ˜ → ˜ y) ∈ F. Using (1), from a = x → y ∈ F and (7) one derives x → ˜ y ∈ F. Thus (2) holds. Relation (3) follows in a similar way by a dual argument. Therefore, (iii) holds.
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161
(iii) ⇒ (i): Suppose that (iii) holds. For all x; y ∈ L, de3ne xRy i6 er (x; y) = (x → y) · (y → x) ∈ F: Then R is a congruence relation of L and F = {x ∈ L=xR1}. Thus (i) holds. Now we establish a connection between the structure of residuated join semilattice monoid and the structure of quantale [23] considered in the commutative case by Goguen [14]. Using this connection we derive a new expression of the Cayley-type representation and an example of homomorphic images in R(). Denition 3.4. A complete residuated join semilattice monoid is an algebra L = (L; ∨; ·; →; →; ˜ 1) such that the following conditions hold: (i) L ∈ R(); (ii) the underlying algebra (L; ∨; 1) is a complete join semilattice, i.e. for every nonempty subset A of L its supremum denoted by sup A or sup(A) exists in the corresponding poset (L; 6), where x6y i6 x ∨ y = y. Let CR() be the class of all complete residuated join semilattice monoids. Denition 3.5. An integral quasi-quantale is a system (L; ∨; ·; 1) such that the following conditions hold: (i) (L; ∨; 1) is a complete join semilattice; (ii) (L; ·; 6; 1) is an integral ordered monoid; (iii) x · (sup A) = sup(x · A) and (sup A) · x = sup(A · x), for all x ∈ L and A ⊆ L such that A = ∅, where ∨ is the binary supremum on L; x · A = {x · a=a ∈ A} and A · x = {a · x=a ∈ A}. Let IqQ be the class of all integral quasi-quantales. An integral quantale is an integral quasi-quantale (L; ∨; ·; 1) with 0, i.e. there exists 0 ∈ L such that 06x, for all x ∈ L. Let IQ be the class of all integral quantales (L; ∨; ·; 0; 1), where 0 is the least element of the underlying poset (L; 6). Example 3.6. (a) Let L = (0; 1] be the real unit interval without 0. Then the system (L; ∨; ·; 1) is an integral quasi-quantale, where x ∨ y = max{x; y}, for all x; y ∈ L and · is the algebraic product on L. This is the strict positive membership space of fuzzy sets. If f : X → [0; 1] is a membership function then the support set of f is de3ned by supp(f) = f−1 (0; 1] i.e. for all x ∈ X we have x ∈ supp(f) i6 f(x)¿0 [26,19,18,27,20]. (b) Let (L; ∨; ·; 1) such that 2.1(i) and (ii) hold. Suppose that there exists a zero element 0 ∈ L, i.e. 06x, for all x ∈ L. Let Id(L) be the set of all ideals of the join semilattice (L; ∨; 1) ordered by the inclusion relation ⊆. There exists a structure of integral quantale on the set Id(L) [23]. For all I; J ∈ Id(L), let I · J be the ideal generated by the set {x · y=x ∈ I and y ∈ J }. If I is a set of ideals from Id(L) then its supremum sup I in the poset (Id(L); ⊆) is the ideal generated by the
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subset I ∈I I . For all I; J ∈ Id(L), the binary join is de3ned by I ∨ J = sup{I; J } which is the ideal generated by the set I ∪ J . Then (Id(L); ∨; ·; 0; 1) is an integral quantale with the underlying poset (Id(L); ⊆), where 0 = {0} and 1 = L. The following proposition shows that the class CR() can be identi3ed with the class IqQ. Proposition 3.7. The following conditions hold: (i) If L = (L; ∨; ·; →; →; ˜ 1) ∈ CR() then (L; ∨; ·; 1) ∈ IqQ. (ii) If (L; ∨; ·; 1) ∈ IqQ then L = (L; ∨; ·; →; →; ˜ 1) ∈ CR(), where → and → ˜ are binary operations on L such that for all a; b ∈ L, a → b = sup{x ∈ L=a · x 6 b};
(34)
a→b ˜ = sup{x ∈ L=x · a 6 b}:
(35)
(iii) There exists a bijection from CR() onto IqQ. Proof. (i) Let L = (L; ∨; ·; →; →; ˜ 1) ∈ CR(). From De3nitions 3.4 and 2.1 it follows that 3.5(i) and (ii) hold. We verify now that 3.5(iii) holds. Let x ∈ L and A ⊆ L such that A = ∅. It is clear that sup(x · A) 6 x · (sup A):
(36)
We have x · a6 sup(x · A) for all a ∈ A. Using Lemma 2.2(iii) one obtains that a 6 x → sup(x · A)
for all a ∈ A;
which implies that sup A6x → sup(x · A). Thus x · (sup A) 6 sup(x · A):
(37)
From (36) and (37) one obtains x · (sup A) = sup(x · A):
(38)
The equation (sup A) · x = sup(A · x)
(39)
follows by a dual argument. Thus 3.5(iii) holds. From De3nition 3.5 it follows that (L; ∨; ·; 1) ∈ IqQ. (ii) Let (L; ∨; ·; 1) ∈ IqQ. For all a; b ∈ L, de3ne a → b and a→b ˜ by (34) and (35). Let a; b ∈ L. From De3nition 3.5(iii), 3.7(1) and 3.7(2) one obtains that a · (a → b) = a · (sup{x=a · x 6 b}) = sup{a · x=a · x 6 b} 6 b; (a→b) ˜ · a = (sup{x=x · a 6 b}) · a = sup{x · a=x · a 6 b} 6 b: From De3nition 3.5 and the precedent relations one obtains that all the conditions from De3nition 2.1 are satis3ed, i.e. L = (L; ∨; ·; →; →; ˜ 1) ∈ R(). Condition 3.5(i) also holds. Thus, from De3nition 3.4 it follows that L ∈ CR(). This completes the proof of 3.7(ii).
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(iii) The correspondence CR() L → (L; ∨; ·; 1) ∈ IqQ is a bijection from CR() to IqQ. Remark 3.8. Let R0 () be the class of all algebras L = (L; ∨; ·; →; →; ˜ 1) ∈ R() such that there exists an element 0 ∈ L with 06x, for all x ∈ L. De3ne a subclass CR0 () ⊆ CR() by CR0 () = CR()∩ R0 (). Using Proposition 3.7 one derives that there exists a bijection from CR0 () to the class IQ of all integral quantales. The following proposition expresses a connection between the classes R() and IQ. Proposition 3.9. Let R0 () and CR0 () be the classes of algebras from R() de:ned in Remark 3.8. Suppose that L = (L; ∨; ·; →; →; ˜ 1) is a residuated join semilattice monoid, i.e. L ∈ R(). Then the following conditions hold: (i) There exist L0 ∈ R0 () and an injective homomorphism from L to L0 . (ii) There exist an integral quantale Id(L0 ) = (Id(L0 ); ∨; ·; 0; 1) ∈ IQ (see Example 3.6(b)) and an injective homomorphism from L into the complete residuated join semilattice monoid with zero Id(L0 ) = (Id(L0 ); ∨; ·; →; →; ˜ 1) ∈ CR0 () associated with Id(L0 ) as in Proposition 3.7(ii). Proof. (i) We consider two cases: Case 1: There exists 0 ∈ L such that 06x, for all x ∈ L. Case 2: For each x ∈ L there exists y ∈ L such that y¡x, i.e. y6x and y = x. In case 1 we let L0 = L ∈ R0 (), which implies that the identity function 1L : L → L is an isomorphism from L onto L0 . In case 2 let 0 ∈= L and de3ne L0 = L ∪ {0}; x · 0 = 0 = 0 · x; 0 → x = 1 = 0→x ˜ and 1 if x = 0; x → 0 = x→0 ˜ = 0 if x = 0 for all x ∈ L0 , which implies that L0 = (L0 ; ∨; ·; →; →; ˜ 1) ∈ R0 () and the inclusion map iL : L → L0 from the subset L ⊆ L0 to L0 is an injective homomorphism from L into L0 . Thus condition 3.9(i) holds. (ii) We prove now that both in cases 1 and 2 Condition 3.9(ii) holds. In case 1, let L0 = L and Id(L0 ) = (Id(L0 ); ∨; ·; 0; 1) ∈ IQ be the integral quantale of all ideals of the bounded join semilattice (L0 ; ∨; 0; 1) de3ned in Example 3.6(b), where L0 = L. Let Id(L0 ) = (Id(L0 ); ∨; ·; →; →; ˜ 1) ∈ CR0 () be the complete residuated join semilattice monoid with zero associated with Id(L) previously de3ned in Proposition 3.7(ii). Let f : L0 → Id(L0 ) be a function de3ned by f(x) = (x] = {y ∈ L=y 6 x} ∈ Id(L0 )
for all x ∈ L0 :
Then f is an injective homomorphism from L0 into Id(L0 ) such that f(0) = 0. In addition f preserves all the supremums existing in L0 . In case 2, condition 3.9(ii) follows from the properties previously presented in case 1 using condition 3.9(i).
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Denition 3.10. A complete join semilattice endomorphism of a complete join semilattice (L; ∨; 1) is a function f : L → L such that f(sup A) = sup f(A)
for all A ⊆ L such that A = ∅:
Consequence 3.11 (Cayley-type representation): Every residuated join semilattice monoid from R() can be embedded into a residuated join semilattice monoid from R() with an underlying set of complete join semilattice endomorphisms. Proof. Let L = (L; ∨ ; ·; → ; →; ˜ 1) ∈ R(). From Proposition 3.9(ii) it follows that there exist a complete residuated join semilattice monoid with zero L∗ = (L∗ ; ∨; ·; →; →; ˜ 1) ∈ CR0 () and an injective homomorphism i : L → L∗ from L into L∗ . From the proof of Consequence 2.11 it follows that there exists an isomorphism in R() from L∗ onto a residuated join semilattice monoid (L∗ ) of complete join semilattice residuated endomorphisms which we denote by : L∗ → (L∗ ). Thus ◦ i : L → (L∗ ) is an injective homomorphism from L into (L∗ ). This shows that 3.11 holds. The following example is suggested by Proposition 3.9(ii) and it expresses a procedure to construct an homomorphic image using the notion of quantic nucleus introduced by Rosenthal [23] for the study of quotient quantales. Example 3.12. Let L = (L; ∨ ; ·; → ; →; ˜ 1) ∈ R(). Suppose that c : L → L is a quantic nucleus on L, i.e. c is a closure operator of the poset (L; 6) such that c(x) · c(y) 6 c(x · y)
for all x; y ∈ L:
˜ c ; 1c ) ∈ R(), where c(L) = {x ∈ L=c(x) = x} ⊆ L is the set of all Then c(L) = (c(L); ∨c ; ·c ; →c ; → c-closed elements of L and for all x; y ∈ c(L): x ∨c y = c(x ∨ y); x ·c y = c(x · y); ˜ 1c = 1: x→ ˜ c y = c(x→y);
x →c y = c(x → y);
x→ ˜ c y = c(x → y);
The restriction of c to its image is a surjective homomorphism from L onto c(L), thus c(L) ∈ H(L). From the 3rst isomorphism theorem with respect to the variety R(), De3nition 3.1 and Theorem 3.3 one derives that the algebra c(L) of c-closed elements of L is isomorphic with the quotient algebra of L with respect to the normal :lter F = {x ∈ L=c(x) = 1} of all c-dense elements of L. 4. Some special subclasses The underlying poset of a residuated join semilattice monoid from R() is not necessarily a lattice (see Example 2.6(c)). One can consider the problem to determine conditions for an algebra from R() such that its underlying poset becomes a lattice.
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Now we present some subclasses of R() with respect to speci3c solutions of the precedent problem. Let RL() be the class of all algebras L = (L; ∨ ; ·; → ; →; ˜ 1) ∈ R() such that the underlying poset (L; 6) of L is a lattice. It is obvious that Boolean algebras [22] can be viewed as particular algebras from RL(). 4.1. Complete structures A complete structure of R() is an algebra of CR() (see De3nition 3.4). For all L ∈ R(), the set of all lower bounds of any two elements subset {x; y} ⊆ L is nonempty, because of x · y6x and x · y6y, thus B{x; y} = {z ∈ L=z 6 x and z 6 y} = ∅: Let L∗ ∈ CR(). For all x; y ∈ L∗ , de3ne x ∧ y = sup LB{x; y}. Then (L∗ ; 6) is a lattice with the binary meet ∧ de3ned by x ∧ y = inf {x; y}. Thus CR()
⊆ RL():
(S1)
We obtain also that f ∈ Res(L∗ ; ∨) i6 f : L∗ → L∗ is a complete join semilattice endomorphism of (L∗ ; ∨; 1) (see De3nition 3.10). This implies that there exists a structure Res(1L∗ ] = (Res(1L∗ ]; ∨; ◦; ⇒; ⇒; ˜ 1L∗ ) ∈ R(); where Res(1L∗ ] = {f ∈ Res(L∗ ; ∨)=f 6 1L∗ }: Consider the corresponding function : L∗ → Res(1L∗ ] from the proof of Cayley-type representation results (Consequences 2.11 and 3.11) de3ned by (a)(x) = a · x
for all a; x ∈ L∗ :
It follows that is an injective homomorphism from L∗ to Res(1L∗ ]. Using Proposition 3.9(ii) we obtain that for any L∈R() there exists a complete structure L∗ ∈ CR() such that L is isomorphic with a subalgebra of Res(1L∗ ]. A particular case is L∗ = [0; 1]. In this case, we let LCF[0; 1] = {f : L∗ → L∗ =f is increasing and left-continuous on L∗ = [0; 1] with f(x) 6 x; ∀x ∈ L∗ }: Then Res(1L∗ ] = LCF[0; 1]. This implies that there exists a complete structure LCF[0; 1] ∈ CR() on the set LCF[0; 1]. Therefore, each structure of residuated join semilattice monoid on the real unit interval [0; 1] is isomorphic with a subalgebra of LCF[0; 1]. The logic of left-continuous commutative t-norms has been considered by Esteva and Godo [8]. A special subclass of CR() is CR0 () = CR() ∩ R0 () (see Remark 3.8) representing the class of all complete structures with a zero element. Thus CR0 ()
⊆ CR():
(S2)
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4.2. Finite structures Let FR() be the class of all L ∈ R() such that L is a 3nite set which we call :nite structures. If L ∈ FR() is a 3nite structure from R() then there exists 0 ∈ L such that 06x, for all x ∈ L and there exists sup A, for all A ⊆ L such that A = ∅ because L is a 3nite set. Thus FR()
⊆ CR0 ():
(S3)
If A ∈ FR() such that the corresponding poset (A; 6) is a chain and the binary operations · and ∧ are di6erent then the set A has the cardinality |A|¿4. An example of 3nite residuated lattice monoid with four elements A4 is any chain with an underlying set A4 = {0; a; b; 1}; 0¡a¡b¡1 and the operation · de3ned by the following table: 0 a b 1
0 0 0 0 0
a 0 0 a a
b 0 0 b b
1 0 a b 1
A functional representation of this 3nite structure with four elements is de3ned by a set S4 = {&; f; g; 1E } of four functions from E to E such that &¡f¡g¡1E , where E = {1; 2; 3} is the chain of three natural numbers 1¡2¡3, 1E (x) = x and &(x) = 1; for all x ∈ E;
f(1) = g(1) = 1;
f(2) = g(2) = 1;
f(3) = 2 and g(3) = 3: With respect to the usual composition of functions ◦ and the natural order 6, the set S4 has a structure of residuated lattice monoid S4 which is isomorphic with the residuated lattice monoid A4 previously de3ned. 4.3. Lattice pseudo-hoops Let L ∈ R() and Res(L; ∨) be the set of all residuated endomorphisms of the join semilattice (L; ∨). Denition 4.1. Let : L → Res(L; ∨) be the function de3ned by (a) = a , for all a ∈ L. The left translation relation on L is a binary relation T on L de3ned by bT a i6 ∃x ∈ L with a (x) = a · x = b: Denition 4:1 . Let : L → Res(L; ∨) be the function de3ned by (a) = a , for all a ∈ L. The right translation relation on L is a binary relation T on L de3ned by bT a i6 ∃x ∈ L with a (x) = x · a = b:
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Lemma 4.2. For all a ∈ L, let a+ : L → L be the residual of a ∈ Res(L; ∨) and a+ : L → L be the residual of a ∈ Res(L; ∨). Consider the two binary relations T and T from De:nitions 4.1 and 4:1 . Then the following conditions hold: (i) The underlying partially ordered monoid (L; ·; 6; 1) is negatively ordered, i.e. a 61L and a 61L , for all a ∈ L. (ii) bT a i8 a · (a → b) = a (a+ (b)) = b. (iii) bT a i8 (a→b) ˜ · a = a (a+ (b)) = b. Proof. (i) Condition 4.2(i) follows from relations a · x6x and x · a6x for all a; x ∈ L. (ii) Suppose that bT a, i.e. ∃x ∈ L with a · x = b. We have a · (a → b)6b. From a·x =b
and
x 6 a → (a · x);
it follows that x6a → b, thus b = a · x6a · (a → b). This implies a · (a → b) = b: If the precedent relation holds then a · x = b, for x = a → b ∈ L, thus bT a. This shows that 4.2(ii) holds. (iii) Condition 4.2(iii) follows in a similar way with 4.2(ii) by a dual argument. Lemma 4.3. Both the left translation relation T and the right translation relation T are order relations on L such that for all a; b ∈ L, bT a or bT a implies b 6 a: Proof. We have that T is reBexive because of relations a · (a → a) = a · 1 = a, for all a ∈ L. Suppose now that cT b and bT a. We verify that cT a, which implies that T is transitive. From Lemma 4.2(ii) it follows that b · (b → c) = c and a · (a → b) = b. This implies c6a · (a → c), using the following relations: c = b · (b → c) = (a · (a → b)) · (b → c) = a · ((a → b) · (b → c)); (a → b) · (b → c) 6 a → c: But a · (a → c)6c, therefore c = a · (a → c). Using Lemma 4.2(ii) this implies cT a. It is clear that bT a implies b6a. Therefore, bT a and aT b implies b6a and a6b, thus a = b. This shows that T is antisymmetric. Thus, T is an order relation such that for all a; b ∈ L, bT a implies b6a. Using a dual argument we derive also that T is an order relation such that for all a; b ∈ L, bT a implies b6a. Denition 4.4. A lattice pseudo-hoop [13] is a residuated join semilattice monoid L ∈ R() such that the underlying partially ordered monoid (L; ·; 6; 1) is the dual of a divisibility monoid [1], i.e. for all a; b ∈ L, b 6 a i6 bT a i6 bT a:
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The following proposition gives two expressions of the notion of lattice pseudo-hoop. Proposition 4.5. Let L ∈ R(). Then the following conditions are equivalent: (i) L is a lattice pseudo-hoop. (ii) L satis:es the following equations, for all a, b ∈ L: a · (a → b) = b · (b → a)
(40)
(a → ˜ b) · a = (b→a) ˜ · b:
(41)
(iii) There exists a binary operation ∧ on L such that the following conditions hold: (L; ∧; ∨) is a lattice;
(42)
a · (a → b) = a ∧ b;
(43)
(a→b) ˜ · a = a ∧ b:
(44)
Proof. (i) ⇔ (ii): The equivalence between conditions 4.5(i) and (ii) follows from De3nition 4.4 and Lemma 4.2. (ii) ⇒ (iii). Suppose that (ii) holds. De3ne a binary operation ∧ by a ∧ b = a · (a → b)
for all a; b ∈ L:
(45)
From (40) and (41) one derives that a · (a → b) = inf {a; b} = (a→b) ˜ · a:
(46)
Relations (45) and (46) imply relations (42)–(44). Thus (iii) holds. (iii) ⇒ (ii): Suppose now that (iii) holds. From (42) we obtain that for all a; b ∈ L, a ∧ b = b ∧ a, thus using (43) and (44) one obtains that (40) and (41) hold. This shows that (ii) holds. Remark 4.6. Let LPH() be the class of all lattice pseudo-hoops. Then LPH()
⊆ RL():
(S4)
˜ 1) such that the The class LPH() can be identi3ed with a variety of algebras (L; ∧; ∨; ·; → ; →; following conditions hold: (LPH1) (LPH2) (LPH3) (LPH4)
L = (L; ∨; ·; →; →; ˜ 1) ∈ R(). (L; ∧; ∨) is a lattice. a · (a → b) = a ∧ b, for all a; b ∈ L. (a → ˜ b) · a = a ∧ b, for all a; b ∈ L.
For each lattice pseudo-hoop its underlying lattice is distributive. Remark 4.7. A pseudo BL-algebra [9] is a lattice pseudo-hoop (L; ∧; ∨; ·; → ; →; ˜ 1) which satis3es the following equations, for all x; y ∈ L: (PL) (x → y) ∨ (y → x) = 1 and (x → ˜ y) ∨ (y → ˜ x) = 1.
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Let PBL() be the class of all pseudo BL-algebras. Then PBL()
⊆ LPH():
(S5)
With respect to the theory of noncommutative fuzzy logic it is natural also to consider the following notion: a Weak BL-algebra (WBL-algebra) is a residuated join semilattice monoid L ∈ R() satisfying the precedent equations (PL). Let WBL() be the class of all WBL-algebras. Then WBL()
⊆ R():
(S6)
An important open problem is to describe subdirectly irreducible WBL-algebras in order to obtain a concrete expression of the subdirect decomposition theorem for this special variety of algebras. We remark that if L ∈ R() such that (L; 6) is a chain then L is a WBL-algebra. 4.4. Cancellative structures Denition 4.8. The class Cancel R() is the class of all cancellative residuated join semilattice monoids, i.e. L ∈ Cancel R() i6 L ∈ R() and the following condition hold, for all a; x; y ∈ L: (CL) a · x = a · y or x · a = y · a implies x = y. De3nition 4.8 shows that Cancel R()
⊆ R():
(S7)
We call a cancellative structure any algebra from Cancel R(). The following lemma gives a characterization of cancellative structures. Lemma 4.9. Let L ∈ R(). The following conditions are equivalent: (i) L ∈ Cancel R(). (ii) a → (a · x) = x = a → ˜ (x · a), for all a; x ∈ L. Proof. (i) ⇒ (ii): Suppose that (i) holds. Let a; x ∈ L. Then the following conditions hold: a · (a → (a · x)) = a · x;
(47)
(a→(x ˜ · a)) · a = x · a:
(48)
Eq. (47) follows from the relations a · (a → (a · x)) 6 a · x
and
x 6 a → (a · x):
Eq. (48) follows by a dual argument. Using De3nition 4.8 (CL) one derives that the two Eqs. (47) and (48) imply 4.9(ii).
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(ii) ⇒ (i): Suppose now that (ii) holds. Let a; x; y ∈ L such that a · x = a · y or x · a = y · a. Using 4.9(ii) one obtains that a · x = a · y implies x = a → (a · x) = a → (a · y) = y; x · a = y · a implies x = a→(x ˜ · a) = a→(y ˜ · a) = y: Therefore, L satis3es the cancellative law 4.8(CL), i.e. 4.9(i) holds. Remark 4.10. Lemma 4.9(ii) expresses the fact that L is a cancellative structure i6 both the family (a )a∈L of left translations and the family (a )a∈L of right translations are families of injective functions such that a+ ◦ a = 1L and + a ◦ a = 1L
for all a ∈ L:
The precedent relations from Remark 4.10 express the facts that in every cancellative structure L, respectively, the residual a+ of a is a retract of a and the residual a+ of a is a retract of a in the set of all increasing functions from L to L. Example 4.11. (a) Let L = (0; 1] ⊆ R with the usual binary join ∨ and algebraic product ·. Then there exists a structure of cancellative commutative structure with the underlying monoid ((0; 1]; ·; 6; 1) such that for all a; x ∈ (0; 1]: 1 if a 6 x; a → x = a→x ˜ = x if x ¡ a: a This structure is used in di6erent applications of fuzzy set theory (see [27]). In addition, this is a complete structure, because sup A exists, for any nonempty subset A ⊆ (0; 1]. Also the corresponding structure ((0; 1]; ∨; ·; →; →; ˜ 1) ∈ R() is a simple algebra, because a subset F ⊆ L is a normal 3lter (see De3nition 3.1 and Theorem 3.3) i6 F = L or F = {1}. In this case, the following speci3c conditions are satis3ed, for all a; b ∈ (0; 1]: a 6 b i6 a · (a → b) = b and a = b; a 6 b i6 (a→b) ˜ · a = b and a = b: (b) The system L[G − ] ∈ R() de3ned in Example 2.6(b) on the negative cone (G − ; +; 6; 0) of a lattice group (G; +; ∧; ∨; 0) is a cancellative structure. In addition, L[G − ] is a lattice pseudo-hoop. We prove in the following proposition that the precedent properties are characteristic for this kind of structures. The following proposition (see [13]) follows using a known result from the study of residuated structures [1,2,10]. It is in connection with the structure from Example 2.6(b) and the properties presented in the previous Example 4.11(b).
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Proposition 4.12. Let L = (L; ∨; ·; →; →; ˜ 1) ∈ R(). The following conditions are equivalent: (i) L is isomorphic to an algebra L[G − ] ∈ R() associated with the negative cone of a lattice group; (ii) L is a cancellative lattice pseudo-hoop. Proof. (i) ⇒ (ii): Suppose that 4.12(i) holds. From the de3nition given in Example 2.6(b) it is clear that the structure L[G − ] is cancellative and a lattice pseudo-hoop. From Lemma 4.9 and Proposition 4.5 it follows that the class Cancel R() ∩ LPH() is equational. Thus, any algebra L ∈ R() isomorphic with L[G − ] is an algebra from Cancel R() ∩ LPH(), i.e. condition 4.12(ii) holds. (ii) ⇒ (i): Suppose now that 4.12(ii) holds. Then the underlying partially ordered monoid (L; ·; 6; 1) of L is isomorphic with the negative cone (G − ; +; 6; 0) of a lattice group (G; +; ∧; ∨; 0) (see [10, Proposition X.1]). Let f : L → G − be an order isomorphism from (L; 6) to (G − ; 6) such that (a) f(x · y) = f(x) + f(y), for all x; y ∈ L, (b) f(1) = 0. Let x; y ∈ L. Using the de3nition of operations → and → ˜ in L[G − ] from (a) and (b) we derive the following relations: f(x) + (f(x) → f(y)) = f(x) + (((−f(x)) + f(y)) ∧ 0) = f(x) + (((−f(x)) + f(y)) ∧ f(1)) = (f(x) + ((−f(x)) + f(y))) ∧ (f(x) + f(1)) = f(y) ∧ f(x · 1) = f(y ∧ x) = f(x ∧ y) = f(x · (x → y)) = f(x) + f(x → y): Thus in L[G − ] we have f(x) + (f(x) → f(y)) = f(x) + f(x → y)
for all x; y ∈ L;
but L[G − ] is cancellative, thus (c) f(x → y) = f(x) → f(y) for all x; y ∈ L. The following relation (d) f(x→y) ˜ = f(x) → ˜ f(y) for all x; y ∈ L follows in a similar way with (c) from (a) and (b) by a dual argument. Conditions (a)–(d) imply that f is an isomorphism from L to L[G − ]. Thus 4.12(i) holds. 5. Extended membership space of commutative residuated structures In order to 3nd an algebraic structure closely related to the structure of the real unit interval suggested by Zadeh [26] and Zimmermann [27] to represent fuzzy sets, an extended membership space of commutative residuated structures must be considered. In this section, we present some elementary properties of an extended membership space in connection with the structures of biresiduated
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algebra [24], L+-algebra [17] and double residuated lattice [21]. It is to show that the structure of residuated join semilattice monoid from R() presented in the precedent sections can be of interest also in the study of commutative fuzzy logic and fuzzy set theory. Suppose that K is a commutative linearly ordered :eld. For all x; y ∈ K, we denote by x + y the sum, by xy the product, by x − y the di8erence of the elements x and y in K and if x = 0, we denote by x−1 the inverse of x and by y=x = yx−1 the division of y by x in K. Denition 5.1. We denote by 6 the order relation on K. Let U (K) = {x ∈ K=06x61} be the unit interval of K. A graded membership chain associated with K is the algebra (U (K); ∧; ∨; →; ·; =; ⊕; ∗; 0; 1) with the carrier set U (K), where ∧; ∨; →; ·; =; ⊕ are binary operations and ∗ is an unary operation de3ned as follows: (i) x ∧ y = min{x; y} (binary meet ∧); (ii) x ∨ y = max{x; y} (binary join ∨); 1 if x6y (iii) x → y = = max{z ∈ U (K)=x ∧ z6y} (G?odel intuitionistic implication →); y if x¿y (iv) x · y = xy (algebraic product ·); 1 if x6y (v) y=x = y = max{z ∈ U (K)=x · z6y} (bounded division /); if x¿y x (vi) x ⊕ y = min{1; x + y} (bounded sum ⊕); (vii) x∗ = 1 − x (complementation ∗). Remark 5.2. On any graded membership chain there exists a new set of binary operations de3ned by: 0 if x6y (i) x − y = = min{z ∈ U (K)=x6y ∨ z} (dual G?odel intuitionistic implication −); x if x¿y (ii) x ⊕p y = x + y − xy (probabilistic sum ⊕p ); 0 if x6y (iii) x p y = x−y = min{z ∈ U (K)=x6y ⊕p z} (probabilistic bounded di8erence p ); if x¿y 1− y (iv) x y = max{0; x + y − 1} (Lukasiewicz conjunction ); (v) x C y = max{0; x − y} = min{z ∈ U (K)=x6y ⊕ z} (bounded di8erence C); (vi) x ⇒ y = min{1; 1 − x + y} = max{z ∈ U (K)=xz6y} (Lukasiewicz implication ⇒). Remark 5.3. The unary operation of complementation ∗ is a decreasing involution of the bounded poset (U (K); 6; 0; 1) such that the following equations hold: (2:1 ) x ∧ y = (x∗ ∨ y∗ )∗ , (2:1 ) x ∨ y = (x∗ ∧ y∗ )∗ , (2:2 ) x → y = (y∗ − x∗ )∗ , (2:2 ) x − y = (y∗ → x∗ )∗ , (2:3 ) x y = (x∗ ⊕ y∗ )∗ , (2:3 ) x ⊕ y = (x∗ y∗ )∗ , (2:4 ) x ⇒ y = (y∗ C x∗ )∗ , (2:4 ) x C y = (y∗ ⇒ x∗ )∗ ,
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(2:5 ) (2:5 ) (2:6 ) (2:6 ) (2:7)
173
x · y = (x∗ ⊕p y∗ )∗ , x ⊕p y = (x∗ · y∗ )∗ , y=x = (y∗ Cp x∗ )∗ , x Cp y = y(x∗ =y∗ )∗ , x∗∗ = x.
Remark 5.4. The system (U (K); ∧; ∨; ∗; 0; 1) is a De Morgan algebra, i.e., (U (K); ∧; ∨; 0; 1) is a bounded distributive lattice verifying Eq. (2.1’). In addition U (K) is a Kleene algebra [5], i.e. the precedent De Morgan algebra satis3es also the following relation: (K) x ∧ x∗ 6y ∨ y∗ . Remark 5.5. From Eqs. (2.1)–(2.7) which we will call the De Morgan equations, it follows that any of the following six pairs of binary operations (∧; ∨); (→; −); ( ; ⊕); (⇒ ; C); (· ; ⊕p ) and (=; Cp ) have components which are dual one to another. Remark 5.6. The following conditions which are also satis3ed in U (K) show that any of the three pairs of operations (∧ ; →); ( ; ⇒) and (·; =) induces on U (K) a structure of commutative residuated l-monoid and any of the three pairs of operations (∨; −); (⊕ ; C) and (⊕p ; Cp ) induces on U (K) a structure of commutative dual residuated l-monoid (see [1,2]): (2:8 ) (2:8 ) (2:9 ) (2:9 ) (2:10 ) (2:10 )
a ∧ x6b i6 x6a → b, a6b ∨ x i6 a − b6x, a x6b i6 x 6 a ⇒ b, a6b ⊕ x i6 a C b6x, a · x6b i6 x6b=a, a6b ⊕p x i6 a Cp b6x.
The following lemmas present some basic properties of U (K) derived from the precedent relations in connection with the known lattice ordered algebraic structures of linear Heyting algebra [3], MValgebra [4,5] and BL-algebra [15]. Lemma 5.7. The algebra (U (K); ∧; ∨; →; 0; 1) is a linear Heyting algebra (see [3]), i.e. its underlying bounded lattice (U (K); ∧; ∨; 0; 1) satis:es condition (2:8 ) and the following equation: (LH) (x → y) ∨ (y → x) = 1. Lemma 5.8. The algebra (U (K); ⊕; ∗; 0) is an MV-algebra (see [5]), i.e. the following equations hold: (MV1) (MV2) (MV3) (MV4) (MV5) (MV6)
x ⊕ y = y ⊕ x, x ⊕ (y ⊕ z) = (x ⊕ y) ⊕ z, x ⊕ 0 = x, x∗∗ = x, x ⊕ 0∗ = 0∗ , (x∗ ⊕ y)∗ ⊕ y = (y∗ ⊕ x)∗ ⊕ x.
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Lemma 5.9. The algebra (U (K); ∧; ∨; ·; =; 0; 1) is a BL-algebra (see [15]), i.e., the following conditions are satis:ed, for all a; b; x; y ∈ U (K): (i) (ii) (iii) (iv) (v)
(U (K); ∧; ∨; 0; 1) is a bounded lattice; (U (K); ·; 1) is a commutative monoid with the unit element 1; · and = form an adjoint pair, i.e., condition (2:10 ) holds; x · (y=x) = x ∧ y; (x=y) ∨ (y=x) = 1.
We present in the following de3nition the notion of linear Heyting MV-algebra (LHMV-algebra) expressing some properties of basic models U (K) derived from De3nition 5.1, Lemmas 5.7 and 5.8. Then we suggest a manner to put together the structure of LHMV-algebra with the structure of BL-algebra. Denition 5.10. A linear Heyting MV-algebra (LHMV-algebra) is an algebra (A; ∨; →; ⊕; ∗; 0) with three binary operations ∨; →; ⊕, an unary operation ∗ and a constant 0 verifying the following equations: (LHMV1) x ∨ y = y ∨ x, (LHMV2) x → (y → z) = (x∗ ∨ y∗ )∗ → z, (LHMV3) (x → y) ∨ (y → x) = 0∗ , (LHMV4) x ⊕ y = y ⊕ x, (LHMV5) x ⊕ (y ⊕ z) = (x ⊕ y) ⊕ z, (LHMV6) x ⊕ 0 = x, (LHMV7) x∗∗ = x, (LHMV8) x ⊕ 0∗ = 0∗ , (LHMV9) (x∗ ⊕ y)∗ ⊕ y = x ∨ y, (LHMV10) a ⊕ (x → y) = (a ⊕ y) ∨ (x → y). Denition 5.11. An LHMV-chain is an LHMV-algebra (A; ∨; →; ⊕; ∗; 0) such that the system (A; 6) is a chain, where 6 is the binary relation on A such that for all x; y ∈ A, x6y
i6
x ∨ y = y:
For example, let K be a linearly-ordered commutative 3eld and U (K) be the unit interval of K together with the operations ∨; →; ⊕ and ∗ de3ned, respectively, by conditions (ii), (iii), (vi) and (vii) from De3nition 5.1. Then (U (K); ∨; →; ⊕; ∗; 0) is an LHMV-chain. The following de3nition introduces a structure including any graded membership chain: (U (K); ∧; ∨; →; ·; =; ⊕; ∗; 0; 1): This de3nition is based on the notions of BL-algebra and L+-algebra [17]. Denition 5.12. A graded membership algebra [25] is a system A = (A; ∧; ∨; →; ·; =; ⊕; ∗; 0; 1)
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such that the following conditions hold: (i) (A; ∨; →; ⊕; ∗; 0) is a LHMV-algebra; (ii) (A; ∧; ∨; ·; =; 0; 1) is a BL-algebra such that 0=x = x → 0, for all x ∈ L; (iii) (A; ·; =; ⊕; C; 0; 1) is a L+-algebra, where C is a binary operation de3ned by x C y = (x∗ ⊕ y)∗ , for all x; y ∈ L. The notions and results presented in this paper can be considered as a basic starting point to develop a theory of generalized sets over a WBL-algebra which is not necessarily a commutative monoid. The study of the structure of WBL-algebra can be also of interest in the elaboration of a comprehensive theory of commutative fuzzy logic based on a suitable notion of graded membership algebra in connection with biresiduated algebras [24], L+-algebras [17] and double residuated lattices [21]. A new interesting possible application can be in the aggregation and ranking of alternatives in multicriteria decision problems [7] formulated in the framework of generalized sets over residuated structures. Acknowledgements The authors would like to thank the referees for their pertinent remarks and suggestions which have helped to realize an improved expression of the work. Special thanks are dues to all the members of the Group of Logic and Universal Algebra from the University of Bucharest. References [1] G. Birkho6, Lattice Theory, 3rd Edition, American Mathematical Society Colloquium Publications, Providence, RI, 1967. [2] T.S. Blyth, M.F. Janowitz, Residuation Theory, Pergamon Press, Oxford, 1972. [3] V. Boicescu, A. Filipoiu, G. Georgescu, S. Rudeanu, Lukasiewicz–Moisil Algebras, North-Holland, Amsterdam, 1991. [4] C.C. Chang, Algebraic analysis of many-valued logics, Trans. Amer. Math. Soc. 88 (1958) 467–490. [5] R. Cignoli, I.M.L. D’Ottaviano, D. Mundici, Algebraic Foundations of Many-Valued Reasoning, Kluwer Academic Publishers, Dordrecht, 2000. [6] A. Di Nola, G. Georgescu, L. LeuVstean, Boolean products of BL-algebras, J. Math. Anal. Appl. 251 (2000) 106–131. [7] D. Dubois, H. Prade, Criteria aggregation and ranking of alternatives in the framework of fuzzy set theory, in: B.R. Gaines, L.A. Zadeh, H.-J. Zimmermann (Eds.), Fuzzy Sets and Decision Analysis, Studies in the Management Sciences, Vol. 20, North-Holland, Amsterdam, 1984, pp. 209–240. [8] F. Esteva, L. Godo, Monoidal t-norm based logic: towards a logic for left-continuous t-norms, Institut d’InvestigaciHo en IntelligHencia Arti3cial (IIIA) Research Report 2001– 02. [9] P. Flondor, G. Georgescu, A. Iorgulescu, Pseudo t-norms and pseudo BL-algebras, Soft Comput. 5 (2001) 355–371. [10] L. Fuchs, Partially Ordered Algebraic Systems, Pergamon Press, Oxford, 1963. [11] G. Georgescu, A. Iorgulescu, Pseudo MV-algebras: a noncommutative extension of MV-algebras, in: Proc. 4th Internat. Symp. on Economic Informatics, Inforec Printing House, Bucharest, May 1999, pp. 961–969. [12] G. Georgescu, A. Iorgulescu, Pseudo BL-algebras: a noncommutative extension of BL-algebras, in: Proc. 5th Internat. Conf. FSTA 2000 on Fuzzy Sets Theory and Its Applications, Lipt. JHan, Slovak Republic, 31 Jan – 4 Feb, 2000, pp. 90 –92. [13] G. Georgescu, L. LeuVstean, Pseudo-hoops, Multiple Valued Logic, submitted (Preprint June 2000).
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[14] J.A. Goguen, The logic of inexact concepts, Synthese 19 (1968– 69) 325–373. [15] P. HHajek, Metamathematics of Fuzzy Logic, Kluwer Academic Publishers, Dordrecht, 1998. [16] L. LeuVstean, Some algebraic properties of noncommutative fuzzy structures, Stud. Inform. Control 9 (2000) 365– 370. [17] F. Montagna, An algebraic approach to propositional fuzzy logic, J. Logic, Language Inform. 9 (2000) 91–124. [18] C.V. Negoita, P. Flondor, M. Sularia, On fuzzy environment in optimization problems, Econom. Comput. Econom. Cybernet. Stud. Res. 1 (1977) 3–15. [19] C.V. Negoita, D.A. Ralescu, Application of Fuzzy Sets to System Analysis, BirkhWauser, Basel, 1975. [20] C.V. Negoita, D.A. Ralescu, Simulation, Knowledge-base Computing and Fuzzy Statistics, Van Nostrand, Princeton, NJ, 1987. [21] E. Orlowska, A. Radzikowska, Information relation and operators based on double residuated lattices, in: H. De Swart (Ed.), Proc. 6th Internat. Conf. on Relational Methods in Computer Science, Oisterwick, 2001, pp. 185 –199. [22] D. Ponasse, J.C. Carrega, AlgHebre et topologie boolHeennes, Masson, Paris, 1979. [23] K. Rosenthal, Quantales and their Applications, Longman Scienti3c & Technical, 1990 (copublished in the United States with Wiley, New York). [24] M. Sularia, On many-valued equivalence and distance functions, Stud. Inform. Control 8 (1999) 329–338. [25] M. Sularia, Graded membership structures, Oral Communication in the Seminar on Fuzzy Sets, ICI Bucharest, May 2002. [26] L.A. Zadeh, Fuzzy sets, Inform. and Control 8 (1965) 338–353. [27] H.-J. Zimmermann, Fuzzy Set Theory and Its Applications, Kluwer-Nijho6 Publishing, Boston, 1985.
On a class of residuated semilattice monoids P. Flondor∗ , M. Sularia Department of Mathematics II, Polytechnic University of Bucharest, Splaiul Independentei St. 313, 77206 Bucharest, Romania Received 22 November 2001; received in revised form 18 July 2002; accepted 5 August 2002
Abstract In this paper, we consider the structure of residuated join semilattice monoid and we study its 3rst properties. The class of residuated join semilattice monoids is a variety of algebras. A representation of this structure as residuated mappings is given. Then we present the notion of normal 3lter together with a characterization theorem of homomorphic images based on this notion. Di6erent classes of residuated join semilattice monoids are introduced. c 2002 Elsevier B.V. All rights reserved. Keywords: Fuzzy logic; Real membership space; Multiple-valued logic; Residuated semilattice monoid; Residuated mapping; Quantale
1. Introduction Residuated monoids [1,2] are proved to be important structures of multiple-valued logic. The real membership space of fuzzy sets introduced by Zadeh is the real unit interval endowed with its usual linear order relation “6” and the three structures of commutative residuated monoid corresponding, respectively, to the binary operations of meet “∧”, bounded sum “⊕” and algebraic product “·” [26,27]. The precedent structures are particular cases of Heyting algebra, MV-algebra and BL-algebra. Starting from this fact di6erent ways to put together these structures have been considered with the aim to improve the mathematical foundation of the notion of fuzzy set [15,8,17,24,21].
∗
Corresponding author. E-mail addresses: [email protected] (P. Flondor), Mircea [email protected] (M. Sularia).
c 2002 Elsevier B.V. All rights reserved. 0165-0114/03/$ - see front matter PII: S 0 1 6 5 - 0 1 1 4 ( 0 2 ) 0 0 3 9 0 - 1
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Following HHajek [15] a BL-algebra is a residuated lattice (L; ∧; ∨; ∗; →; 0; 1) such that (i) x ∧ y = x ∗ (x → y), (ii) (x → y) ∨ (y → x) = 1, for all x; y ∈ L. It is to say that in [15] the notion of BL-algebra is considered basic for fuzzy logic (in the narrow sense). After the extension of the notion of MV-algebra [4,5] to the noncommutative case by Georgescu and Iorgulescu [11], a noncommutative extension of BL-algebras was introduced [12,16]. The new fact emerging from this extension (noncommutativity apart) is that condition (i) is not to be expected in the general case (this is due to the continuity of T-norms and a theorem of Aczel [9]). Another way to put this is that “∧” is not determined by “*” and “→” (in other terms “∧” is somewhat independent of “*” and “→”). From the logical point of view, we interpret “*” as (a kind of) conjunction (may be commutative) and “∨” as disjunction (“∧” is another conjunction). It seems natural to consider a “minimal” structure with only one conjunction “*” (may be noncommutative), disjunction “∨” (of semilattice type) and implication “→” (two of them in the noncommutative case) as a possible starting point for a theory of not necessarily commutative fuzzy logic. We call such a structure a WBL-algebra (we do not consider (ii) as necessary for the beginning). One can consider a WBL-algebra as the “compact” part of a quantale [23] (see below). This connection with the theory of quantales is viewed as a “completion”. As for some concrete “models” of logic to be developed over such structures, we can consider the logic of actions; after all applying actions is a noncommutative process. That is why we consider the composition of functions as the basic model for the noncommutative conjunction. The idea to represent such structures as residuated mappings comes from the possible applications in the theory of actions and causality.
2. Residuated semilattice monoids We recall that an integral ordered monoid [1,10] is an ordered monoid (L; ·; 6; 1) such that its unity element 1 is a greatest element, i.e. L is a set, · is a binary operation on L called multiplication, 6 is an order relation on L and 1 ∈ L is a constant such that the following conditions hold, for every element a ∈ L: 1. (L; ·; 1) is a monoid with the unity element 1; 2. the left translation a and the right translation a on L given by a (x) = a · x and a (x) = x · a are isotone mappings, i.e. for all x; y ∈ L, if x6y then a · x6a · y and x · a6y · a; 3. 1 = max L, i.e. (∀x ∈ L)x61. Denition 2.1. Let K() be the class of all algebras of type = (2; 2; 2; 2; 0). A residuated join semilattice monoid is a system L = (L; ∨; ·; →; →; ˜ 1) ∈ K(), such that the following conditions hold: (i) (L; ∨) is a join semilattice, where ∨ is the binary supremum on L;
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(ii) (L; ·; 6; 1) is an integral ordered monoid, where 6 is the unique order relation on L associated with ∨ and de3ned by x6y i6 x ∨ y = y; (iii) for all a; b ∈ L, the element a → b ∈ L is the right residual of b by a, i.e. a → b = max{x ∈ L=a · x 6 b}; (iv) for all a; b ∈ L, the element a→b ˜ ∈ L is the left residual of b by a, i.e. a→b ˜ = max{x ∈ L=x · a 6 b}: Lemma 2.2. Let L = (L; ∨; ·; →; →; ˜ 1) be a residuated join semilattice monoid. Then the following conditions hold, for all a; b; x; y ∈ L: (i) (ii) (iii) (iv) (v) (vi) (vii)
x · y6x and x · y6y; x · (x → y)6y and (x → ˜ y) · x6y; x6a → b i8 a · x6b; x6a→b ˜ i8 x · a6b; x6y i8 x → y = 1 i8 x → ˜ y = 1; a · (x ∨ y) = (a · x) ∨ (a · y); (x ∨ y) · a = (x · a) ∨ (y · a).
Proof. Let a; b; x; y ∈ L. Conditions 2.2(i)–(v) are the consequences of De3nition 2.1 which are easy to prove. We show that Eq. 2.2(vi) holds. From x6x ∨ y and y6x ∨ y it follows that a · x 6 a · (x ∨ y)
and
a · y 6 a · (x ∨ y):
This implies that (a · x) ∨ (a · y)6a · (x ∨ y). Suppose now that z ∈ L veri3es a · x6z and a · y6z. Using 2.2(iii) this implies that x6a → z and y6a → z, thus x ∨ y6a → z. Using again 2.2(iii), from the precedent relation it follows that a · (x ∨ y)6z. Thus, Eq. 2.2(vi) is veri3ed. In a similar way, one shows that Eq. 2.2(vii) also holds. Lemma 2.3. Let L = (L; ∨; ·; →; →; ˜ 1) ∈ K(). Then the following conditions are equivalent: (i) L is a residuated join semilattice monoid; (ii) L satis:es the following equations, for all x; y; z ∈ L: x ∨ x = x;
(1)
x ∨ y = y ∨ x;
(2)
(x ∨ y) ∨ z = x ∨ (y ∨ z);
(3)
x ∨ 1 = 1;
(4)
(x · y) · z = x · (y · z);
(5)
x · 1 = x;
(6)
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1 · x = x;
(7)
z · (x ∨ y) = (z · x) ∨ (z · y);
(8)
(x ∨ y) · z = (x · z) ∨ (y · z);
(9)
(x · (x → y)) ∨ y = y;
(10)
((x→y) ˜ · x) ∨ y = y;
(11)
x → (x ∨ y) = 1;
(12)
x→(x ˜ ∨ y) = 1;
(13)
x → (y → z) = (y · x) → z;
(14)
x→(y ˜ →z) ˜ = (x · y)→z: ˜
(15)
Proof. (i) ⇒ (ii): Suppose that L veri3es condition 2.3(i). We show that all Eqs. (1)–(15) are valid in L. Eqs. (1)–(9) follow from De3nition 2.1(i) and (ii) and Lemma 2.2(vi) and (vii). Eqs. (10)–(13) follow from Lemma 2.2(ii) and (v). De3nition 2.1(ii) and Lemma 2.2(ii) imply (y · x) · (x → (y → z)) = y · (x · (x → (y → z))) 6 y · (y → z) 6 z: Using Lemma 2.2(iii) this implies x → (y → z) 6 (y · x) → z: Using again De3nition 2.1(ii) and Lemma 2.2(ii) and (iii) one obtains that y · (x · ((y · x) → z)) = (y · x) · ((y · x) → z) 6 z; which imply (y · x) → z 6 x → (y → z): Therefore, Eq. (14) holds. One can also verify that Eq. (15) holds. This completes the proof of condition 2.3(ii). (ii) ⇒ (i): Suppose that condition 2.3(ii) holds. We verify that all conditions 2.1(i)–(iv) are valid in L. Using Eqs. (1)–(9) one obtains that 2.1(i) and (ii) are satis3ed. We show that condition 2.1(iii) holds. Suppose that x6a → b. Using Eq. (10) one obtains that a·x6a·(a → b)6b, therefore a·x6b. Now suppose that a · x6b, i.e. b = (a · x) ∨ b. Using Eqs. (12) and (14) this implies x → (a → b) = (a · x) → b = (a · x) → ((a · x) ∨ b) = 1: The precedent relations imply x = x · 1 = x · (x → (a → b))6a → b. Thus condition 2.1(iii) holds. In a similar way, one obtains that condition 2.1(iv) also holds. This completes the proof of 2.3(i). Denition 2.4. Let R() be the class of all residuated join semilattice monoids and L; M ∈ R(). (i) P() is the algebraic category associated with the class R().
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(ii) Hom[L; M] is the set of all morphisms from L to M in the category P(); (L) is the set of all subalgebras of L; H(L) is the set of all homomorphic images of L and Is(L) is the set of all M which are isomorphic with L. (iii) i∈I Li is the direct product of a family (Li )i∈I of algebras from K() and j : i∈I Li → Lj is the canonical projection of rank j, for all j ∈ I . Consequence 2.5. The class R() is a variety of algebras from K(), i.e. the following conditions hold: (i)
(L) ∪ H(L) ∪ Is(L) ⊆ R(), for all L ∈ R(). (ii) i∈I Li ∈ R(), for any family (Li )i∈I of algebras from R(). Proof. From Lemma 2.3 it follows that R() is an equational class of algebras from K(). Therefore, both conditions 2.5(i) and (ii) hold. Example 2.6. (a) Let L = [0; 1] be the real unit interval and a; b ∈ L such that 0¡a¡b¡1. For all x; y ∈ L, de3ne x ∨ y = max{x; y}; x ∧ y = min{x; y}; x·y =
0 if x 6 a and y 6 b; x ∧ y if a ¡ x or b ¡ y;
1 if x 6 y; x → y = b if y ¡ x 6 a; y if y ¡ x and a ¡ x; if x 6 y; 1 x→y ˜ = a ∨ y if y ¡ x 6 b; y if y ¡ x and b ¡ x: Then L[0; 1] = ([0; 1]; ∨; ·; →; →; ˜ 1) ∈ R() (see [9]). (b) The negative cone (G − ; +; 6; 0) of any lattice group (G; +; ∧; ∨; 0) has a structure of residuated join semilattice monoid, L[G − ] = (G − ; ∨; +; →; →; ˜ 0) ∈ R(), where G − = {x ∈ G=x60} such that x → y = ((−x) + y) ∧ 0
and
x→y ˜ = (y + (−x)) ∧ 0; for all x; y ∈ G − :
(c) Let L be a set of points from the real plane R2 de3ned by L = {(x; x) ∈ R2 =0 6 x ¡ 1} ∪ {a; b; c; 1};
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where a = (1; 2); b = (2; 1); c = (2; 2); 1 = (3; 3). De3ne four binary operations ∨; ·; → and → ˜ on L such that for all (x; y); (x ; y ); u; v ∈ L, (x; y) ∨ (x ; y ) = (max{x; x }; max{y; y }) u · 1 = u = 1 · u;
u · v = v · u = (0; 0); if u = 1 and v = 1; 1; if u ∨ v = v; u → v = u→v ˜ = v; if u = 1; c; if u = 1 and u ∨ v = v: Then (L; ∨; ·; →; →; ˜ 1) ∈ R() and the underlying poset (L; 6; 1) is not a lattice: the set of all lower bounds of the two elements set {a; b} is de3ned by [0; 1) = {(x; x) ∈ R2 =06x¡1} and [0; 1) is a subset of L without a greatest element. Now we introduce speci3c residuated mappings on a join semilattice. Denition 2.7. Let f : L → L be an endomorphism of a join semilattice (L; ∨), i.e. f(x ∨ y) = f(x) ∨ f(y), for all x; y ∈ L. We call f a residuated endomorphism if there exists a function g : L → L such that the following conditions hold, for all x; y ∈ L: (i) x6y implies g(x)6g(y); (ii) f(g(x))6x6g(f(x)). Let Res(L; ∨) the set of all residuated endomorphisms of (L; ∨). If f ∈ Res(L; ∨) then the function g satisfying 2.7(i) and (ii) is unique. We denote this unique function by f+ and call it the residual of f [2]. If f ∈ Res(L; ∨) then the following conditions hold, for all ∈ L: • f+ () = max{x ∈ L=f(x)6}; • f() = min{x ∈ L=6f+ (x)}. Example 2.8. Let L = (L; ∨; ·; →; →; ˜ 1) ∈ R(). The left translation a and the right translation a given by a (x) = a · x and a (x) = x · a are residuated endomorphisms of (L; ∨) such that a+ = ra and a+ = la , where ra is the right residual and la is the left residual by a ∈ L given by ra (x) = a → x and la (x) = a→x. ˜ The following lemma shows that a residuated join semilattice monoid can be associated with some families of residuated endomorphisms (see De3nition 2.7). Then a representation theorem for algebras from R() will be proved. Lemma 2.9. Let T = (L; ∨; 1); ; be an upper bounded join semilattice (L; ∨; 1) together with two functions and from L to Res(L; ∨) such that the following conditions hold, for all a; b ∈ L: (C1) (1)(a) = a = (a)(1) and (1)(a) = a = (a)(1);
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(C2) a6b i8 (a)(x)6(b)(x), for all x ∈ L i8 (a)(x)6(b)(x), for all x ∈ L; (C3) (a) ◦ (b) = (b) ◦ (a), where ◦ is the usual composition of functions. There exists a structure of residuated join semilattice monoid L[T] = (L[T]; ∨; ·; →; →; ˜ 1) associated with T such that L[T] = L and the following relations hold: (i) (x)(y) = x · y = (y)(x); (ii) x → y = + (x)(y); (iii) x → ˜ y = + (x)(y). Proof. De3ne three binary operations ·; → and → ˜ on L by x · y = (x)(y);
x → y = + (x)(y)
and
x→y ˜ = + (x)(y)
for all x; y ∈ L, where + (x) : L → L is the residual of (x) ∈ Res(L; ∨) and + (x) : L → L is the residual of (x) ∈ Res(L; ∨). Using (C1) and (C3) it follows that the following relations holds, for all x; y; z ∈ A: x · y = (x)(y) = (x)((y)(1)) = ((x) ◦ (y))(1) = ((y) ◦ (x))(1) = (y)((x)(1)) = (y)(x); x · 1 = (x)(1) =x = (1)(x) = 1 · x; x · (y · z) = (x)(y · z) = (x)((z)(y)) = ((x) ◦ (z))(y) = ((z) ◦ (x))(y) = (z)((x)(y)) = (z)(x · y) = (x · y) · z: Suppose now that x6y and a ∈ L. Using (i) and (C2) it follows that x · a = (x)(a) 6 (y)(a) = y · a
and
a · x = (x)(a) 6 (y)(a) = a · y:
From the precedent properties it follows that both conditions (i) and (ii) from De3nition 2.1 hold. Let a; x; b ∈ L. From De3nition 2.7 one obtains that a · (a → b) = (a)(+ (a)(b)) 6 b; (a→b) ˜ · a = (a)(+ (a)(b)) 6 b; a · x 6 b implies (a)(x) 6 b; thus x 6 + (a)(b) = a → b; x · a 6 b implies (a)(x) 6 b; thus x 6 + (a)(b) = a→b: ˜ This completes the proof of Lemma 2.9.
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Theorem 2.10 (Representation of a residuated join semilattice monoid). Let L=(L; ∨; ·; →; →; ˜ 1) be a residuated join semilattice monoid. Then there exists a system T = (L; ∨; 1); ; such that conditions (C1)–(C3) from Lemma 2.9 hold and L = L[T]. Proof. From Example 2.8 it follows that one can de3ne a function : L → Res(L; ∨) by (a) = a and a function : L → Res(L; ∨) by (a) = a . Then the system T = (L; ∨; 1); ; satis3es all conditions (C1)–(C3) from Lemma 2.9 and L = L[T]. Consequence 2.11 (Cayley-type representation): Let L ∈ R() and 1L : L → L be the identity function on L. Then there exists an isomorphism from L onto a residuated join semilattice monoid (L) of residuated endomorphisms such that its underlying monoid is a submonoid of (Res(L; ∨); ◦; 1L ). Proof. There exists an order relation 6 on Res(L; ∨) such that for all f; g ∈ Res(L; ∨), f 6 g i6 f(x) 6 g(x)
for all x ∈ L
(16)
and the supremum of the two element subset {f; g} exists such that sup{f; g} = f ∨ g ∈ Res(L; ∨);
(17)
where f ∨ g : L → L is de3ned by (f ∨ g)(x) = f(x) ∨ g(x)
for all x ∈ L:
(18)
The residual of f ∨ g is the function (f ∨ g)+ : L → L de3ned by (f ∨ g)+ (x) = f+ (x) ∨ g+ (x)
for all x ∈ L:
(19)
Let Res(1L ] be the subset of Res(L; ∨) de3ned by Res(1L ] = {f ∈ Res(L; ∨)=f 6 1L }:
(20)
(Res(1L ]; ∨) is a join semilattice;
(21)
(Res(1L ]; ◦; 6; 1L ) is an integral ordered monoid:
(22)
Then
From Theorem 2.10 it follows that there exists a system T = (L; ∨; 1); ; such that conditions (C1)–(C3) from Lemma 2.9 hold and L = L[T]. Let (L) be the image set of the function : L → Res(L; ∨). Then the following conditions hold for all x; y ∈ L: (L) = {(a)=a ∈ L} = {f ∈ Res(1L ]=f(x) = f(1) · x for all x ∈ L} ⊆ Res(1L ];
(23)
(x ∨ y) = (x) ∨ (y);
(24)
(x · y) = (x) ◦ (y);
(25)
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(x → y) = max{f ∈ (L)=(x) ◦ f 6 (y)};
(26)
(x→y) ˜ = max{f ∈ (L)=f ◦ (x) 6 (y)}:
(27)
From 2.9 (C1) it follows that the function : L → Res(L; ∨) is injective: a = b implies (a)(1) = a = b = (b)(1); thus (a) = (b): From relations (21)–(27) it follows that there exists a structure of a residuated join semilattice monoid (L) = ((L); ∨; ◦; ⇒; ⇒; ˜ 1L ) ∈ R(); with the underlying poset ((L); 6) such that for all x; y ∈ L, (x) ⇒ (y) = (x → y); (x) ⇒ ˜ (y) = (x→y): ˜ The restriction of the function to its image set (L) de3ned by the correspondence L x → (x) ∈ (L) is an isomorphism from L onto (L) in the class R(). This completes the proof of Consequence 2.11. Consequence 2.12. Let (L; ∨; 1) be an upper bounded join semilattice and Res(1L ] be the set of all residuated endomorphisms f ∈ Res(L; ∨) such that f61L . Then the following conditions are equivalent: (i) There exists a structure L = (L; ∨; ·; →; →; ˜ 1) ∈ R() with the underlying lattice (L; ∨; 1). (ii) There exists a function : L → Res(1L ] such that the following conditions hold, for all x; y ∈ L: (x ∨ y) = (x) ∨ (y);
(28)
(1)(x) = x = (x)(1);
(29)
((x)(y)) = (x) ◦ (y):
(30)
If one of conditions (i) and (ii) holds then there exists a structure of residuated join semilattice monoid (L) with the underlying set (L) ⊆ Res(1L ] such that the restriction of the function to its image set (L) is an isomorphism from L onto (L). Proof. (i) ⇒ (ii): Suppose that (i) holds. From Theorem 2.10 it follows that there exists a system T = (L; ∨; 1); ; such that conditions (C1)–(C3) from Lemma 2.9 hold and L = L[T]. From (C1) it follows that (29) holds. From Lemmas 2.9(i), 2.2(i) and 2.3(10) it follows that for all x; y; z ∈ L, the following relations hold: (x)(y) = x · y 6 y = 1L (y);
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(x ∨ y)(z) = (x ∨ y) · z = (x · z) ∨ (y · z) = (x)(z) ∨ (y)(z) = ((x) ∨ (y))(z); ((x)(y))(z) = (x · y)(z) = (x · y) · z = x · (y · z) = (x)(y · z) = (x)((y)(z)) = ((x) ◦ (y))(z): Therefore, condition (ii) holds. (ii) ⇒ (i): Suppose that (ii) holds. De3ne a function : L → Res(1L ] by (x)(y) = (y)(x)
for all x; y ∈ L:
(31)
We verify that the system T = (L; ∨; 1); ; satis3es all conditions (C1)–(C3) from Lemma 2.9, which implies that condition (i) holds. Let a; b ∈ L. Then (1)(a) = (a)(1) = a = (1)(a) = (a)(1); i.e. condition 2.9(C1) holds. From (28) and (29) it follows that a 6 b i6 (a)(x) 6 (b)(x)
for all x ∈ L:
(32)
Let x ∈ L. Then using relation (31) one obtains (a ∨ b)(x) = (x)(a ∨ b) = (x)(a) ∨ (x)(b) = (a)(x) ∨ (b)(x) = ((a) ∨ (b))(x): Thus (a ∨ b) = (a) ∨ (b), which implies a 6 b i6 (a)(x) 6 (b)(x)
for all x ∈ L:
(33)
From (32) and (33) it follows that condition 2.9(C2) holds. Using relations (30) and (31) one obtains ((a) ◦ (b))(x) = (a)((b)(x)) = (a)((x)((b)) = ((a) ◦ (x))(b) = ((a)(x))(b) = ((b) ◦ (a))(x); for all a; b; x ∈ L. This implies that condition 2.9(C3) holds. Thus, equivalence (i) ⇔ (ii) is proved. The proof of the second part of consequence 2.12 is included in the proof of Consequence 2.11. Comment. Let R∗ () be the class of all (L) ∈ R() previously de3ned in the proof of consequence 2.11 for every L ∈ R(). Using the precedent results, one can prove that there exists an equivalence of categories from P() to the algebraic category P∗ () associated with the class R∗ (). Let = (2; 2; 2; 2; 2; 0). For every L = (L; ∨; ·; →; →; ˜ 1) ∈ R(), let L = (L; ∨; ∗l ; →; ∗r ; →; ˜ 1) be an algebra of type , where ∗l and ∗r are binary operations on L de3ned by x ∗l y = x · y = y ∗r x
for all x; y ∈ L:
We remark also that there exists an isomorphism of categories from P() to the algebraic category P( ) associated with the class R( ) of all algebras L , for any L ∈ R(). We have HomP() [L; M] = HomP( ) [L ; M ]
for all L; M ∈ P():
We shall use in the sequel the following duality principle: let e be an equation having the set of symbols of operations included in {∨; ∗l ; →; 1}; let eo be the equation obtained from e by replacing
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each occurrence of ∗l in e by ∗r and each occurrence of → in e by →; ˜ if e holds in R( ) then eo holds in R( ). 3. Homomorphic images In this section, we present a characterization theorem for homomorphic images of L in the algebraic category P() introduced in De3nition 2.4. We will use the usual notions of congruence relation and semilattice 3lter. Denition 3.1. Let L = (L; ∨; ·; →; →; ˜ 1) be a residuated join semilattice monoid. A normal :lter on L is a subset F of L with the property that there exists a congruence relation R of L such that F = {x ∈ L=xR1}: Lemma 3.2. Let R be a congruence relation of L and FR = {x ∈ L=xR1} be the normal :lter associated with R. For all x; y ∈ L, de:ne er (x; y) = (x → y) · (y → x) and el (x; y) = (x → ˜ y) · (y→x). ˜ Then the following conditions hold: (i) xRy i8 er (x; y) ∈ FR ; (ii) xRy i8 el (x; y) ∈ FR ; (iii) For any normal :lter F on L there exists an unique congruence relation R such that F = {x ∈ L=xR1}: Proof. (i) Let x; y ∈ L. Suppose that xRy. From yRy; yRx and xRx it follows that x → yRy → y
and
y → xRx → x;
but 1 = y → y = x → x, thus er (x; y)R1, i.e. er (x; y) ∈ FR . Suppose now that er (x; y) ∈ FR , i.e. (x → y) · (y → x)R1: From x → yRx → y; 1 = (x → y)·(y → x) → (x → y); 1 → (x → y) = x → y, one obtains that x → yR1, but xRx, thus x · (x → y)Rx. We have also yRy. Then (x · (x → y)) ∨ yRx ∨ y, but using Eq. (10) [Lemma 2.3(ii)] we have (x · (x → y)) ∨ y = y, thus yRx ∨ y. Also we have xRx ∨ y, therefore xRy. (ii) The equivalence (ii) follows in a similar way with (i). (iii) Let F be a normal 3lter and R; R be congruence relations such that F = {x ∈ L=xR1} = {x ∈ L=xR 1}: From (i) it follows that xRy i6 er (x; y) ∈ FR = F = FR i6 xR y. Therefore, R = R . The following theorem presents an algebraic characterization of normal 3lters. Theorem 3.3. Let F ⊆ L. The following conditions are equivalent: (i) F is a normal :lter of L.
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(ii) F satis:es: (FN1) F is a :lter of the semilattice (L; ∨); (FN2) F is a submonoid of (L; ·); (FN3) er (a · x; x · a) ∈ F and el (a · x; x · a) ∈ F, for all a ∈ F and x ∈ L. (iii) F satis:es: (FN1∗ ) 1 ∈ F; (FN2∗ ) x ∈ F and x → y ∈ F implies y ∈ F; (FN3∗ ) x → y ∈ F i8 x → ˜ y ∈ F. Proof. (i) ⇒ (ii): Suppose that (i) holds, i.e. F = {x ∈ L=xR1}, for some congruence relation R. It is clear that F satis3es (FN1) and (FN2). We veri3es now the 3rst part of (FN3). Let a ∈ F and x ∈ L. It follows that aR1; a · xR1 · x
and
x · aRx · 1
but x · 1 = x = 1 · x and 1 = x → x, thus (a · x) → (x · a)R1
and
(x · a) → (a · x)R1:
This implies that er (a · x; x · a) ∈ F. The relation el (a · x; x · a) ∈ F from the second part of (FN3) follows in a similar way by a dual argument. Thus (ii) holds. (ii) ⇒ (iii): Suppose that (ii) holds. We have F = ∅. Let x0 ∈ F. From x0 61 and (FN1) it follows that (FN1∗ ) holds. Let x; y ∈ L such that x ∈ F and x → y ∈ F. From (FN2) it follows that x · (x → y) ∈ F, but x · (x → y)6y, thus using (FN1) one obtains that y ∈ F. Therefore, condition (FN2∗ ) holds. One obtains in a similar way the following relation, for all x; y ∈ L: (1) x ∈ F and x → ˜ y ∈ F implies y ∈ F. We verify now that (FN3∗ ) also holds, i.e. for all x; y ∈ L: (2) x → y ∈ F implies x → ˜ y ∈ F. (3) x → ˜ y ∈ F implies x → y ∈ F. For (2) let x; y ∈ L such that a = x → y ∈ F. From (FN3) and (FN1∗ ) it follows that (4) u = (a · x)→(x ˜ · a) ∈ F. (5) v = (x · a)→y ˜ = 1 ∈ F. From Eqs. (15) and (11) [Lemma 2.3(ii)] one obtains that (6) v→(u ˜ →((a ˜ · x)→y)) ˜ = 1 ∈ F. Using (1), relations (4)–(6) imply (a · x)→y ˜ ∈ F, but (a · x)→y ˜ = a→(x ˜ → ˜ y), thus (7) a→(x ˜ → ˜ y) ∈ F. Using (1), from a = x → y ∈ F and (7) one derives x → ˜ y ∈ F. Thus (2) holds. Relation (3) follows in a similar way by a dual argument. Therefore, (iii) holds.
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(iii) ⇒ (i): Suppose that (iii) holds. For all x; y ∈ L, de3ne xRy i6 er (x; y) = (x → y) · (y → x) ∈ F: Then R is a congruence relation of L and F = {x ∈ L=xR1}. Thus (i) holds. Now we establish a connection between the structure of residuated join semilattice monoid and the structure of quantale [23] considered in the commutative case by Goguen [14]. Using this connection we derive a new expression of the Cayley-type representation and an example of homomorphic images in R(). Denition 3.4. A complete residuated join semilattice monoid is an algebra L = (L; ∨; ·; →; →; ˜ 1) such that the following conditions hold: (i) L ∈ R(); (ii) the underlying algebra (L; ∨; 1) is a complete join semilattice, i.e. for every nonempty subset A of L its supremum denoted by sup A or sup(A) exists in the corresponding poset (L; 6), where x6y i6 x ∨ y = y. Let CR() be the class of all complete residuated join semilattice monoids. Denition 3.5. An integral quasi-quantale is a system (L; ∨; ·; 1) such that the following conditions hold: (i) (L; ∨; 1) is a complete join semilattice; (ii) (L; ·; 6; 1) is an integral ordered monoid; (iii) x · (sup A) = sup(x · A) and (sup A) · x = sup(A · x), for all x ∈ L and A ⊆ L such that A = ∅, where ∨ is the binary supremum on L; x · A = {x · a=a ∈ A} and A · x = {a · x=a ∈ A}. Let IqQ be the class of all integral quasi-quantales. An integral quantale is an integral quasi-quantale (L; ∨; ·; 1) with 0, i.e. there exists 0 ∈ L such that 06x, for all x ∈ L. Let IQ be the class of all integral quantales (L; ∨; ·; 0; 1), where 0 is the least element of the underlying poset (L; 6). Example 3.6. (a) Let L = (0; 1] be the real unit interval without 0. Then the system (L; ∨; ·; 1) is an integral quasi-quantale, where x ∨ y = max{x; y}, for all x; y ∈ L and · is the algebraic product on L. This is the strict positive membership space of fuzzy sets. If f : X → [0; 1] is a membership function then the support set of f is de3ned by supp(f) = f−1 (0; 1] i.e. for all x ∈ X we have x ∈ supp(f) i6 f(x)¿0 [26,19,18,27,20]. (b) Let (L; ∨; ·; 1) such that 2.1(i) and (ii) hold. Suppose that there exists a zero element 0 ∈ L, i.e. 06x, for all x ∈ L. Let Id(L) be the set of all ideals of the join semilattice (L; ∨; 1) ordered by the inclusion relation ⊆. There exists a structure of integral quantale on the set Id(L) [23]. For all I; J ∈ Id(L), let I · J be the ideal generated by the set {x · y=x ∈ I and y ∈ J }. If I is a set of ideals from Id(L) then its supremum sup I in the poset (Id(L); ⊆) is the ideal generated by the
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subset I ∈I I . For all I; J ∈ Id(L), the binary join is de3ned by I ∨ J = sup{I; J } which is the ideal generated by the set I ∪ J . Then (Id(L); ∨; ·; 0; 1) is an integral quantale with the underlying poset (Id(L); ⊆), where 0 = {0} and 1 = L. The following proposition shows that the class CR() can be identi3ed with the class IqQ. Proposition 3.7. The following conditions hold: (i) If L = (L; ∨; ·; →; →; ˜ 1) ∈ CR() then (L; ∨; ·; 1) ∈ IqQ. (ii) If (L; ∨; ·; 1) ∈ IqQ then L = (L; ∨; ·; →; →; ˜ 1) ∈ CR(), where → and → ˜ are binary operations on L such that for all a; b ∈ L, a → b = sup{x ∈ L=a · x 6 b};
(34)
a→b ˜ = sup{x ∈ L=x · a 6 b}:
(35)
(iii) There exists a bijection from CR() onto IqQ. Proof. (i) Let L = (L; ∨; ·; →; →; ˜ 1) ∈ CR(). From De3nitions 3.4 and 2.1 it follows that 3.5(i) and (ii) hold. We verify now that 3.5(iii) holds. Let x ∈ L and A ⊆ L such that A = ∅. It is clear that sup(x · A) 6 x · (sup A):
(36)
We have x · a6 sup(x · A) for all a ∈ A. Using Lemma 2.2(iii) one obtains that a 6 x → sup(x · A)
for all a ∈ A;
which implies that sup A6x → sup(x · A). Thus x · (sup A) 6 sup(x · A):
(37)
From (36) and (37) one obtains x · (sup A) = sup(x · A):
(38)
The equation (sup A) · x = sup(A · x)
(39)
follows by a dual argument. Thus 3.5(iii) holds. From De3nition 3.5 it follows that (L; ∨; ·; 1) ∈ IqQ. (ii) Let (L; ∨; ·; 1) ∈ IqQ. For all a; b ∈ L, de3ne a → b and a→b ˜ by (34) and (35). Let a; b ∈ L. From De3nition 3.5(iii), 3.7(1) and 3.7(2) one obtains that a · (a → b) = a · (sup{x=a · x 6 b}) = sup{a · x=a · x 6 b} 6 b; (a→b) ˜ · a = (sup{x=x · a 6 b}) · a = sup{x · a=x · a 6 b} 6 b: From De3nition 3.5 and the precedent relations one obtains that all the conditions from De3nition 2.1 are satis3ed, i.e. L = (L; ∨; ·; →; →; ˜ 1) ∈ R(). Condition 3.5(i) also holds. Thus, from De3nition 3.4 it follows that L ∈ CR(). This completes the proof of 3.7(ii).
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(iii) The correspondence CR() L → (L; ∨; ·; 1) ∈ IqQ is a bijection from CR() to IqQ. Remark 3.8. Let R0 () be the class of all algebras L = (L; ∨; ·; →; →; ˜ 1) ∈ R() such that there exists an element 0 ∈ L with 06x, for all x ∈ L. De3ne a subclass CR0 () ⊆ CR() by CR0 () = CR()∩ R0 (). Using Proposition 3.7 one derives that there exists a bijection from CR0 () to the class IQ of all integral quantales. The following proposition expresses a connection between the classes R() and IQ. Proposition 3.9. Let R0 () and CR0 () be the classes of algebras from R() de:ned in Remark 3.8. Suppose that L = (L; ∨; ·; →; →; ˜ 1) is a residuated join semilattice monoid, i.e. L ∈ R(). Then the following conditions hold: (i) There exist L0 ∈ R0 () and an injective homomorphism from L to L0 . (ii) There exist an integral quantale Id(L0 ) = (Id(L0 ); ∨; ·; 0; 1) ∈ IQ (see Example 3.6(b)) and an injective homomorphism from L into the complete residuated join semilattice monoid with zero Id(L0 ) = (Id(L0 ); ∨; ·; →; →; ˜ 1) ∈ CR0 () associated with Id(L0 ) as in Proposition 3.7(ii). Proof. (i) We consider two cases: Case 1: There exists 0 ∈ L such that 06x, for all x ∈ L. Case 2: For each x ∈ L there exists y ∈ L such that y¡x, i.e. y6x and y = x. In case 1 we let L0 = L ∈ R0 (), which implies that the identity function 1L : L → L is an isomorphism from L onto L0 . In case 2 let 0 ∈= L and de3ne L0 = L ∪ {0}; x · 0 = 0 = 0 · x; 0 → x = 1 = 0→x ˜ and 1 if x = 0; x → 0 = x→0 ˜ = 0 if x = 0 for all x ∈ L0 , which implies that L0 = (L0 ; ∨; ·; →; →; ˜ 1) ∈ R0 () and the inclusion map iL : L → L0 from the subset L ⊆ L0 to L0 is an injective homomorphism from L into L0 . Thus condition 3.9(i) holds. (ii) We prove now that both in cases 1 and 2 Condition 3.9(ii) holds. In case 1, let L0 = L and Id(L0 ) = (Id(L0 ); ∨; ·; 0; 1) ∈ IQ be the integral quantale of all ideals of the bounded join semilattice (L0 ; ∨; 0; 1) de3ned in Example 3.6(b), where L0 = L. Let Id(L0 ) = (Id(L0 ); ∨; ·; →; →; ˜ 1) ∈ CR0 () be the complete residuated join semilattice monoid with zero associated with Id(L) previously de3ned in Proposition 3.7(ii). Let f : L0 → Id(L0 ) be a function de3ned by f(x) = (x] = {y ∈ L=y 6 x} ∈ Id(L0 )
for all x ∈ L0 :
Then f is an injective homomorphism from L0 into Id(L0 ) such that f(0) = 0. In addition f preserves all the supremums existing in L0 . In case 2, condition 3.9(ii) follows from the properties previously presented in case 1 using condition 3.9(i).
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Denition 3.10. A complete join semilattice endomorphism of a complete join semilattice (L; ∨; 1) is a function f : L → L such that f(sup A) = sup f(A)
for all A ⊆ L such that A = ∅:
Consequence 3.11 (Cayley-type representation): Every residuated join semilattice monoid from R() can be embedded into a residuated join semilattice monoid from R() with an underlying set of complete join semilattice endomorphisms. Proof. Let L = (L; ∨ ; ·; → ; →; ˜ 1) ∈ R(). From Proposition 3.9(ii) it follows that there exist a complete residuated join semilattice monoid with zero L∗ = (L∗ ; ∨; ·; →; →; ˜ 1) ∈ CR0 () and an injective homomorphism i : L → L∗ from L into L∗ . From the proof of Consequence 2.11 it follows that there exists an isomorphism in R() from L∗ onto a residuated join semilattice monoid (L∗ ) of complete join semilattice residuated endomorphisms which we denote by : L∗ → (L∗ ). Thus ◦ i : L → (L∗ ) is an injective homomorphism from L into (L∗ ). This shows that 3.11 holds. The following example is suggested by Proposition 3.9(ii) and it expresses a procedure to construct an homomorphic image using the notion of quantic nucleus introduced by Rosenthal [23] for the study of quotient quantales. Example 3.12. Let L = (L; ∨ ; ·; → ; →; ˜ 1) ∈ R(). Suppose that c : L → L is a quantic nucleus on L, i.e. c is a closure operator of the poset (L; 6) such that c(x) · c(y) 6 c(x · y)
for all x; y ∈ L:
˜ c ; 1c ) ∈ R(), where c(L) = {x ∈ L=c(x) = x} ⊆ L is the set of all Then c(L) = (c(L); ∨c ; ·c ; →c ; → c-closed elements of L and for all x; y ∈ c(L): x ∨c y = c(x ∨ y); x ·c y = c(x · y); ˜ 1c = 1: x→ ˜ c y = c(x→y);
x →c y = c(x → y);
x→ ˜ c y = c(x → y);
The restriction of c to its image is a surjective homomorphism from L onto c(L), thus c(L) ∈ H(L). From the 3rst isomorphism theorem with respect to the variety R(), De3nition 3.1 and Theorem 3.3 one derives that the algebra c(L) of c-closed elements of L is isomorphic with the quotient algebra of L with respect to the normal :lter F = {x ∈ L=c(x) = 1} of all c-dense elements of L. 4. Some special subclasses The underlying poset of a residuated join semilattice monoid from R() is not necessarily a lattice (see Example 2.6(c)). One can consider the problem to determine conditions for an algebra from R() such that its underlying poset becomes a lattice.
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Now we present some subclasses of R() with respect to speci3c solutions of the precedent problem. Let RL() be the class of all algebras L = (L; ∨ ; ·; → ; →; ˜ 1) ∈ R() such that the underlying poset (L; 6) of L is a lattice. It is obvious that Boolean algebras [22] can be viewed as particular algebras from RL(). 4.1. Complete structures A complete structure of R() is an algebra of CR() (see De3nition 3.4). For all L ∈ R(), the set of all lower bounds of any two elements subset {x; y} ⊆ L is nonempty, because of x · y6x and x · y6y, thus B{x; y} = {z ∈ L=z 6 x and z 6 y} = ∅: Let L∗ ∈ CR(). For all x; y ∈ L∗ , de3ne x ∧ y = sup LB{x; y}. Then (L∗ ; 6) is a lattice with the binary meet ∧ de3ned by x ∧ y = inf {x; y}. Thus CR()
⊆ RL():
(S1)
We obtain also that f ∈ Res(L∗ ; ∨) i6 f : L∗ → L∗ is a complete join semilattice endomorphism of (L∗ ; ∨; 1) (see De3nition 3.10). This implies that there exists a structure Res(1L∗ ] = (Res(1L∗ ]; ∨; ◦; ⇒; ⇒; ˜ 1L∗ ) ∈ R(); where Res(1L∗ ] = {f ∈ Res(L∗ ; ∨)=f 6 1L∗ }: Consider the corresponding function : L∗ → Res(1L∗ ] from the proof of Cayley-type representation results (Consequences 2.11 and 3.11) de3ned by (a)(x) = a · x
for all a; x ∈ L∗ :
It follows that is an injective homomorphism from L∗ to Res(1L∗ ]. Using Proposition 3.9(ii) we obtain that for any L∈R() there exists a complete structure L∗ ∈ CR() such that L is isomorphic with a subalgebra of Res(1L∗ ]. A particular case is L∗ = [0; 1]. In this case, we let LCF[0; 1] = {f : L∗ → L∗ =f is increasing and left-continuous on L∗ = [0; 1] with f(x) 6 x; ∀x ∈ L∗ }: Then Res(1L∗ ] = LCF[0; 1]. This implies that there exists a complete structure LCF[0; 1] ∈ CR() on the set LCF[0; 1]. Therefore, each structure of residuated join semilattice monoid on the real unit interval [0; 1] is isomorphic with a subalgebra of LCF[0; 1]. The logic of left-continuous commutative t-norms has been considered by Esteva and Godo [8]. A special subclass of CR() is CR0 () = CR() ∩ R0 () (see Remark 3.8) representing the class of all complete structures with a zero element. Thus CR0 ()
⊆ CR():
(S2)
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4.2. Finite structures Let FR() be the class of all L ∈ R() such that L is a 3nite set which we call :nite structures. If L ∈ FR() is a 3nite structure from R() then there exists 0 ∈ L such that 06x, for all x ∈ L and there exists sup A, for all A ⊆ L such that A = ∅ because L is a 3nite set. Thus FR()
⊆ CR0 ():
(S3)
If A ∈ FR() such that the corresponding poset (A; 6) is a chain and the binary operations · and ∧ are di6erent then the set A has the cardinality |A|¿4. An example of 3nite residuated lattice monoid with four elements A4 is any chain with an underlying set A4 = {0; a; b; 1}; 0¡a¡b¡1 and the operation · de3ned by the following table: 0 a b 1
0 0 0 0 0
a 0 0 a a
b 0 0 b b
1 0 a b 1
A functional representation of this 3nite structure with four elements is de3ned by a set S4 = {&; f; g; 1E } of four functions from E to E such that &¡f¡g¡1E , where E = {1; 2; 3} is the chain of three natural numbers 1¡2¡3, 1E (x) = x and &(x) = 1; for all x ∈ E;
f(1) = g(1) = 1;
f(2) = g(2) = 1;
f(3) = 2 and g(3) = 3: With respect to the usual composition of functions ◦ and the natural order 6, the set S4 has a structure of residuated lattice monoid S4 which is isomorphic with the residuated lattice monoid A4 previously de3ned. 4.3. Lattice pseudo-hoops Let L ∈ R() and Res(L; ∨) be the set of all residuated endomorphisms of the join semilattice (L; ∨). Denition 4.1. Let : L → Res(L; ∨) be the function de3ned by (a) = a , for all a ∈ L. The left translation relation on L is a binary relation T on L de3ned by bT a i6 ∃x ∈ L with a (x) = a · x = b: Denition 4:1 . Let : L → Res(L; ∨) be the function de3ned by (a) = a , for all a ∈ L. The right translation relation on L is a binary relation T on L de3ned by bT a i6 ∃x ∈ L with a (x) = x · a = b:
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Lemma 4.2. For all a ∈ L, let a+ : L → L be the residual of a ∈ Res(L; ∨) and a+ : L → L be the residual of a ∈ Res(L; ∨). Consider the two binary relations T and T from De:nitions 4.1 and 4:1 . Then the following conditions hold: (i) The underlying partially ordered monoid (L; ·; 6; 1) is negatively ordered, i.e. a 61L and a 61L , for all a ∈ L. (ii) bT a i8 a · (a → b) = a (a+ (b)) = b. (iii) bT a i8 (a→b) ˜ · a = a (a+ (b)) = b. Proof. (i) Condition 4.2(i) follows from relations a · x6x and x · a6x for all a; x ∈ L. (ii) Suppose that bT a, i.e. ∃x ∈ L with a · x = b. We have a · (a → b)6b. From a·x =b
and
x 6 a → (a · x);
it follows that x6a → b, thus b = a · x6a · (a → b). This implies a · (a → b) = b: If the precedent relation holds then a · x = b, for x = a → b ∈ L, thus bT a. This shows that 4.2(ii) holds. (iii) Condition 4.2(iii) follows in a similar way with 4.2(ii) by a dual argument. Lemma 4.3. Both the left translation relation T and the right translation relation T are order relations on L such that for all a; b ∈ L, bT a or bT a implies b 6 a: Proof. We have that T is reBexive because of relations a · (a → a) = a · 1 = a, for all a ∈ L. Suppose now that cT b and bT a. We verify that cT a, which implies that T is transitive. From Lemma 4.2(ii) it follows that b · (b → c) = c and a · (a → b) = b. This implies c6a · (a → c), using the following relations: c = b · (b → c) = (a · (a → b)) · (b → c) = a · ((a → b) · (b → c)); (a → b) · (b → c) 6 a → c: But a · (a → c)6c, therefore c = a · (a → c). Using Lemma 4.2(ii) this implies cT a. It is clear that bT a implies b6a. Therefore, bT a and aT b implies b6a and a6b, thus a = b. This shows that T is antisymmetric. Thus, T is an order relation such that for all a; b ∈ L, bT a implies b6a. Using a dual argument we derive also that T is an order relation such that for all a; b ∈ L, bT a implies b6a. Denition 4.4. A lattice pseudo-hoop [13] is a residuated join semilattice monoid L ∈ R() such that the underlying partially ordered monoid (L; ·; 6; 1) is the dual of a divisibility monoid [1], i.e. for all a; b ∈ L, b 6 a i6 bT a i6 bT a:
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The following proposition gives two expressions of the notion of lattice pseudo-hoop. Proposition 4.5. Let L ∈ R(). Then the following conditions are equivalent: (i) L is a lattice pseudo-hoop. (ii) L satis:es the following equations, for all a, b ∈ L: a · (a → b) = b · (b → a)
(40)
(a → ˜ b) · a = (b→a) ˜ · b:
(41)
(iii) There exists a binary operation ∧ on L such that the following conditions hold: (L; ∧; ∨) is a lattice;
(42)
a · (a → b) = a ∧ b;
(43)
(a→b) ˜ · a = a ∧ b:
(44)
Proof. (i) ⇔ (ii): The equivalence between conditions 4.5(i) and (ii) follows from De3nition 4.4 and Lemma 4.2. (ii) ⇒ (iii). Suppose that (ii) holds. De3ne a binary operation ∧ by a ∧ b = a · (a → b)
for all a; b ∈ L:
(45)
From (40) and (41) one derives that a · (a → b) = inf {a; b} = (a→b) ˜ · a:
(46)
Relations (45) and (46) imply relations (42)–(44). Thus (iii) holds. (iii) ⇒ (ii): Suppose now that (iii) holds. From (42) we obtain that for all a; b ∈ L, a ∧ b = b ∧ a, thus using (43) and (44) one obtains that (40) and (41) hold. This shows that (ii) holds. Remark 4.6. Let LPH() be the class of all lattice pseudo-hoops. Then LPH()
⊆ RL():
(S4)
˜ 1) such that the The class LPH() can be identi3ed with a variety of algebras (L; ∧; ∨; ·; → ; →; following conditions hold: (LPH1) (LPH2) (LPH3) (LPH4)
L = (L; ∨; ·; →; →; ˜ 1) ∈ R(). (L; ∧; ∨) is a lattice. a · (a → b) = a ∧ b, for all a; b ∈ L. (a → ˜ b) · a = a ∧ b, for all a; b ∈ L.
For each lattice pseudo-hoop its underlying lattice is distributive. Remark 4.7. A pseudo BL-algebra [9] is a lattice pseudo-hoop (L; ∧; ∨; ·; → ; →; ˜ 1) which satis3es the following equations, for all x; y ∈ L: (PL) (x → y) ∨ (y → x) = 1 and (x → ˜ y) ∨ (y → ˜ x) = 1.
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Let PBL() be the class of all pseudo BL-algebras. Then PBL()
⊆ LPH():
(S5)
With respect to the theory of noncommutative fuzzy logic it is natural also to consider the following notion: a Weak BL-algebra (WBL-algebra) is a residuated join semilattice monoid L ∈ R() satisfying the precedent equations (PL). Let WBL() be the class of all WBL-algebras. Then WBL()
⊆ R():
(S6)
An important open problem is to describe subdirectly irreducible WBL-algebras in order to obtain a concrete expression of the subdirect decomposition theorem for this special variety of algebras. We remark that if L ∈ R() such that (L; 6) is a chain then L is a WBL-algebra. 4.4. Cancellative structures Denition 4.8. The class Cancel R() is the class of all cancellative residuated join semilattice monoids, i.e. L ∈ Cancel R() i6 L ∈ R() and the following condition hold, for all a; x; y ∈ L: (CL) a · x = a · y or x · a = y · a implies x = y. De3nition 4.8 shows that Cancel R()
⊆ R():
(S7)
We call a cancellative structure any algebra from Cancel R(). The following lemma gives a characterization of cancellative structures. Lemma 4.9. Let L ∈ R(). The following conditions are equivalent: (i) L ∈ Cancel R(). (ii) a → (a · x) = x = a → ˜ (x · a), for all a; x ∈ L. Proof. (i) ⇒ (ii): Suppose that (i) holds. Let a; x ∈ L. Then the following conditions hold: a · (a → (a · x)) = a · x;
(47)
(a→(x ˜ · a)) · a = x · a:
(48)
Eq. (47) follows from the relations a · (a → (a · x)) 6 a · x
and
x 6 a → (a · x):
Eq. (48) follows by a dual argument. Using De3nition 4.8 (CL) one derives that the two Eqs. (47) and (48) imply 4.9(ii).
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(ii) ⇒ (i): Suppose now that (ii) holds. Let a; x; y ∈ L such that a · x = a · y or x · a = y · a. Using 4.9(ii) one obtains that a · x = a · y implies x = a → (a · x) = a → (a · y) = y; x · a = y · a implies x = a→(x ˜ · a) = a→(y ˜ · a) = y: Therefore, L satis3es the cancellative law 4.8(CL), i.e. 4.9(i) holds. Remark 4.10. Lemma 4.9(ii) expresses the fact that L is a cancellative structure i6 both the family (a )a∈L of left translations and the family (a )a∈L of right translations are families of injective functions such that a+ ◦ a = 1L and + a ◦ a = 1L
for all a ∈ L:
The precedent relations from Remark 4.10 express the facts that in every cancellative structure L, respectively, the residual a+ of a is a retract of a and the residual a+ of a is a retract of a in the set of all increasing functions from L to L. Example 4.11. (a) Let L = (0; 1] ⊆ R with the usual binary join ∨ and algebraic product ·. Then there exists a structure of cancellative commutative structure with the underlying monoid ((0; 1]; ·; 6; 1) such that for all a; x ∈ (0; 1]: 1 if a 6 x; a → x = a→x ˜ = x if x ¡ a: a This structure is used in di6erent applications of fuzzy set theory (see [27]). In addition, this is a complete structure, because sup A exists, for any nonempty subset A ⊆ (0; 1]. Also the corresponding structure ((0; 1]; ∨; ·; →; →; ˜ 1) ∈ R() is a simple algebra, because a subset F ⊆ L is a normal 3lter (see De3nition 3.1 and Theorem 3.3) i6 F = L or F = {1}. In this case, the following speci3c conditions are satis3ed, for all a; b ∈ (0; 1]: a 6 b i6 a · (a → b) = b and a = b; a 6 b i6 (a→b) ˜ · a = b and a = b: (b) The system L[G − ] ∈ R() de3ned in Example 2.6(b) on the negative cone (G − ; +; 6; 0) of a lattice group (G; +; ∧; ∨; 0) is a cancellative structure. In addition, L[G − ] is a lattice pseudo-hoop. We prove in the following proposition that the precedent properties are characteristic for this kind of structures. The following proposition (see [13]) follows using a known result from the study of residuated structures [1,2,10]. It is in connection with the structure from Example 2.6(b) and the properties presented in the previous Example 4.11(b).
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Proposition 4.12. Let L = (L; ∨; ·; →; →; ˜ 1) ∈ R(). The following conditions are equivalent: (i) L is isomorphic to an algebra L[G − ] ∈ R() associated with the negative cone of a lattice group; (ii) L is a cancellative lattice pseudo-hoop. Proof. (i) ⇒ (ii): Suppose that 4.12(i) holds. From the de3nition given in Example 2.6(b) it is clear that the structure L[G − ] is cancellative and a lattice pseudo-hoop. From Lemma 4.9 and Proposition 4.5 it follows that the class Cancel R() ∩ LPH() is equational. Thus, any algebra L ∈ R() isomorphic with L[G − ] is an algebra from Cancel R() ∩ LPH(), i.e. condition 4.12(ii) holds. (ii) ⇒ (i): Suppose now that 4.12(ii) holds. Then the underlying partially ordered monoid (L; ·; 6; 1) of L is isomorphic with the negative cone (G − ; +; 6; 0) of a lattice group (G; +; ∧; ∨; 0) (see [10, Proposition X.1]). Let f : L → G − be an order isomorphism from (L; 6) to (G − ; 6) such that (a) f(x · y) = f(x) + f(y), for all x; y ∈ L, (b) f(1) = 0. Let x; y ∈ L. Using the de3nition of operations → and → ˜ in L[G − ] from (a) and (b) we derive the following relations: f(x) + (f(x) → f(y)) = f(x) + (((−f(x)) + f(y)) ∧ 0) = f(x) + (((−f(x)) + f(y)) ∧ f(1)) = (f(x) + ((−f(x)) + f(y))) ∧ (f(x) + f(1)) = f(y) ∧ f(x · 1) = f(y ∧ x) = f(x ∧ y) = f(x · (x → y)) = f(x) + f(x → y): Thus in L[G − ] we have f(x) + (f(x) → f(y)) = f(x) + f(x → y)
for all x; y ∈ L;
but L[G − ] is cancellative, thus (c) f(x → y) = f(x) → f(y) for all x; y ∈ L. The following relation (d) f(x→y) ˜ = f(x) → ˜ f(y) for all x; y ∈ L follows in a similar way with (c) from (a) and (b) by a dual argument. Conditions (a)–(d) imply that f is an isomorphism from L to L[G − ]. Thus 4.12(i) holds. 5. Extended membership space of commutative residuated structures In order to 3nd an algebraic structure closely related to the structure of the real unit interval suggested by Zadeh [26] and Zimmermann [27] to represent fuzzy sets, an extended membership space of commutative residuated structures must be considered. In this section, we present some elementary properties of an extended membership space in connection with the structures of biresiduated
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algebra [24], L+-algebra [17] and double residuated lattice [21]. It is to show that the structure of residuated join semilattice monoid from R() presented in the precedent sections can be of interest also in the study of commutative fuzzy logic and fuzzy set theory. Suppose that K is a commutative linearly ordered :eld. For all x; y ∈ K, we denote by x + y the sum, by xy the product, by x − y the di8erence of the elements x and y in K and if x = 0, we denote by x−1 the inverse of x and by y=x = yx−1 the division of y by x in K. Denition 5.1. We denote by 6 the order relation on K. Let U (K) = {x ∈ K=06x61} be the unit interval of K. A graded membership chain associated with K is the algebra (U (K); ∧; ∨; →; ·; =; ⊕; ∗; 0; 1) with the carrier set U (K), where ∧; ∨; →; ·; =; ⊕ are binary operations and ∗ is an unary operation de3ned as follows: (i) x ∧ y = min{x; y} (binary meet ∧); (ii) x ∨ y = max{x; y} (binary join ∨); 1 if x6y (iii) x → y = = max{z ∈ U (K)=x ∧ z6y} (G?odel intuitionistic implication →); y if x¿y (iv) x · y = xy (algebraic product ·); 1 if x6y (v) y=x = y = max{z ∈ U (K)=x · z6y} (bounded division /); if x¿y x (vi) x ⊕ y = min{1; x + y} (bounded sum ⊕); (vii) x∗ = 1 − x (complementation ∗). Remark 5.2. On any graded membership chain there exists a new set of binary operations de3ned by: 0 if x6y (i) x − y = = min{z ∈ U (K)=x6y ∨ z} (dual G?odel intuitionistic implication −); x if x¿y (ii) x ⊕p y = x + y − xy (probabilistic sum ⊕p ); 0 if x6y (iii) x p y = x−y = min{z ∈ U (K)=x6y ⊕p z} (probabilistic bounded di8erence p ); if x¿y 1− y (iv) x y = max{0; x + y − 1} (Lukasiewicz conjunction ); (v) x C y = max{0; x − y} = min{z ∈ U (K)=x6y ⊕ z} (bounded di8erence C); (vi) x ⇒ y = min{1; 1 − x + y} = max{z ∈ U (K)=xz6y} (Lukasiewicz implication ⇒). Remark 5.3. The unary operation of complementation ∗ is a decreasing involution of the bounded poset (U (K); 6; 0; 1) such that the following equations hold: (2:1 ) x ∧ y = (x∗ ∨ y∗ )∗ , (2:1 ) x ∨ y = (x∗ ∧ y∗ )∗ , (2:2 ) x → y = (y∗ − x∗ )∗ , (2:2 ) x − y = (y∗ → x∗ )∗ , (2:3 ) x y = (x∗ ⊕ y∗ )∗ , (2:3 ) x ⊕ y = (x∗ y∗ )∗ , (2:4 ) x ⇒ y = (y∗ C x∗ )∗ , (2:4 ) x C y = (y∗ ⇒ x∗ )∗ ,
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(2:5 ) (2:5 ) (2:6 ) (2:6 ) (2:7)
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x · y = (x∗ ⊕p y∗ )∗ , x ⊕p y = (x∗ · y∗ )∗ , y=x = (y∗ Cp x∗ )∗ , x Cp y = y(x∗ =y∗ )∗ , x∗∗ = x.
Remark 5.4. The system (U (K); ∧; ∨; ∗; 0; 1) is a De Morgan algebra, i.e., (U (K); ∧; ∨; 0; 1) is a bounded distributive lattice verifying Eq. (2.1’). In addition U (K) is a Kleene algebra [5], i.e. the precedent De Morgan algebra satis3es also the following relation: (K) x ∧ x∗ 6y ∨ y∗ . Remark 5.5. From Eqs. (2.1)–(2.7) which we will call the De Morgan equations, it follows that any of the following six pairs of binary operations (∧; ∨); (→; −); ( ; ⊕); (⇒ ; C); (· ; ⊕p ) and (=; Cp ) have components which are dual one to another. Remark 5.6. The following conditions which are also satis3ed in U (K) show that any of the three pairs of operations (∧ ; →); ( ; ⇒) and (·; =) induces on U (K) a structure of commutative residuated l-monoid and any of the three pairs of operations (∨; −); (⊕ ; C) and (⊕p ; Cp ) induces on U (K) a structure of commutative dual residuated l-monoid (see [1,2]): (2:8 ) (2:8 ) (2:9 ) (2:9 ) (2:10 ) (2:10 )
a ∧ x6b i6 x6a → b, a6b ∨ x i6 a − b6x, a x6b i6 x 6 a ⇒ b, a6b ⊕ x i6 a C b6x, a · x6b i6 x6b=a, a6b ⊕p x i6 a Cp b6x.
The following lemmas present some basic properties of U (K) derived from the precedent relations in connection with the known lattice ordered algebraic structures of linear Heyting algebra [3], MValgebra [4,5] and BL-algebra [15]. Lemma 5.7. The algebra (U (K); ∧; ∨; →; 0; 1) is a linear Heyting algebra (see [3]), i.e. its underlying bounded lattice (U (K); ∧; ∨; 0; 1) satis:es condition (2:8 ) and the following equation: (LH) (x → y) ∨ (y → x) = 1. Lemma 5.8. The algebra (U (K); ⊕; ∗; 0) is an MV-algebra (see [5]), i.e. the following equations hold: (MV1) (MV2) (MV3) (MV4) (MV5) (MV6)
x ⊕ y = y ⊕ x, x ⊕ (y ⊕ z) = (x ⊕ y) ⊕ z, x ⊕ 0 = x, x∗∗ = x, x ⊕ 0∗ = 0∗ , (x∗ ⊕ y)∗ ⊕ y = (y∗ ⊕ x)∗ ⊕ x.
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Lemma 5.9. The algebra (U (K); ∧; ∨; ·; =; 0; 1) is a BL-algebra (see [15]), i.e., the following conditions are satis:ed, for all a; b; x; y ∈ U (K): (i) (ii) (iii) (iv) (v)
(U (K); ∧; ∨; 0; 1) is a bounded lattice; (U (K); ·; 1) is a commutative monoid with the unit element 1; · and = form an adjoint pair, i.e., condition (2:10 ) holds; x · (y=x) = x ∧ y; (x=y) ∨ (y=x) = 1.
We present in the following de3nition the notion of linear Heyting MV-algebra (LHMV-algebra) expressing some properties of basic models U (K) derived from De3nition 5.1, Lemmas 5.7 and 5.8. Then we suggest a manner to put together the structure of LHMV-algebra with the structure of BL-algebra. Denition 5.10. A linear Heyting MV-algebra (LHMV-algebra) is an algebra (A; ∨; →; ⊕; ∗; 0) with three binary operations ∨; →; ⊕, an unary operation ∗ and a constant 0 verifying the following equations: (LHMV1) x ∨ y = y ∨ x, (LHMV2) x → (y → z) = (x∗ ∨ y∗ )∗ → z, (LHMV3) (x → y) ∨ (y → x) = 0∗ , (LHMV4) x ⊕ y = y ⊕ x, (LHMV5) x ⊕ (y ⊕ z) = (x ⊕ y) ⊕ z, (LHMV6) x ⊕ 0 = x, (LHMV7) x∗∗ = x, (LHMV8) x ⊕ 0∗ = 0∗ , (LHMV9) (x∗ ⊕ y)∗ ⊕ y = x ∨ y, (LHMV10) a ⊕ (x → y) = (a ⊕ y) ∨ (x → y). Denition 5.11. An LHMV-chain is an LHMV-algebra (A; ∨; →; ⊕; ∗; 0) such that the system (A; 6) is a chain, where 6 is the binary relation on A such that for all x; y ∈ A, x6y
i6
x ∨ y = y:
For example, let K be a linearly-ordered commutative 3eld and U (K) be the unit interval of K together with the operations ∨; →; ⊕ and ∗ de3ned, respectively, by conditions (ii), (iii), (vi) and (vii) from De3nition 5.1. Then (U (K); ∨; →; ⊕; ∗; 0) is an LHMV-chain. The following de3nition introduces a structure including any graded membership chain: (U (K); ∧; ∨; →; ·; =; ⊕; ∗; 0; 1): This de3nition is based on the notions of BL-algebra and L+-algebra [17]. Denition 5.12. A graded membership algebra [25] is a system A = (A; ∧; ∨; →; ·; =; ⊕; ∗; 0; 1)
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such that the following conditions hold: (i) (A; ∨; →; ⊕; ∗; 0) is a LHMV-algebra; (ii) (A; ∧; ∨; ·; =; 0; 1) is a BL-algebra such that 0=x = x → 0, for all x ∈ L; (iii) (A; ·; =; ⊕; C; 0; 1) is a L+-algebra, where C is a binary operation de3ned by x C y = (x∗ ⊕ y)∗ , for all x; y ∈ L. The notions and results presented in this paper can be considered as a basic starting point to develop a theory of generalized sets over a WBL-algebra which is not necessarily a commutative monoid. The study of the structure of WBL-algebra can be also of interest in the elaboration of a comprehensive theory of commutative fuzzy logic based on a suitable notion of graded membership algebra in connection with biresiduated algebras [24], L+-algebras [17] and double residuated lattices [21]. A new interesting possible application can be in the aggregation and ranking of alternatives in multicriteria decision problems [7] formulated in the framework of generalized sets over residuated structures. Acknowledgements The authors would like to thank the referees for their pertinent remarks and suggestions which have helped to realize an improved expression of the work. Special thanks are dues to all the members of the Group of Logic and Universal Algebra from the University of Bucharest. References [1] G. Birkho6, Lattice Theory, 3rd Edition, American Mathematical Society Colloquium Publications, Providence, RI, 1967. [2] T.S. Blyth, M.F. Janowitz, Residuation Theory, Pergamon Press, Oxford, 1972. [3] V. Boicescu, A. Filipoiu, G. Georgescu, S. Rudeanu, Lukasiewicz–Moisil Algebras, North-Holland, Amsterdam, 1991. [4] C.C. Chang, Algebraic analysis of many-valued logics, Trans. Amer. Math. Soc. 88 (1958) 467–490. [5] R. Cignoli, I.M.L. D’Ottaviano, D. Mundici, Algebraic Foundations of Many-Valued Reasoning, Kluwer Academic Publishers, Dordrecht, 2000. [6] A. Di Nola, G. Georgescu, L. LeuVstean, Boolean products of BL-algebras, J. Math. Anal. Appl. 251 (2000) 106–131. [7] D. Dubois, H. Prade, Criteria aggregation and ranking of alternatives in the framework of fuzzy set theory, in: B.R. Gaines, L.A. Zadeh, H.-J. Zimmermann (Eds.), Fuzzy Sets and Decision Analysis, Studies in the Management Sciences, Vol. 20, North-Holland, Amsterdam, 1984, pp. 209–240. [8] F. Esteva, L. Godo, Monoidal t-norm based logic: towards a logic for left-continuous t-norms, Institut d’InvestigaciHo en IntelligHencia Arti3cial (IIIA) Research Report 2001– 02. [9] P. Flondor, G. Georgescu, A. Iorgulescu, Pseudo t-norms and pseudo BL-algebras, Soft Comput. 5 (2001) 355–371. [10] L. Fuchs, Partially Ordered Algebraic Systems, Pergamon Press, Oxford, 1963. [11] G. Georgescu, A. Iorgulescu, Pseudo MV-algebras: a noncommutative extension of MV-algebras, in: Proc. 4th Internat. Symp. on Economic Informatics, Inforec Printing House, Bucharest, May 1999, pp. 961–969. [12] G. Georgescu, A. Iorgulescu, Pseudo BL-algebras: a noncommutative extension of BL-algebras, in: Proc. 5th Internat. Conf. FSTA 2000 on Fuzzy Sets Theory and Its Applications, Lipt. JHan, Slovak Republic, 31 Jan – 4 Feb, 2000, pp. 90 –92. [13] G. Georgescu, L. LeuVstean, Pseudo-hoops, Multiple Valued Logic, submitted (Preprint June 2000).
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