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Residuation in lattice effect algebras
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Residuation in lattice effect algebras ✩ Ivan Chajda a , Helmut Länger b,a,∗ a Palacký University Olomouc, Faculty of Science, Department of Algebra and Geometry, 17. listopadu 12, 771 46 Olomouc, Czech Republic b TU Wien, Faculty of Mathematics and Geoinformation, Institute of Discrete Mathematics and Geometry, Wiedner Hauptstraße 8-10,
1040 Vienna, Austria Received 14 May 2019; received in revised form 29 October 2019; accepted 13 November 2019
Abstract We introduce the concept of a quasiresiduated lattice and prove that every lattice effect algebra can be organized into a commutative quasiresiduated lattice with divisibility. Also conversely, every such lattice can be converted into a lattice effect algebra and every lattice effect algebra can be reconstructed from its assigned quasiresiduated lattice. We apply this method also for lattice pseudoeffect algebras introduced by Dvureˇcenskij and Vetterlein. We show that every good lattice pseudoeffect algebra can be organized into a (possibly non-commutative) quasiresiduated lattice with divisibility and conversely, every such lattice can be converted into a lattice pseudoeffect algebra. Moreover, also a good lattice pseudoeffect algebra can be reconstructed from the assigned quasiresiduated lattice. © 2019 Elsevier B.V. All rights reserved. Keywords: Lattice effect algebra; Lattice pseudoeffect algebra; Quasiresiduated lattice; Quasiadjointness; Divisibility
1. Introduction In order to axiomatize quantum logic effects in a Hilbert space, Foulis and Bennett ([6]) introduced the so-called effect algebras. These are partial algebras with one partial binary operation which can be converted into bounded posets in general and into lattices in particular cases. It turns out that effect algebras form a successful axiomatization of the logic of quantum mechanics, but we suppose that there exists a connection with a kind of substructural logics whose algebraic semantics is based on residuated lattices, see e.g. [7]. Residuated lattices are used for modeling the structures of truth values of fuzzy logics, see [1]. As mentioned above, effect algebras form an algebraic axiomatization of the logic of quantum mechanics. Therefore, it is a natural question if the logic of quantum mechanics can be recognized as a kind of fuzzy logic. In order to answer this question, we try to find a certain modification of residuation for effect algebras. An attempt in this direction was already done in [2] where the so-called conditional residuation ✩ Support of the research by ÖAD, project CZ 02/2019, and support of the research of the first author by IGA, project PˇrF 2019 015, is gratefully acknowledged. * Corresponding author. E-mail addresses: [email protected] (I. Chajda), [email protected] (H. Länger).
https://doi.org/10.1016/j.fss.2019.11.008 0165-0114/© 2019 Elsevier B.V. All rights reserved.
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was introduced. A disadvantage of this approach is that the axioms of residuated structures are reflected only in the case when the terms used in adjointness are defined. This is an essential restriction which prevents the development of this theory. The aim of the present paper is to introduce the more general concept of quasiresiduation and to show that lattice effect algebras and lattice pseudoeffect algebras satisfy this concept. Pseudoeffect algebras were introduced by Dvureˇcenskij and Vetterlein ([5]). 2. Effect algebras We start with the following definition. Definition 2.1. An effect algebra is a partial algebra E = (E, +, , 0, 1) of type (2, 1, 0, 0) where (E, , 0, 1) is an algebra and + is a partial operation satisfying the following conditions for all x, y, z ∈ E: (E1) (E2) (E3) (E4)
x + y is defined if and only if so is y + x and in this case x + y = y + x, (x + y) + z is defined if and only if so is x + (y + z) and in this case (x + y) + z = x + (y + z), x is the unique u ∈ E with x + u = 1, if 1 + x is defined then x = 0.
On E a binary relation ≤ can be defined by x ≤ y if there exists some z ∈ E with x + z = y (x, y ∈ E). Then (E, ≤, 0, 1) is a bounded poset and ≤ is called the induced order of E. If (E, ≤) is a lattice then E is called a lattice effect algebra. In the sequel we will use the properties of effect algebras listed in the following lemma. Lemma 2.2. (see [4], [5]) If E = (E, +, , 0, 1) is an effect algebra, ≤ its induced order and a, b, c ∈ E then the following hold: (i) (ii) (iii) (iv) (v) (vi) (vii)
a = a, a ≤ b implies b ≤ a , a + b is defined if and only if a ≤ b , if a ≤ b and b + c is defined then a + c is defined and a + c ≤ b + c, if a ≤ b then a + (a + b ) = b, a + 0 = 0 + a = a, 0 = 1 and 1 = 0.
Let us introduce the following concept. A partial monoid is a partial algebra A = (A, , 1) of type (2, 0) where 1 ∈ A and is a partial operation satisfying the following conditions for all x, y, z ∈ A: (i) (x y) z is defined if and only if so is x (y z) and in this case (x y) z = x (y z), (ii) x 1 = 1 x = x. The partial monoid A is called commutative if it satisfies the following condition for all x, y ∈ A: (iii) x y is defined if and only if so is y x and in this case x y = y x, The authors already introduced a certain modification of residuation for sectionally pseudocomplemented lattices, see [3]. For lattice effect algebras, we introduce another version of residuation called quasiresiduation.
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Definition 2.3. A commutative quasiresiduated lattice is a partial algebra C = (C, ∨, ∧, , →, 0, 1) of type (2, 2, 2, 2, 0, 0) where (C, ∨, ∧, 0, 1) is a bounded lattice, is a partial and → a full operation satisfying the following conditions for all x, y, z ∈ C: (C1) (C, , 1) is a commutative partial monoid where x y is defined if and only if x ≤ y, (C2) x = x, and x ≤ y implies y ≤ x , (C3) (x ∨ y ) y ≤ y ∧ z if and only if x ∨ y ≤ y → z. Here x is an abbreviation for x → 0. The commutative quasiresiduated lattice C is called divisible if x ≤ y implies y (y → x) = x for all x, y ∈ C. Note that the terms in (C3) are everywhere defined. In case y ≤ x and z ≤ y condition (C3) reduces to x y ≤ z if and only if x ≤ y → z which is usual adjointness. Therefore condition (C3) will be called commutative quasiadjointness. Hence, contrary to the similar concept in [2], in commutative quasiadjointness we have only everywhere defined terms in C although C is a partial algebra. Our aim is to show that every lattice effect algebra can be organized into a commutative quasiresiduated lattice. Theorem 2.4. Let E = (E, +, , 0, 1) be a lattice effect algebra with lattice operations ∨ and ∧ and put x y := (x + y ) if and only if x ≤ y, x → y := (x ∧ y) + x (x, y ∈ E). Then C(E) := (E, ∨, ∧, , →, 0, 1) is a divisible commutative quasiresiduated lattice. Proof. Let a, b, c ∈ E. Obviously, (E, ∨, ∧, 0, 1) is a bounded lattice and (C1) and (C2) hold. If (a ∨ b ) b ≤ b ∧ c then ((a ∨ b ) + b ) ≤ b ∧ c and b ≤ a ∨ b and hence a ∨ b = b + ((a ∨ b ) + b ) ≤ b + (b ∧ c) = b → c. If, conversely, a ∨ b ≤ b → c then a ∨ b ≤ (b ∧ c) + b and b ≤ (b ∧ c) and hence (a ∨ b ) b = (b + (a ∨ b ) ) ≤ (b + ((b ∧ c) + b ) ) = (b ∧ c) = b ∧ c proving (C3). If a ≤ b then b ≤ a and hence b (b → a) = (b + (a + b ) ) = a = a proving divisibility.
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Example 2.5. Let E denote the 6-element set {0, a, a , b, b , 1} and define + and as follows: + 0 a a b b 1
0 0 a a b b 1
a a a a − 1 1 − − − − − − −
b b 1 b b 1 − − − − − − − 1 − 1 − − − − −
x 0 a a b b 1
x 1 a a b b 0
Then E := (E, +, , 0, 1) is a lattice effect algebra with lattice operations ∨ and ∧ and the Hasse diagram of (E, ≤) is depicted in Fig. 1.
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u Q A Q A Q A QQ Au b u u Qu b a Q a A Q Q A Q A Q QA u 0 Fig. 1. A lattice effect algebra.
Now C(E) = (E, ∨, ∧, , →, 0, 1) where and → are defined as follows is a divisible commutative quasiresiduated lattice: 0 a a b b 1
0 − − − − − 0
a a − − − 0 0 − − − − − a a
b − − − − 0 b
b − − − 0 − b
1 0 a a b b 1
→ 0 a a b b 1
0 1 a a b b 0
a 1 1 a b b a
a 1 a 1 b b a
b 1 a a 1 b b
b 1 a a b 1 b
1 1 1 1 1 1 1
Remark 2.6. Let us mention that Definition 2.3 can be modified in such a way that it contains only everywhere defined operations. Namely, if we put x ⊗ y := (x ∨ y ) y for all x, y ∈ C then ⊗ is everywhere defined and satisfies the identities x ⊗ 1 ≈ 1 ⊗ x ≈ x, and commutative quasiadjointness can then be expressed in the form x ⊗ y ≤ y ∧ z if and only if x ∨ y ≤ y → z. This means that our definition of commutative quasiresiduation differs from that of usual residuation only in the point that y occurs on the right-hand side of x ⊗ y ≤ y ∧ z and y on the left-hand side of x ∨ y ≤ y → z. On the other hand, using this version, divisibility cannot be easily defined. Moreover, since in lattice effect algebras we have y → z = (y ∧ z) + y = y → (y ∧ z), commutative quasiadjointness can be rewritten in the form (x ∨ y ) y ≤ y ∧ z if and only if x ∨ y ≤ y → (y ∧ z) which corresponds to usual adjointness if we abbreviate x ∨ y by X and y ∧ z by Z, i.e. X y ≤ Z if and only if X ≤ y → Z. We can prove also the converse. Theorem 2.7. Let C = (C, ∨, ∧, , →, 0, 1) be a commutative quasiresiduated lattice and put x + y := (x y ) if and only if x ≤ y , x := x → 0 (x, y ∈ C). Then E(C) := (C, +, , 0, 1) is a lattice effect algebra whose order coincides with that in C.
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Proof. Let a, b ∈ C. It is easy to see that (E1), (E2) and (E4) hold. Since 0 ∨ a = a ≤ a = a → 0 we have a a = a (0 ∨ a ) ≤ a ∧ 0 = 0, i.e. a a = 0. If, conversely, a b = 0 then a ≤ b and hence a (b ∨ a ) = a b = 0 ≤ a ∧ 0 whence b = b ∨ a ≤ a → 0 = a showing b = a . Hence a b = 0 if and only if b = a . Now the following are equivalent: a + b = 1, a b = 0, a = b , b = a. This shows (E3). Moreover, the following are equivalent: a ≤ b holds in E(C), a + b is defined, a b is defined, a ≤ b holds in C. Since (C, ∨, ∧) is a lattice and the partial order relations in C and E(C) coincide, E(C) is a lattice effect algebra. 2 If we consider the commutative quasiresiduated lattice C(E) from Example 2.5 then E(C(E)) = E. We show that every lattice effect algebra can be reconstructed from its assigned quasiresiduated lattice. Theorem 2.8. Let E be a lattice effect algebra. Then E(C(E)) = E. Proof. Let E = (E, +, , 0, 1) with lattice operations ∨ and ∧, C(E) = (E, ∨, ∧, , →, 0, 1), E(C(E)) = (E, ⊕, ∗ , 0, 1) and a, b ∈ E. Then a ∗ = a → 0 = (a ∧ 0) + a = 0 + a = a . Moreover, the following are equivalent: a ⊕ b is defined, a ≤ b in C(E), a ≤ b in E and in this case a ⊕ b = (a ∗ b∗ )∗ = (a b ) = (a + b ) = a + b.
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The following theorem characterizes those commutative quasiresiduated lattices that can be reconstructed from their assigned lattice effect algebras. Theorem 2.9. Let C = (C, ∨, ∧, , →, 0, 1) be a commutative quasiresiduated lattice. Then C(E(C)) = C if and only if C satisfies the identity ((x ∧ y) → 0) x ≈ (x → y) → 0. Proof. If E(C) = (C, +, , 0, 1), C(E(C)) = (C, ∨, ∧, ⊗, ⇒, 0, 1) and a, b ∈ C then a ⊗ b = (a + b ) = (a b ) = a b if a ≤ b, a ⇒ b = (a ∧ b) + a = ((a ∧ b) a ) = ((a ∧ b) a) .
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Remark 2.10. If C is a commutative quasiresiduated lattice that is not divisible then because of Theorem 2.4, C(E(C)) = C. 3. Pseudoeffect algebras Now we turn our attention to a more general case. The following concept was introduced by Dvureˇcenskij and Vetterlein ([5]). Definition 3.1. A pseudoeffect algebra is a partial algebra P = (P , +,− ,∼ , 0, 1) of type (2, 1, 1, 0, 0) where (P ,− ,∼ , 0, 1) is an algebra and + is a partial operation satisfying the following conditions for all x, y, z ∈ P : (P1) (P2) (P3) (P4)
If x + y is defined then there exist u, w ∈ P with u + x = y + w = x + y, (x + y) + z is defined if and only if x + (y + z) is defined, and in this case (x + y) + z = x + (y + z), x − is the unique u ∈ P with u + x = 1, and x ∼ is the unique w ∈ P with x + w = 1, if 1 + x or x + 1 is defined then x = 0.
The pseudoeffect algebra P is called good if (x − + y − )∼ = (x ∼ + y ∼ )− for all x, y ∈ P with x ∼ ≤ y. On P a binary relation ≤ can be defined by x ≤ y if there exists some z ∈ E with x + z = y (x, y ∈ P ). Then (P , ≤, 0, 1) is a bounded poset and ≤ is called the induced order of P. If (P , ≤) is a lattice then P is called a lattice pseudoeffect algebra. Example 3.2. (cf. [5]) If P := {(0, x, y) | x, y ∈ Z, x, y ≥ 0} ∪ {(1, x, y) | x, y ∈ Z, x, y ≤ 0} and (0, x, y) + (0, z, u) := (0, x + z, y + u), (0, x, y) + (1, v, w) := (1, y + v, x + w) if v ≤ −y and w ≤ −x, (1, v, w) + (0, x, y) := (1, v + x, w + y) if x ≤ −v and y ≤ −w, (0, x, y)− := (1, −x, −y), (1, v, w)− := (0, −w, −v),
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(0, x, y)∼ := (1, −y, −x), (1, v, w)∼ := (0, −v, −w) ((0, x, y), (0, z, u), (1, v, w) ∈ P ) then P := (P , +,− ,∼ , (0, 0, 0), (1, 0, 0)) is a good lattice pseudoeffect algebra and we have (x, y, z) ≤ (u, v, w) if and only if either x < u or (x = u and y ≤ v and z ≤ w) ((x, y, z), (u, v, w) ∈ P ). For our investigation we need the following results taken from [5]. Lemma 3.3. If P = (P , +,− ,∼ , 0, 1) is a pseudoeffect algebra, ≤ its induced order and a, b, c ∈ P then (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix)
a ∼− = a −∼ = a, the following are equivalent: a ≤ b, b− ≤ a − , b∼ ≤ a ∼ , a + b is defined if and only if a ≤ b− , if a ≤ b and b + c is defined then a + c is defined and a + c ≤ b + c, if a ≤ b and c + b is defined then c + a is defined and c + a ≤ c + b, if a ≤ b then a + (b− + a)∼ = (a + b∼ )− + a = b, a + 0 = 0 + a = a, 0− = 0∼ = 1 and 1− = 1∼ = 0, the following are equivalent: • a ≤ b, • there exists some d ∈ P with a + d = b, • there exists some e ∈ P with e + a = b.
Since pseudoeffect algebras are more general than effect algebras, we must define a quasiresiduated lattice for the case when the partial operation is not commutative and the mapping x → x − is not an involution. Definition 3.4. A quasiresiduated lattice is a partial algebra Q = (Q, ∨, ∧, , →, ;, 0, 1) of type (2, 2, 2, 2, 2, 0, 0) where (Q, ∨, ∧, 0, 1) is a bounded lattice, is a partial and → and ; are full operations satisfying the following conditions for all x, y, z ∈ Q: (Q1) (Q2) (Q3) (Q4) (Q5)
(Q, , 1) is a partial monoid where x y is defined if and only if x ∼ ≤ y, x −∼ = x ∼− = x, and x ≤ y implies y − ≤ x − and y ∼ ≤ x ∼ , (x ∨ y − ) y ≤ y ∧ z if and only if x ∨ y − ≤ y → z, y (x ∨ y ∼ ) ≤ y ∧ z if and only if x ∨ y ∼ ≤ y ; z, (x − y − )∼ = (x ∼ y ∼ )− .
Here x − and x ∼ are abbreviations for x → 0 and x ; 0, respectively. The quasiresiduated lattice Q is called divisible if x ≤ y implies (y → x) y = y (y ; x) = x for all x, y ∈ Q. Note that the terms in (Q3) and (Q4) are everywhere defined. In case y − ≤ x and z ≤ y condition (Q3) reduces to x y ≤ z if and only if x ≤ y → z which is usual adjointness. Analogously, in case y ∼ ≤ x and z ≤ y condition (Q4) reduces to y x ≤ z if and only if x ≤ y ; z which is usual adjointness if is commutative. Therefore conditions (Q3) and (Q4) will be called quasiadjointness. Hence, contrary to the similar concept in [2], in quasiadjointness we have only everywhere defined terms in Q although Q is a partial algebra.
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Similarly as for effect algebras, we prove that every good lattice pseudoeffect algebra can be organized into a quasiresiduated lattice which, however, need not be commutative. Theorem 3.5. Let P = (P , +,− ,∼ , 0, 1) be a good lattice pseudoeffect algebra with lattice operations ∨ and ∧ and put x y := (x − + y − )∼ if and only if x ∼ ≤ y, x → y := x − + (x ∧ y), x ; y := (x ∧ y) + x ∼ (x, y ∈ P ). Then Q(P) := (P , ∨, ∧, , →, ;, 0, 1) is a divisible quasiresiduated lattice. Proof. Let a, b, c ∈ P . Obviously, (P , ∨, ∧, 0, 1) is a bounded lattice and (Q1), (Q2) and (Q5) hold. If (a ∨ b− ) b ≤ b ∧ c then ((a ∨ b− )− + b− )∼ ≤ b ∧ c and b− ≤ a ∨ b− and hence a ∨ b− = b− + ((a ∨ b− )− + b− )∼ ≤ b− + (b ∧ c) = b → c. If, conversely, a ∨ b− ≤ b → c then a ∨ b− ≤ b− + (b ∧ c) and b− ≤ (b ∧ c)− and hence (a ∨ b− ) b = ((a ∨ b− )− + b− )∼ ≤ ((b− + (b ∧ c))− + b− )∼ = (b ∧ c)−∼ = b ∧ c proving (Q3). If b (a ∨ b∼ ) ≤ b ∧ c then (b∼ + (a ∨ b∼ )∼ )− ≤ b ∧ c and b∼ ≤ a ∨ b∼ and hence a ∨ b∼ = (b∼ + (a ∨ b∼ )∼ )− + b∼ ≤ (b ∧ c) + b∼ = b ; c. If, conversely, a ∨ b∼ ≤ b ; c then a ∨ b∼ ≤ (b ∧ c) + b∼ and b∼ ≤ (b ∧ c)∼ and hence b (a ∨ b∼ ) = (b∼ + (a ∨ b∼ )∼ )− ≤ (b∼ + ((b ∧ c) + b∼ )∼ )− = (b ∧ c)∼− = b ∧ c proving (Q4). If a ≤ b then b− ≤ a − and b∼ ≤ a ∼ and hence by Lemma 3.3 (vi) (b → a) b = ((b− + a)− + b− )∼ = a −∼ = a, b (b ; a) = (b∼ + (a + b∼ )∼ )− = a ∼− = a proving divisibility.
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Example 3.6. If P denotes the good lattice pseudoeffect algebra from Example 3.2 and ∨ and ∧ denote its lattice operations then Q(P) = (P , ∨, ∧, , →, ;, (0, 0, 0), (1, 0, 0)) where , → and ; are defined as follows: (0, x, y) (1, v, w) := (0, x + w, y + v) if x ≥ −w and y ≥ −v, (1, v, w) (0, x, y) := (0, v + x, w + y) if x ≥ −v and y ≥ −w, (1, v, w) (1, r, s) := (1, v + r, w + s), (0, x, y) → (0, z, u) := (1, min(z − x, 0), min(u − y, 0)), (0, x, y) → (1, v, w) := (1, 0, 0), (1, v, w) → (0, x, y) := (0, x − w, y − v), (1, v, w) → (1, r, s) := (1, min(r − v, 0), min(s − w, 0)), (0, x, y) ; (0, z, u) := (1, min(u − y, 0), min(z − x, 0)), (0, x, y) ; (1, v, w) := (1, 0, 0), (1, v, w) ; (0, x, y) := (0, x − v, y − w), (1, v, w) ; (1, r, s) := (1, min(r − v, 0), min(s − w, 0)) ((0, x, y), (0, z, u), (1, v, w), (1, r, s) ∈ P ). We can prove also the converse of Theorem 3.5.
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Theorem 3.7. Let Q = (Q, ∨, ∧, , →, ;, 0, 1) be a quasiresiduated lattice and put x + y := (x − y − )∼ if and only if x ≤ y − , x − := x → 0, x ∼ := x ; 0 (x, y ∈ Q). Then P (Q) := (Q, +,− ,∼ , 0, 1) is a good lattice pseudoeffect algebra whose order coincides with that in Q. Proof. Let a, b, c ∈ Q. It is easy to see that (P2) and (P4) hold. Since 0 ∨ a∼ = a∼ ≤ a∼ = a ; 0 we have by (Q4) a a ∼ = a (0 ∨ a ∼ ) ≤ a ∧ 0 = 0, i.e. a a ∼ = 0. If, conversely, a b = 0 then a ∼ ≤ b and hence a (b ∨ a ∼ ) = a b = 0 ≤ a ∧ 0 whence b = b ∨ a∼ ≤ a ; 0 = a∼ showing b = a ∼ . Hence a b = 0 if and only if b = a ∼ . Since 0 ∨ a− = a− ≤ a− = a → 0 we have by (Q3) a − a = (0 ∨ a − ) a ≤ a ∧ 0 = 0, i.e. a − a = 0. If, conversely, b a = 0 then b∼ ≤ a, i.e. a − ≤ b, and hence (b ∨ a − ) a = b a = 0 ≤ a ∧ 0 whence b = b ∨ a− ≤ a → 0 = a− showing b = a − . Hence b a = 0 if and only if b = a − . Now the following are equivalent: a + b = 1, −
a b− = 0, a = b− , b = a∼. This shows (P3). Since a (1 ∨ a ∼ ) = a 1 = a ≤ a ∧ a we have 1 = 1 ∨ a ∼ ≤ a ; a, i.e. a ; a = 1. If a ≤ b− then because of b− ∨ a ≤ a − ; a − we have a − b− = a − (b− ∨ a) ≤ a − ∧ a − = a − whence a = a −∼ ≤ (a − b− )∼ = a + b
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showing that a + (a + b)∼ is defined. Now in case a ≤ b− the following are equivalent: c = (a + (a + b)∼ )− , c + (a + (a + b)∼ ) = 1, (c + a) + (a + b)∼ = 1, c + a = a + b. Since (1 ∨ a − ) a = 1 a = a ≤ a ∧ a we have 1 = 1 ∨ a − ≤ a → a, i.e. a → a = 1. If b ≤ a ∼ then because of a ∼ ∨ b ≤ b∼ → b∼ we have a ∼ b∼ = (a ∼ ∨ b) b∼ ≤ b∼ ∧ b∼ = b∼ whence b = b∼− ≤ (a ∼ b∼ )− = (a − b− )∼ = a + b showing that (a + b)− + b is defined. Now in case b ≤ a ∼ , i.e. a ≤ b− the following are equivalent: c = ((a + b)− + b)∼ , ((a + b)− + b) + c = 1, (a + b)− + (b + c) = 1, b + c = a + b. This shows (P1). Now the following are equivalent: a ≤ b holds in P (Q), a + b∼ is defined, a − b is defined, a ≤ b holds in Q. Since (Q, ∨, ∧) is a lattice and the partial order relations in Q and P (Q) coincide, P (Q) is a good lattice pseudoeffect algebra. 2 As in the case of effect algebras, also every good lattice pseudoeffect algebra can be reconstructed from its assigned quasiresiduated lattice. Theorem 3.8. Let P be a good lattice pseudoeffect algebra. Then P (Q(P)) = P. Proof. Let P = (P , +,− ,∼ , 0, 1) with lattice operations ∨ and ∧, Q(P) = (P , ∨, ∧, , →, ;, 0, 1), P (Q(P)) = (P , ⊕, ∗ , + , 0, 1) and a, b ∈ P . Then a ∗ = a → 0 = a − + (a ∧ 0) = a − + 0 = a − , a + = a ; 0 = (a ∧ 0) + a ∼ = 0 + a ∼ = a ∼ .
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Moreover, the following are equivalent: a ⊕ b is defined, a ≤ b− in Q(P), a ≤ b− in P and in this case a ⊕ b = (a ∗ b∗ )+ = (a − b− )∼ = (a ∼ b∼ )− = a + b. 2 The following theorem characterizes those quasiresiduated lattices that can be reconstructed from their assigned good lattice pseudoeffect algebras. Theorem 3.9. Let Q = (Q, ∨, ∧, , →, ;, 0, 1) be a quasiresiduated lattice. Then Q(P (Q)) = Q if and only if Q satisfies the identities x ((x ∧ y) ; 0) ≈ (x → y) ; 0, ((x ∧ y) → 0) x ≈ (x ; y) → 0. Proof. If P (Q) = (Q, +,− ,∼ , 0, 1), Q(P (Q)) = (Q, ∨, ∧, ⊗, ⇒, , 0, 1) and a, b ∈ C then a ⊗ b = (a − + b− )∼ = (a ∼ + b∼ )− = (a ∼− b∼− )∼− = a b if a ∼ ≤ b, a ⇒ b = a − + (a ∧ b) = (a −− (a ∧ b)− )∼ = (a −∼ (a ∧ b)∼ )− = (a (a ∧ b)∼ )− , a b = (a ∧ b) + a ∼ = ((a ∧ b)− a ∼− )∼ = ((a ∧ b)− a)∼ .
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Remark 3.10. If Q is a quasiresiduated lattice that is not divisible then because of Theorem 3.5, Q(P (Q)) = Q. 4. Conclusion Although lattice effect algebras cannot be expressed as residuated structures, we found out that they can be represented as so-called commutative quasiresiduated lattices. Similarly, we have shown that lattice pseudoeffect algebras can be represented by means of (not necessarily commutative) quasiresiduated lattices. Hence, a certain connection between lattice (pseudo-)effect algebras and fuzzy logics in a broad sense exists. The next interesting task would be to develop this kind of fuzzy logic whose structure of truth values is a quasiresiduated lattice. This may be investigated in future. Acknowledgement The authors are very grateful to the anonymous referees whose valuable suggestions helped to increase the quality of the paper. References [1] [2] [3] [4] [5] [6] [7]
R. Bˇelohlávek, Fuzzy Relational Systems. Foundations and Principles, Springer, New York, ISBN 978-1-4613-5168-9, 2002. I. Chajda, R. Halaš, Effect algebras are conditionally residuated structures, Soft Comput. 15 (2011) 1383–1387. I. Chajda, J. Kühr, H. Länger, Relatively residuated lattices and posets, Math. Slovaca (2020), in press, http://arxiv.org/abs/1901.06664. A. Dvureˇcenskij, S. Pulmannová, New Trends in Quantum Structures, Kluwer, Dordrecht, ISBN 0-7923-6471-6, 2000. A. Dvureˇcenskij, T. Vetterlein, Pseudoeffect algebras. I. Basic properties, Int. J. Theor. Phys. 40 (2001) 685–701. D.J. Foulis, M.K. Bennett, Effect algebras and unsharp quantum logics, Found. Phys. 24 (1994) 1331–1352. N. Galatos, P. Jipsen, T. Kowalski, H. Ono, Residuated Lattices: An Algebraic Glimpse at Substructural Logics, Elsevier, Amsterdam, ISBN 978-0-444-52141-5, 2007.
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Available online at www.sciencedirect.com
ScienceDirect Fuzzy Sets and Systems ••• (••••) •••–••• www.elsevier.com/locate/fss
Residuation in lattice effect algebras ✩ Ivan Chajda a , Helmut Länger b,a,∗ a Palacký University Olomouc, Faculty of Science, Department of Algebra and Geometry, 17. listopadu 12, 771 46 Olomouc, Czech Republic b TU Wien, Faculty of Mathematics and Geoinformation, Institute of Discrete Mathematics and Geometry, Wiedner Hauptstraße 8-10,
1040 Vienna, Austria Received 14 May 2019; received in revised form 29 October 2019; accepted 13 November 2019
Abstract We introduce the concept of a quasiresiduated lattice and prove that every lattice effect algebra can be organized into a commutative quasiresiduated lattice with divisibility. Also conversely, every such lattice can be converted into a lattice effect algebra and every lattice effect algebra can be reconstructed from its assigned quasiresiduated lattice. We apply this method also for lattice pseudoeffect algebras introduced by Dvureˇcenskij and Vetterlein. We show that every good lattice pseudoeffect algebra can be organized into a (possibly non-commutative) quasiresiduated lattice with divisibility and conversely, every such lattice can be converted into a lattice pseudoeffect algebra. Moreover, also a good lattice pseudoeffect algebra can be reconstructed from the assigned quasiresiduated lattice. © 2019 Elsevier B.V. All rights reserved. Keywords: Lattice effect algebra; Lattice pseudoeffect algebra; Quasiresiduated lattice; Quasiadjointness; Divisibility
1. Introduction In order to axiomatize quantum logic effects in a Hilbert space, Foulis and Bennett ([6]) introduced the so-called effect algebras. These are partial algebras with one partial binary operation which can be converted into bounded posets in general and into lattices in particular cases. It turns out that effect algebras form a successful axiomatization of the logic of quantum mechanics, but we suppose that there exists a connection with a kind of substructural logics whose algebraic semantics is based on residuated lattices, see e.g. [7]. Residuated lattices are used for modeling the structures of truth values of fuzzy logics, see [1]. As mentioned above, effect algebras form an algebraic axiomatization of the logic of quantum mechanics. Therefore, it is a natural question if the logic of quantum mechanics can be recognized as a kind of fuzzy logic. In order to answer this question, we try to find a certain modification of residuation for effect algebras. An attempt in this direction was already done in [2] where the so-called conditional residuation ✩ Support of the research by ÖAD, project CZ 02/2019, and support of the research of the first author by IGA, project PˇrF 2019 015, is gratefully acknowledged. * Corresponding author. E-mail addresses: [email protected] (I. Chajda), [email protected] (H. Länger).
https://doi.org/10.1016/j.fss.2019.11.008 0165-0114/© 2019 Elsevier B.V. All rights reserved.
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was introduced. A disadvantage of this approach is that the axioms of residuated structures are reflected only in the case when the terms used in adjointness are defined. This is an essential restriction which prevents the development of this theory. The aim of the present paper is to introduce the more general concept of quasiresiduation and to show that lattice effect algebras and lattice pseudoeffect algebras satisfy this concept. Pseudoeffect algebras were introduced by Dvureˇcenskij and Vetterlein ([5]). 2. Effect algebras We start with the following definition. Definition 2.1. An effect algebra is a partial algebra E = (E, +, , 0, 1) of type (2, 1, 0, 0) where (E, , 0, 1) is an algebra and + is a partial operation satisfying the following conditions for all x, y, z ∈ E: (E1) (E2) (E3) (E4)
x + y is defined if and only if so is y + x and in this case x + y = y + x, (x + y) + z is defined if and only if so is x + (y + z) and in this case (x + y) + z = x + (y + z), x is the unique u ∈ E with x + u = 1, if 1 + x is defined then x = 0.
On E a binary relation ≤ can be defined by x ≤ y if there exists some z ∈ E with x + z = y (x, y ∈ E). Then (E, ≤, 0, 1) is a bounded poset and ≤ is called the induced order of E. If (E, ≤) is a lattice then E is called a lattice effect algebra. In the sequel we will use the properties of effect algebras listed in the following lemma. Lemma 2.2. (see [4], [5]) If E = (E, +, , 0, 1) is an effect algebra, ≤ its induced order and a, b, c ∈ E then the following hold: (i) (ii) (iii) (iv) (v) (vi) (vii)
a = a, a ≤ b implies b ≤ a , a + b is defined if and only if a ≤ b , if a ≤ b and b + c is defined then a + c is defined and a + c ≤ b + c, if a ≤ b then a + (a + b ) = b, a + 0 = 0 + a = a, 0 = 1 and 1 = 0.
Let us introduce the following concept. A partial monoid is a partial algebra A = (A, , 1) of type (2, 0) where 1 ∈ A and is a partial operation satisfying the following conditions for all x, y, z ∈ A: (i) (x y) z is defined if and only if so is x (y z) and in this case (x y) z = x (y z), (ii) x 1 = 1 x = x. The partial monoid A is called commutative if it satisfies the following condition for all x, y ∈ A: (iii) x y is defined if and only if so is y x and in this case x y = y x, The authors already introduced a certain modification of residuation for sectionally pseudocomplemented lattices, see [3]. For lattice effect algebras, we introduce another version of residuation called quasiresiduation.
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Definition 2.3. A commutative quasiresiduated lattice is a partial algebra C = (C, ∨, ∧, , →, 0, 1) of type (2, 2, 2, 2, 0, 0) where (C, ∨, ∧, 0, 1) is a bounded lattice, is a partial and → a full operation satisfying the following conditions for all x, y, z ∈ C: (C1) (C, , 1) is a commutative partial monoid where x y is defined if and only if x ≤ y, (C2) x = x, and x ≤ y implies y ≤ x , (C3) (x ∨ y ) y ≤ y ∧ z if and only if x ∨ y ≤ y → z. Here x is an abbreviation for x → 0. The commutative quasiresiduated lattice C is called divisible if x ≤ y implies y (y → x) = x for all x, y ∈ C. Note that the terms in (C3) are everywhere defined. In case y ≤ x and z ≤ y condition (C3) reduces to x y ≤ z if and only if x ≤ y → z which is usual adjointness. Therefore condition (C3) will be called commutative quasiadjointness. Hence, contrary to the similar concept in [2], in commutative quasiadjointness we have only everywhere defined terms in C although C is a partial algebra. Our aim is to show that every lattice effect algebra can be organized into a commutative quasiresiduated lattice. Theorem 2.4. Let E = (E, +, , 0, 1) be a lattice effect algebra with lattice operations ∨ and ∧ and put x y := (x + y ) if and only if x ≤ y, x → y := (x ∧ y) + x (x, y ∈ E). Then C(E) := (E, ∨, ∧, , →, 0, 1) is a divisible commutative quasiresiduated lattice. Proof. Let a, b, c ∈ E. Obviously, (E, ∨, ∧, 0, 1) is a bounded lattice and (C1) and (C2) hold. If (a ∨ b ) b ≤ b ∧ c then ((a ∨ b ) + b ) ≤ b ∧ c and b ≤ a ∨ b and hence a ∨ b = b + ((a ∨ b ) + b ) ≤ b + (b ∧ c) = b → c. If, conversely, a ∨ b ≤ b → c then a ∨ b ≤ (b ∧ c) + b and b ≤ (b ∧ c) and hence (a ∨ b ) b = (b + (a ∨ b ) ) ≤ (b + ((b ∧ c) + b ) ) = (b ∧ c) = b ∧ c proving (C3). If a ≤ b then b ≤ a and hence b (b → a) = (b + (a + b ) ) = a = a proving divisibility.
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Example 2.5. Let E denote the 6-element set {0, a, a , b, b , 1} and define + and as follows: + 0 a a b b 1
0 0 a a b b 1
a a a a − 1 1 − − − − − − −
b b 1 b b 1 − − − − − − − 1 − 1 − − − − −
x 0 a a b b 1
x 1 a a b b 0
Then E := (E, +, , 0, 1) is a lattice effect algebra with lattice operations ∨ and ∧ and the Hasse diagram of (E, ≤) is depicted in Fig. 1.
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u Q A Q A Q A QQ Au b u u Qu b a Q a A Q Q A Q A Q QA u 0 Fig. 1. A lattice effect algebra.
Now C(E) = (E, ∨, ∧, , →, 0, 1) where and → are defined as follows is a divisible commutative quasiresiduated lattice: 0 a a b b 1
0 − − − − − 0
a a − − − 0 0 − − − − − a a
b − − − − 0 b
b − − − 0 − b
1 0 a a b b 1
→ 0 a a b b 1
0 1 a a b b 0
a 1 1 a b b a
a 1 a 1 b b a
b 1 a a 1 b b
b 1 a a b 1 b
1 1 1 1 1 1 1
Remark 2.6. Let us mention that Definition 2.3 can be modified in such a way that it contains only everywhere defined operations. Namely, if we put x ⊗ y := (x ∨ y ) y for all x, y ∈ C then ⊗ is everywhere defined and satisfies the identities x ⊗ 1 ≈ 1 ⊗ x ≈ x, and commutative quasiadjointness can then be expressed in the form x ⊗ y ≤ y ∧ z if and only if x ∨ y ≤ y → z. This means that our definition of commutative quasiresiduation differs from that of usual residuation only in the point that y occurs on the right-hand side of x ⊗ y ≤ y ∧ z and y on the left-hand side of x ∨ y ≤ y → z. On the other hand, using this version, divisibility cannot be easily defined. Moreover, since in lattice effect algebras we have y → z = (y ∧ z) + y = y → (y ∧ z), commutative quasiadjointness can be rewritten in the form (x ∨ y ) y ≤ y ∧ z if and only if x ∨ y ≤ y → (y ∧ z) which corresponds to usual adjointness if we abbreviate x ∨ y by X and y ∧ z by Z, i.e. X y ≤ Z if and only if X ≤ y → Z. We can prove also the converse. Theorem 2.7. Let C = (C, ∨, ∧, , →, 0, 1) be a commutative quasiresiduated lattice and put x + y := (x y ) if and only if x ≤ y , x := x → 0 (x, y ∈ C). Then E(C) := (C, +, , 0, 1) is a lattice effect algebra whose order coincides with that in C.
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Proof. Let a, b ∈ C. It is easy to see that (E1), (E2) and (E4) hold. Since 0 ∨ a = a ≤ a = a → 0 we have a a = a (0 ∨ a ) ≤ a ∧ 0 = 0, i.e. a a = 0. If, conversely, a b = 0 then a ≤ b and hence a (b ∨ a ) = a b = 0 ≤ a ∧ 0 whence b = b ∨ a ≤ a → 0 = a showing b = a . Hence a b = 0 if and only if b = a . Now the following are equivalent: a + b = 1, a b = 0, a = b , b = a. This shows (E3). Moreover, the following are equivalent: a ≤ b holds in E(C), a + b is defined, a b is defined, a ≤ b holds in C. Since (C, ∨, ∧) is a lattice and the partial order relations in C and E(C) coincide, E(C) is a lattice effect algebra. 2 If we consider the commutative quasiresiduated lattice C(E) from Example 2.5 then E(C(E)) = E. We show that every lattice effect algebra can be reconstructed from its assigned quasiresiduated lattice. Theorem 2.8. Let E be a lattice effect algebra. Then E(C(E)) = E. Proof. Let E = (E, +, , 0, 1) with lattice operations ∨ and ∧, C(E) = (E, ∨, ∧, , →, 0, 1), E(C(E)) = (E, ⊕, ∗ , 0, 1) and a, b ∈ E. Then a ∗ = a → 0 = (a ∧ 0) + a = 0 + a = a . Moreover, the following are equivalent: a ⊕ b is defined, a ≤ b in C(E), a ≤ b in E and in this case a ⊕ b = (a ∗ b∗ )∗ = (a b ) = (a + b ) = a + b.
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The following theorem characterizes those commutative quasiresiduated lattices that can be reconstructed from their assigned lattice effect algebras. Theorem 2.9. Let C = (C, ∨, ∧, , →, 0, 1) be a commutative quasiresiduated lattice. Then C(E(C)) = C if and only if C satisfies the identity ((x ∧ y) → 0) x ≈ (x → y) → 0. Proof. If E(C) = (C, +, , 0, 1), C(E(C)) = (C, ∨, ∧, ⊗, ⇒, 0, 1) and a, b ∈ C then a ⊗ b = (a + b ) = (a b ) = a b if a ≤ b, a ⇒ b = (a ∧ b) + a = ((a ∧ b) a ) = ((a ∧ b) a) .
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Remark 2.10. If C is a commutative quasiresiduated lattice that is not divisible then because of Theorem 2.4, C(E(C)) = C. 3. Pseudoeffect algebras Now we turn our attention to a more general case. The following concept was introduced by Dvureˇcenskij and Vetterlein ([5]). Definition 3.1. A pseudoeffect algebra is a partial algebra P = (P , +,− ,∼ , 0, 1) of type (2, 1, 1, 0, 0) where (P ,− ,∼ , 0, 1) is an algebra and + is a partial operation satisfying the following conditions for all x, y, z ∈ P : (P1) (P2) (P3) (P4)
If x + y is defined then there exist u, w ∈ P with u + x = y + w = x + y, (x + y) + z is defined if and only if x + (y + z) is defined, and in this case (x + y) + z = x + (y + z), x − is the unique u ∈ P with u + x = 1, and x ∼ is the unique w ∈ P with x + w = 1, if 1 + x or x + 1 is defined then x = 0.
The pseudoeffect algebra P is called good if (x − + y − )∼ = (x ∼ + y ∼ )− for all x, y ∈ P with x ∼ ≤ y. On P a binary relation ≤ can be defined by x ≤ y if there exists some z ∈ E with x + z = y (x, y ∈ P ). Then (P , ≤, 0, 1) is a bounded poset and ≤ is called the induced order of P. If (P , ≤) is a lattice then P is called a lattice pseudoeffect algebra. Example 3.2. (cf. [5]) If P := {(0, x, y) | x, y ∈ Z, x, y ≥ 0} ∪ {(1, x, y) | x, y ∈ Z, x, y ≤ 0} and (0, x, y) + (0, z, u) := (0, x + z, y + u), (0, x, y) + (1, v, w) := (1, y + v, x + w) if v ≤ −y and w ≤ −x, (1, v, w) + (0, x, y) := (1, v + x, w + y) if x ≤ −v and y ≤ −w, (0, x, y)− := (1, −x, −y), (1, v, w)− := (0, −w, −v),
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(0, x, y)∼ := (1, −y, −x), (1, v, w)∼ := (0, −v, −w) ((0, x, y), (0, z, u), (1, v, w) ∈ P ) then P := (P , +,− ,∼ , (0, 0, 0), (1, 0, 0)) is a good lattice pseudoeffect algebra and we have (x, y, z) ≤ (u, v, w) if and only if either x < u or (x = u and y ≤ v and z ≤ w) ((x, y, z), (u, v, w) ∈ P ). For our investigation we need the following results taken from [5]. Lemma 3.3. If P = (P , +,− ,∼ , 0, 1) is a pseudoeffect algebra, ≤ its induced order and a, b, c ∈ P then (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix)
a ∼− = a −∼ = a, the following are equivalent: a ≤ b, b− ≤ a − , b∼ ≤ a ∼ , a + b is defined if and only if a ≤ b− , if a ≤ b and b + c is defined then a + c is defined and a + c ≤ b + c, if a ≤ b and c + b is defined then c + a is defined and c + a ≤ c + b, if a ≤ b then a + (b− + a)∼ = (a + b∼ )− + a = b, a + 0 = 0 + a = a, 0− = 0∼ = 1 and 1− = 1∼ = 0, the following are equivalent: • a ≤ b, • there exists some d ∈ P with a + d = b, • there exists some e ∈ P with e + a = b.
Since pseudoeffect algebras are more general than effect algebras, we must define a quasiresiduated lattice for the case when the partial operation is not commutative and the mapping x → x − is not an involution. Definition 3.4. A quasiresiduated lattice is a partial algebra Q = (Q, ∨, ∧, , →, ;, 0, 1) of type (2, 2, 2, 2, 2, 0, 0) where (Q, ∨, ∧, 0, 1) is a bounded lattice, is a partial and → and ; are full operations satisfying the following conditions for all x, y, z ∈ Q: (Q1) (Q2) (Q3) (Q4) (Q5)
(Q, , 1) is a partial monoid where x y is defined if and only if x ∼ ≤ y, x −∼ = x ∼− = x, and x ≤ y implies y − ≤ x − and y ∼ ≤ x ∼ , (x ∨ y − ) y ≤ y ∧ z if and only if x ∨ y − ≤ y → z, y (x ∨ y ∼ ) ≤ y ∧ z if and only if x ∨ y ∼ ≤ y ; z, (x − y − )∼ = (x ∼ y ∼ )− .
Here x − and x ∼ are abbreviations for x → 0 and x ; 0, respectively. The quasiresiduated lattice Q is called divisible if x ≤ y implies (y → x) y = y (y ; x) = x for all x, y ∈ Q. Note that the terms in (Q3) and (Q4) are everywhere defined. In case y − ≤ x and z ≤ y condition (Q3) reduces to x y ≤ z if and only if x ≤ y → z which is usual adjointness. Analogously, in case y ∼ ≤ x and z ≤ y condition (Q4) reduces to y x ≤ z if and only if x ≤ y ; z which is usual adjointness if is commutative. Therefore conditions (Q3) and (Q4) will be called quasiadjointness. Hence, contrary to the similar concept in [2], in quasiadjointness we have only everywhere defined terms in Q although Q is a partial algebra.
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Similarly as for effect algebras, we prove that every good lattice pseudoeffect algebra can be organized into a quasiresiduated lattice which, however, need not be commutative. Theorem 3.5. Let P = (P , +,− ,∼ , 0, 1) be a good lattice pseudoeffect algebra with lattice operations ∨ and ∧ and put x y := (x − + y − )∼ if and only if x ∼ ≤ y, x → y := x − + (x ∧ y), x ; y := (x ∧ y) + x ∼ (x, y ∈ P ). Then Q(P) := (P , ∨, ∧, , →, ;, 0, 1) is a divisible quasiresiduated lattice. Proof. Let a, b, c ∈ P . Obviously, (P , ∨, ∧, 0, 1) is a bounded lattice and (Q1), (Q2) and (Q5) hold. If (a ∨ b− ) b ≤ b ∧ c then ((a ∨ b− )− + b− )∼ ≤ b ∧ c and b− ≤ a ∨ b− and hence a ∨ b− = b− + ((a ∨ b− )− + b− )∼ ≤ b− + (b ∧ c) = b → c. If, conversely, a ∨ b− ≤ b → c then a ∨ b− ≤ b− + (b ∧ c) and b− ≤ (b ∧ c)− and hence (a ∨ b− ) b = ((a ∨ b− )− + b− )∼ ≤ ((b− + (b ∧ c))− + b− )∼ = (b ∧ c)−∼ = b ∧ c proving (Q3). If b (a ∨ b∼ ) ≤ b ∧ c then (b∼ + (a ∨ b∼ )∼ )− ≤ b ∧ c and b∼ ≤ a ∨ b∼ and hence a ∨ b∼ = (b∼ + (a ∨ b∼ )∼ )− + b∼ ≤ (b ∧ c) + b∼ = b ; c. If, conversely, a ∨ b∼ ≤ b ; c then a ∨ b∼ ≤ (b ∧ c) + b∼ and b∼ ≤ (b ∧ c)∼ and hence b (a ∨ b∼ ) = (b∼ + (a ∨ b∼ )∼ )− ≤ (b∼ + ((b ∧ c) + b∼ )∼ )− = (b ∧ c)∼− = b ∧ c proving (Q4). If a ≤ b then b− ≤ a − and b∼ ≤ a ∼ and hence by Lemma 3.3 (vi) (b → a) b = ((b− + a)− + b− )∼ = a −∼ = a, b (b ; a) = (b∼ + (a + b∼ )∼ )− = a ∼− = a proving divisibility.
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Example 3.6. If P denotes the good lattice pseudoeffect algebra from Example 3.2 and ∨ and ∧ denote its lattice operations then Q(P) = (P , ∨, ∧, , →, ;, (0, 0, 0), (1, 0, 0)) where , → and ; are defined as follows: (0, x, y) (1, v, w) := (0, x + w, y + v) if x ≥ −w and y ≥ −v, (1, v, w) (0, x, y) := (0, v + x, w + y) if x ≥ −v and y ≥ −w, (1, v, w) (1, r, s) := (1, v + r, w + s), (0, x, y) → (0, z, u) := (1, min(z − x, 0), min(u − y, 0)), (0, x, y) → (1, v, w) := (1, 0, 0), (1, v, w) → (0, x, y) := (0, x − w, y − v), (1, v, w) → (1, r, s) := (1, min(r − v, 0), min(s − w, 0)), (0, x, y) ; (0, z, u) := (1, min(u − y, 0), min(z − x, 0)), (0, x, y) ; (1, v, w) := (1, 0, 0), (1, v, w) ; (0, x, y) := (0, x − v, y − w), (1, v, w) ; (1, r, s) := (1, min(r − v, 0), min(s − w, 0)) ((0, x, y), (0, z, u), (1, v, w), (1, r, s) ∈ P ). We can prove also the converse of Theorem 3.5.
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Theorem 3.7. Let Q = (Q, ∨, ∧, , →, ;, 0, 1) be a quasiresiduated lattice and put x + y := (x − y − )∼ if and only if x ≤ y − , x − := x → 0, x ∼ := x ; 0 (x, y ∈ Q). Then P (Q) := (Q, +,− ,∼ , 0, 1) is a good lattice pseudoeffect algebra whose order coincides with that in Q. Proof. Let a, b, c ∈ Q. It is easy to see that (P2) and (P4) hold. Since 0 ∨ a∼ = a∼ ≤ a∼ = a ; 0 we have by (Q4) a a ∼ = a (0 ∨ a ∼ ) ≤ a ∧ 0 = 0, i.e. a a ∼ = 0. If, conversely, a b = 0 then a ∼ ≤ b and hence a (b ∨ a ∼ ) = a b = 0 ≤ a ∧ 0 whence b = b ∨ a∼ ≤ a ; 0 = a∼ showing b = a ∼ . Hence a b = 0 if and only if b = a ∼ . Since 0 ∨ a− = a− ≤ a− = a → 0 we have by (Q3) a − a = (0 ∨ a − ) a ≤ a ∧ 0 = 0, i.e. a − a = 0. If, conversely, b a = 0 then b∼ ≤ a, i.e. a − ≤ b, and hence (b ∨ a − ) a = b a = 0 ≤ a ∧ 0 whence b = b ∨ a− ≤ a → 0 = a− showing b = a − . Hence b a = 0 if and only if b = a − . Now the following are equivalent: a + b = 1, −
a b− = 0, a = b− , b = a∼. This shows (P3). Since a (1 ∨ a ∼ ) = a 1 = a ≤ a ∧ a we have 1 = 1 ∨ a ∼ ≤ a ; a, i.e. a ; a = 1. If a ≤ b− then because of b− ∨ a ≤ a − ; a − we have a − b− = a − (b− ∨ a) ≤ a − ∧ a − = a − whence a = a −∼ ≤ (a − b− )∼ = a + b
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showing that a + (a + b)∼ is defined. Now in case a ≤ b− the following are equivalent: c = (a + (a + b)∼ )− , c + (a + (a + b)∼ ) = 1, (c + a) + (a + b)∼ = 1, c + a = a + b. Since (1 ∨ a − ) a = 1 a = a ≤ a ∧ a we have 1 = 1 ∨ a − ≤ a → a, i.e. a → a = 1. If b ≤ a ∼ then because of a ∼ ∨ b ≤ b∼ → b∼ we have a ∼ b∼ = (a ∼ ∨ b) b∼ ≤ b∼ ∧ b∼ = b∼ whence b = b∼− ≤ (a ∼ b∼ )− = (a − b− )∼ = a + b showing that (a + b)− + b is defined. Now in case b ≤ a ∼ , i.e. a ≤ b− the following are equivalent: c = ((a + b)− + b)∼ , ((a + b)− + b) + c = 1, (a + b)− + (b + c) = 1, b + c = a + b. This shows (P1). Now the following are equivalent: a ≤ b holds in P (Q), a + b∼ is defined, a − b is defined, a ≤ b holds in Q. Since (Q, ∨, ∧) is a lattice and the partial order relations in Q and P (Q) coincide, P (Q) is a good lattice pseudoeffect algebra. 2 As in the case of effect algebras, also every good lattice pseudoeffect algebra can be reconstructed from its assigned quasiresiduated lattice. Theorem 3.8. Let P be a good lattice pseudoeffect algebra. Then P (Q(P)) = P. Proof. Let P = (P , +,− ,∼ , 0, 1) with lattice operations ∨ and ∧, Q(P) = (P , ∨, ∧, , →, ;, 0, 1), P (Q(P)) = (P , ⊕, ∗ , + , 0, 1) and a, b ∈ P . Then a ∗ = a → 0 = a − + (a ∧ 0) = a − + 0 = a − , a + = a ; 0 = (a ∧ 0) + a ∼ = 0 + a ∼ = a ∼ .
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Moreover, the following are equivalent: a ⊕ b is defined, a ≤ b− in Q(P), a ≤ b− in P and in this case a ⊕ b = (a ∗ b∗ )+ = (a − b− )∼ = (a ∼ b∼ )− = a + b. 2 The following theorem characterizes those quasiresiduated lattices that can be reconstructed from their assigned good lattice pseudoeffect algebras. Theorem 3.9. Let Q = (Q, ∨, ∧, , →, ;, 0, 1) be a quasiresiduated lattice. Then Q(P (Q)) = Q if and only if Q satisfies the identities x ((x ∧ y) ; 0) ≈ (x → y) ; 0, ((x ∧ y) → 0) x ≈ (x ; y) → 0. Proof. If P (Q) = (Q, +,− ,∼ , 0, 1), Q(P (Q)) = (Q, ∨, ∧, ⊗, ⇒, , 0, 1) and a, b ∈ C then a ⊗ b = (a − + b− )∼ = (a ∼ + b∼ )− = (a ∼− b∼− )∼− = a b if a ∼ ≤ b, a ⇒ b = a − + (a ∧ b) = (a −− (a ∧ b)− )∼ = (a −∼ (a ∧ b)∼ )− = (a (a ∧ b)∼ )− , a b = (a ∧ b) + a ∼ = ((a ∧ b)− a ∼− )∼ = ((a ∧ b)− a)∼ .
2
Remark 3.10. If Q is a quasiresiduated lattice that is not divisible then because of Theorem 3.5, Q(P (Q)) = Q. 4. Conclusion Although lattice effect algebras cannot be expressed as residuated structures, we found out that they can be represented as so-called commutative quasiresiduated lattices. Similarly, we have shown that lattice pseudoeffect algebras can be represented by means of (not necessarily commutative) quasiresiduated lattices. Hence, a certain connection between lattice (pseudo-)effect algebras and fuzzy logics in a broad sense exists. The next interesting task would be to develop this kind of fuzzy logic whose structure of truth values is a quasiresiduated lattice. This may be investigated in future. Acknowledgement The authors are very grateful to the anonymous referees whose valuable suggestions helped to increase the quality of the paper. References [1] [2] [3] [4] [5] [6] [7]
R. Bˇelohlávek, Fuzzy Relational Systems. Foundations and Principles, Springer, New York, ISBN 978-1-4613-5168-9, 2002. I. Chajda, R. Halaš, Effect algebras are conditionally residuated structures, Soft Comput. 15 (2011) 1383–1387. I. Chajda, J. Kühr, H. Länger, Relatively residuated lattices and posets, Math. Slovaca (2020), in press, http://arxiv.org/abs/1901.06664. A. Dvureˇcenskij, S. Pulmannová, New Trends in Quantum Structures, Kluwer, Dordrecht, ISBN 0-7923-6471-6, 2000. A. Dvureˇcenskij, T. Vetterlein, Pseudoeffect algebras. I. Basic properties, Int. J. Theor. Phys. 40 (2001) 685–701. D.J. Foulis, M.K. Bennett, Effect algebras and unsharp quantum logics, Found. Phys. 24 (1994) 1331–1352. N. Galatos, P. Jipsen, T. Kowalski, H. Ono, Residuated Lattices: An Algebraic Glimpse at Substructural Logics, Elsevier, Amsterdam, ISBN 978-0-444-52141-5, 2007.