Normal forms and free algebras for some extensions of MTL

Fuzzy Sets and Systems 159 (2008) 1131 – 1152 www.elsevier.com/locate/fss

Normal forms and free algebras for some extensions of MTL Stefano Aguzzolia,∗ , Brunella Gerlab a Dept. Computer Science, Università degli Studi di Milano, Via Comelico 39-41, 20135 Milano, Italy b DICOM, Università dell’Insubria, Via Mazzini 5, 21100 Varese, Italy

Available online 8 December 2007

Abstract We introduce a semantical definition of minterms and maxterms which generalizes the usual notion in Boolean logic to a class of many-valued logics. We apply this notion to get normal forms for logics G, NM, NMG. Then we obtain a combinatorial description of the n-generated free algebras in the varieties constituting the algebraic semantics of those logics. Specifically, we represent via combinatorial posets the embedding of the n-generated free algebra into the direct product of all n-generated chains in the variety. © 2008 Elsevier B.V. All rights reserved. Keywords: Nilpotent minimum and Gödel logic; MTL; WNM; Algebraic semantics; Normal forms; Free algebras

1. Introduction In classical propositional logic, each formula over n variables is associated with a truth table, i.e. with a function from {0, 1}n to {0, 1}. The free Boolean algebra over n generators is the algebra of formulas with n variables, up to logical equivalence, and the functional completeness result for Boolean propositional logic states that the free Boolean algebra over n generators is the direct product of 2n copies of the Boolean algebra {0, 1}. A standard way to obtain a formula associated with a given function f : {0, 1}n → {0, 1} is to write it in either disjunctive or conjunctive normal form. Disjunctive normal forms are nothing else than sums (∨) of minterms, where a minterm is a formula having a truth table that takes value 1 over exactly one point in {0, 1}n . In classical propositional logic, minterms are products (∧) of variables or negation of variables. Dually, conjunctive normal forms are product of maxterms, each maxterm being the sum of variables or negation of variables. If we enlarge the set of truth values, the situation changes. In particular, if the set of truth values is the interval [0, 1] there is no hope to establish any functional completeness result if we consider a propositional logic built on a finite or denumerable alphabet. Even in the case the free algebras are finite, they are not in general direct product of chains. The problem of characterizing the set of truth functions of a given logic is hence interesting. This amounts to finding concrete representations of free algebras in the variety that constitutes the algebraic semantics of the logic. The logics we shall deal with in this paper are schematic extensions of monoidal t-norm based logic (MTL, [11]) that is the logic of all left-continuous t-norms and their residua. In particular we shall focus on some logics in the hierarchy of extensions of weak nilpotent minimum logic (WNM, [11]) which in turn is a schematic extension of MTL whose corresponding algebraic variety is locally finite. ∗ Corresponding author.

E-mail addresses: [email protected] (S. Aguzzoli), [email protected] (B. Gerla). 0165-0114/$ - see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2007.12.003

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More specifically, we shall deal with the following extensions of WNM: Gödel logic G based on Gödel t-norm [15,16], NM logic based on Nilpotent Minimum t-norm [11,13] and NMG logic based on the ordinal sum of Nilpotent Minimum and Gödel t-norms [11,22]. Free algebras in the variety associated with Gödel logic have been studied in [17,14]. In [9] authors give representations of finitely generated Gödel algebras based on combinatorial description of prime spectra. A functional representation of free NM algebras is given in [23], while [7] offers a characterization through a purely algebraic approach. Ref. [1] extends to NM the approach of [9]. In this paper we start from an elementary analysis of finitely generated chains to get poset representation theorems for finitely generated free algebras. Then we derive functional representation theorems and standard completeness theorems from poset representation of free algebras. We further give here proofs of results already announced in conference papers [3–5], where we have followed the different route of obtaining poset representations of finitely generated free algebras from functional representation theorems—which in turn derive from standard completeness theorems. We also fix a couple of bugs occurred in those papers and simplify certain constructions. We introduce a semantical definition of minterms and maxterms which generalizes the usual notion of Boolean logic to a class of many-valued logics. We apply this notion to get normal forms for logics G, NM, NMG. We represent via suitably constructed combinatorial posets the embedding of the n-generated free algebra into the direct product of all n-generated chains in the variety giving the algebraic semantics of the logics we deal with. 2. MTL and some of its schematic extensions Given a left-continuous t-norm  and its associated residuum x → y = max{z|z  x y}, the algebra [0, 1] = ([0, 1], , →, ∧, 0) where x ∧ y = min(x, y) is called a standard algebra. Monoidal t-norm based logic (MTL, for short) is introduced and axiomatized in [11]. The propositional language is built over the constant 0 and the binary connectives , →, ∧, whose standard interpretations (i.e. over [0, 1]) are a left-continuous t-norm, its residuum and minimum, respectively (throughout the paper we shall use the same symbols for connectives and operations interpreting them). Usual derived connectives are the binary ∨, defined as ((x → y) → y) ∧ ((y → x) → x) and interpreted as maximum; binary ↔ defined as x ↔ y := (x → y) ∧ (y → x); the unary negation ¬, defined as ¬x := x → 0; the constant 1 defined as ¬0. The algebraic counterpart of MTL is the variety V(MTL) of MTL-algebras. The ∨, ∧ reduct of each MTL-algebra is a lattice. MTL turns out to be the logic of all left-continuous t-norms and their residua, in the sense that V(MTL) is generated by the set of standard algebras [18]. If  is a formula with variables among x1 , . . . , xn , A is an MTL-algebra and a1 , . . . , an ∈ A, we denote by A (a1 , . . . , an ) the element of A obtained by interpreting variables xi in ai and the formula  in A. If A is generated by elements a1 , . . . , an then we shall write A for denoting A (a1 , . . . , an ). In particular xiA = ai . Each schematic extension L of MTL determines a subvariety V(L) of V(MTL) such that a formula  with n variables is a tautology of L if and only if for every algebra A = A, A , →A , ∧A , 0A in V(L) and for every a1 , . . . , an ∈ A, A (a1 , . . . , an ) = 1A . We denote by ≡L the logical equivalence relation between formulas. That is  ≡L  iff for every A ∈ V(L) and a1 , . . . , an ∈ A, ( ↔ )A = 1A , i.e.  ↔  is an L-tautology. The Lindenbaum algebra L(L) of logic L is the algebra in V(L) whose elements are equivalence classes of formulas of L with respect to equivalence relation ≡L . We denote the free n-generated algebra in a variety V(L) by Fn (L). In this context, n-generated free algebras are the algebras of classes of logically equivalent formulas (w.r.t. L) over a fixed set of n many propositional variables, i.e. algebras of truth functions. An element of a free algebra Fn (L) will be denoted either by Fn (L) or by  when n and L will be clear from the context. Weak nilpotent minimum logic (WNM for short) is the schematic extension of MTL with the axiom ¬(  ) ∨ (( ∧ ) → (  )). Operations in a WNM chain can be described starting from its negation as ⎧ ⎨ 0 if x ¬y, x  y = x if x > ¬y, x y, (1) ⎩ y if x > ¬y, x > y, ⎧ if x y, ⎨1 if x > y, y > ¬x, x→y= y (2) ⎩ ¬x if x > y, y ¬x.

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Since negation is order reversing, if x > ¬x and y > ¬y then x  y = min(x, y). From (1), (2) and the validity of ¬¬¬x = ¬x in MTL, it follows that for each WNM chain C generated by {c1 , . . . , cn } and for any element c ∈ C, there is a formula  in the set Fn = {0, x1 , . . . , xn , ¬x1 , . . . , ¬xn , ¬¬x1 , . . . , ¬¬xn , 1} of basic formulas, such that c = C (c1 , . . . , cn ). Every WNM chain C is locally finite and since in each subvariety of MTL the subdirectly irreducible members are chains, in [20,21] it is proved that the variety V(WNM) is locally finite, i.e. each finitely generated WNM algebra is finite. We refer the reader to [19] for further background on WNM. For sake of simplicity, elements xi , ¬xi and ¬¬xi of Fn are called 0-negs, 1-negs and 2-negs, respectively. Definition 2.1. Let C be a chain generated by g1 , . . . , gn . Display C as {c0 < c1 < · · · < ck }. Let Wi = { ∈ Fn |C (g1 , . . . , gn ) = ci } be the set of all formulas in Fn which are interpreted in ci . The map ci → Wi induces over W0 < W1 < · · · < Wk , a structure of WNM chain isomorphic to C. W0 < W1 < · · · < Wk is called an ordered partition of the set Fn . The sets Wi are called blocks. Throughout the paper, n-generated chains will be identified with ordered partitions of Fn . If the chain C is identified with an ordered partition of Fn , the block of C containing the element xi will be denoted by xiC . Analogously, the bottom and top elements of C are, respectively, denoted by 0C and 1C . For instance, consider the WNM chain C = {0 < 13 < 23 < 1} equipped with the involutive negation ¬c = 1 − c and let g1 = 13 , g2 = 23 be its generators. Then its representation as an ordered partition of F2 is {{0} < {x1 , ¬¬x1 , ¬x2 } < {¬x1 , x2 , ¬¬x2 } < {1}}; in this chain x1C = {x1 , ¬¬x1 , ¬x2 } is the block containing x1 . Note that in this case C can be 1-generated by g1 (or by g2 ); we can then represent C as ordered partition of F1 as {{0} < {x1 , ¬¬x1 } < {¬x1 } < {1}} (or {{0} < {¬x1 } < {x1 , ¬¬x1 } < {1}}, resp.). Gödel logic (G for short, [15]) is the schematic extension of MTL (and WNM, too) given by adding the idempotency axiom  → (  ). This axiom forces conjunction of Gödel chains to be the minimum, and implication to be  1 if x y, x→y= y otherwise. Hence ¬x = x → 0 is equal to 1 if x = 0 and it is equal to 0 otherwise. The variety V(G) of Gödel algebras is generated by the standard Gödel algebra [0, 1], , →, ∧, 0 , defined by x  y = x ∧ y. Nilpotent minimum logic (NM for short, [11,13]) is the schematic extension of WNM obtained by adding the involutiveness axiom ¬¬ → . Hence, in any NM algebra ¬¬x = x. Involutiveness of negation implies validity of the DeMorgan laws (x∧y) ≡NM ¬(¬x∨¬y), (x∨y) ≡NM ¬(¬x∧¬y), and interdefinability of connectives: (x → y) ≡NM ¬(x  ¬y), (x  y) ≡NM ¬(x → ¬y). The variety V(NM) is generated by the standard algebra [0, 1], , →, ∧, 0 , determined by setting x → 0 = ¬x := 1 − x. Every other NM standard algebra is NM isomorphic to this one. The logic NMG (introduced in [22]) is the schematic extension of WNM obtained by adding the axiom (¬¬ → ) ∨ (( ∧ ) → (  )). In [22] it is proved that V(NMG) is generated by the standard algebra [0, 1], , →, ∧, 0

given by  min(x, y) if x + y > 21 , xy = 0 otherwise,  1 if x y, x→y= max( 21 − x, y) otherwise. Each NMG standard algebra is the ordinal sum of the NM and G standard algebras (the choice of the middle idempotent at 21 is immaterial, any 0 < t < 1 would do). The interest for the logic NMG lies in the fact that it is a non-trivial case of a logic whose standard semantics is given by an ordinal sum.

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3. Preliminary notions on poset representation We introduce some notions on partially ordered sets (posets, for short), needed for our results (see [10] for background). The (disjoint) union, or horizontal sum A B of two disjoint posets A and B is the poset over A ∪ B formed by defining x y if and only if either x, y ∈ A and x y in A, or x, y ∈ B and x y in B. The linear sum, or vertical sum A ⊕ B of two disjoint posets A and B is the poset over A ∪ B defined by taking the following ordered relation: x y if and only if either x, y ∈ A and x y in A, or x, y ∈ B and x y in B, or x ∈ A and y ∈ B. A special case of vertical sum is the lifting construction A⊥ := {⊥} ⊕ A, with ⊥ ∈ / A. In the following we shall always assume that posets involved in horizontal and vertical sums are disjoint, by taking isomorphic copies of operands when necessary. By nA we denote A A · · · A n-times. We write 1 to denote the poset containing only one element. A chain is a totally ordered poset. A chain with n elements is isomorphic to 1 ⊕ · · · ⊕ 1, n times, which we denote by n. For every chain C = {a1 , . . . , au } with a1 < a2 < · · · < au we write C = {a1 < a2 < · · · < au }. A poset P ,  is an antichain if for all x, y ∈ P , x y implies x = y. An antichain with n elements is isomorphic to n1. The dual of a poset P ,  is a poset P j ,  j where P j = P and x  j y if and only if y x. A subposet of a poset P ,  is a subset of P equipped with the restriction of . Each poset A,  is order isomorphic to a poset o(A),  obtained by replacing each element of A with a copy of 1. We call o(A) the type of A, since o(A) retains only the order theoretic information about A. Definition 3.1. A finite poset A is nice if its type o(A) is described using only operations 1, and ⊕. Definition 3.2. Let A be a finite poset. A branch B of A is a maximal chain in A. We denote by B(A) the set of branches of A. A section over A is a maximal antichain of A. Hence a section contains exactly one element pB for every branch B ∈ B(A). We shall denote a section over A by [pB ]B∈B(A) , for pB ∈ B. The set of sections over A is denoted by S(A). For any two sections [pB ]B∈B(A) and [qB ]B∈B(A) we set [pB ]B∈B(A) [qB ]B∈B(A) if and only if for every B ∈ B(A), pB qB . Then S(A),  is a bounded lattice. Note that if A = A1 A2 , then S(A) = S(A1 ) × S(A2 ) and if A = A1 ⊕ A2 then S(A) = S(A1 ) ⊕ S(A2 ). Moreover, the number of elements of a nice poset and of the set of its sections is easily computed from its type: |1| = 1, |S(1)| = 1, |A B| = |A| + |B|, |S(A B)| = |S(A)| · |S(B)|, |A ⊕ B| = |A| + |B|, |S(A ⊕ B)| = |S(A)| + |S(B)|. Definition 3.3. The common prefix of two chains C1 and C2 is a chain P (possibly empty) such that C1 = P ⊕ C1 and C2 = P ⊕ C2 . If bottom elements of C1 and C2 are distinct then P is the longest common prefix of C1 and C2 . Definition 3.4. Let S(A) be the set of sections over a finite poset A. Let B be a branch of A and let p ∈ B. A semantical minterm for (B, p) is the smallest section B,p in S(A) such that B,p ∩ B = {p}. A semantical maxterm for (B, p) is the greatest section B,p in S(A) such that B,p ∩ B = {p}. Theorem 3.5. Let [pB ]B∈B(A) be an element of S(A). Then   [pB ]B∈B(A) = B,pB = B,pB . B∈B(A)

B∈B(A)

Throughout the paper, we shall mainly deal with posets whose elements are subsets of the disjoint union of many copies of a certain set S and whose branches are made of mutually disjoint subsets of some copy of S. Given such a

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branch B and an element x ∈ S we shall write x ∈∈ B to mean that x is an element of some element of B. We shall also take the liberty to denote by B,x and B,x the semantical minterms and maxterms B,p and B,p where p is the element of B containing x, whenever convenient. The same liberty will be taken when dealing with syntactical minterms and maxterms which are introduced in the next section. 4. Free algebras and normal forms In classical propositional logic, a truth table of a formula with n variables is a section over the nice poset 2n (2) and the n free Boolean algebra over n generators has 22 elements. For the logics we are going to deal with, combinatorial aspects are more interesting since the algebras of sections representing the free algebras are not direct product of chains. Let L be an axiomatic extension of WNM and let CnL (x1 , . . . , xn ) be the set of all chains in the variety V(L) generated by {x1 , . . . , xn }. Chains in CnL are represented as ordered partitions of Fn according to Definition 2.1 and every C ∈ CnL is generated by the set of blocks {xiC |1i n}. Suppose we are given a chain C = {c1 < c2 < · · · < ck }, represented as the ordered partition W1 < · · · < Wk of Fn , and let C  = C/R, for some congruence R. Then elements of C  are equivalence classes [ci ]R , that is, subsets of {c1 , . . . , ck }. By abuse of notation we still represent C  as an ordered partition W1 < · · · < Wh of Fn , for Wi being the union of all blocks containing R-equivalent elements of C. Let KnL (x1 , . . . , xn ) denote the set of all (ordered partition representation) of chains generated by {x1 , . . . , xn } that are not proper quotients of elements of CnL (x1 , . . . , xn ). We shall write Kn and Cn whenever L and x1 , . . . , xn will be clear from the context. Lemma 4.1. The map  ∈ Fn (L) → (C )C∈Kn ∈



C

C∈Kn

is a monomorphism. Proof. The map sending each  in (C )C∈Kn is the unique homomorphism given by the universal property of the free algebra Fn (L) that extends the map sending each generator xi in the element (xiC )C∈Kn ∈ C∈Kn C where xiC is the unique element of C containing xi . In order to prove that this map is a monomorphism, let (x1 , . . . , xn ) and (x1 , . . . , xn ) be two L-formulas. If   =  then, since the subdirectly irreducible elements in the variety V(L) are chains, there exists a chain C  ∈ Cn   such that C  = C . If C  is not a proper quotient of a chain in Cn then the claim is proved. Otherwise, we can write C  = C/∼ with ∼ proper congruence of C and C not proper quotient of any other chain in CnL . Then also C  = C , hence (C )C∈Kn  = (C )C∈Kn .  Lemma 4.2. For any congruence R on a chain C = {c0 < · · · < ck } ∈ V(L), we have that if ci cj and ci Rck then cj Rck and whenever [ci ]R  = {ci } then ci Rck or ¬ci Rck . Proof. If ci cj and ci Rck then (ci → cj )R(ck → cj ) hence ck Rcj (by definition in Eq. (2), since ck is the top element of C). Let ci  = cj and ci Rcj . Suppose without loss of generality that ci < cj . Then (ci → ci )R(cj → ci ) hence ck R(¬cj ∨ ci ). Two cases are then possible: either ci < ¬cj hence ck R¬cj , that together with ¬cj ¬ci gives ¬ci Rck , or ¬cj ci hence ck Rci .  The following definition will be used when stating results about representation of free algebras as algebras of functions from [0, 1]n to [0, 1]. Definition 4.3. Let A be a standard algebra in V(L). With each chain C = {W0 < · · · < Wk } ∈ CnL we associate a subset DC ⊆ [0, 1]n , as follows: DC is the set of all (y1 , . . . , yn ) ∈ [0, 1]n such that the chain generated by y1 , . . . , yn as a subalgebra of A is isomorphic to a quotient of C via the map yi → xiC . We denote by DnL the set {DC |C ∈ KnL }. It

is easy to see that DnL = [0, 1]n .

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In the next sections we shall describe free algebras of varieties associated with logical systems as algebras of sections over a suitable poset. Suppose to have proved that for a logic L, the free algebra Fn (L) is isomorphic to S(Ln ) for some finite poset Ln , via the map sending  in [B ]B∈B(Ln ) (cf. Theorems 5.9, 6.10, 7.15). Then, if B is a branch of S(Ln ) and p ∈ B, a  S(L ) syntactical minterm B,p is a formula of L such that the section B,pn = [B B,p ]B  ∈B(Ln ) is a semantical minterm for S(L )



(B, p), and analogously, a syntactical maxterm is a formula B,p such that B,pn = [B B,p ]B  ∈B(Ln ) is a semantical maxterm for (B, p). We have as a consequence of Theorem 3.5: Corollary 4.4. If Fn (L) is isomorphic to S(Ln ) and (x1 , . . . , xn ) is a formula of L, then    ≡L B,B ≡L B,B . B∈B(Ln )

B∈B(Ln )

Since we have the constant 0 in our language (and the derived connective 1 = ¬0), we can improve our sum-of-minterms and product-of-maxterms normal forms as follows: Definition 4.5. If Fn (L) is isomorphic to S(Ln ) and (x1 , . . . , xn ) is a formula of L, then: • If  ≡L 0 then NF() := 0, • otherwise:  NF() :=

B,B .

B∈B(Ln ), B =0

Analogously, • If  ≡L 1 then NF() := 1, • otherwise:  NF() := B∈B(Ln

B,B .

), B =1

Trivially, Theorem 4.6. NF() ≡L  ≡L NF(). Minterms of the form B,0 and maxterms of the form B,1 are called null. Note that Definition 4.5 generalize the classical sum-of-minterms and product-of-maxterms normal form of Boolean propositional logic. Definition 4.7. An element a  = 0 of a bounded lattice M is join-irreducible if a = b ∨ c for some b, c ∈ M implies a = b or a = c. Dually, an element a  = 1 of M is meet-irreducible if a = b ∧ c for some b, c ∈ M implies a = b or a = c. Proposition 4.8. In each finite lattice S(A) join-irreducible elements and non-null semantical minterms coincide. The same holds for meet-irreducible elements and non-null semantical maxterms. By Proposition 4.8, the techniques to find syntactical min/max-terms yield procedures to find join/meet-irreducible elements of free algebras. Proposition 4.9. Suppose Fn (L) is isomorphic to S(Ln ) for some finite poset Ln = D1 D2 · · · Dk . Then each n-generated directly indecomposable algebra in the variety V(L) is isomorphic to a quotient of S(Di ), for some i ∈ {1, . . . , k}. Proof. Fn (L) is isomorphic to the direct product of finite lattices S(D1 ) × S(D2 ) × · · · × S(Dk ).



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5. Poset representation for G In Gödel chains, negation and double negation of elements do not bring any new information. Indeed, if we represent Gödel chains as C = {W0 < · · · < Wk },

(3)

where {Wi }ki=0 is an ordered partition of {0, x1 , . . . , xn , 1} and 0 ∈ W0 , 1 ∈ Wk , there is only one way to add negation and double negation of elements in C respecting axioms of Gödel algebras, i.e. for every xj , adding ¬xj to W0 and ¬¬xj to Wk if xj ∈ Wi with i > 0, and adding ¬xj to Wk and ¬¬xj to W0 if xj ∈ W0 . Gödel negation is defined on C as ¬W0 = Wk and ¬Wi = W0 for i > 0. We let CnG be the set of all Gödel chains as represented in (3). Lemma 5.1. C = {W0 < · · · < Wk } ∈ KnG if and only if Wk = {1}. Proof. By Lemma 4.2, Wk = H ∪ {1} for H  = ∅ if and only if C is a proper quotient of C  = {W0 < · · · < Wk−1 < H < {1}}.  Lemma 5.2. For any formula (x1 , . . . , xn ), (C )C∈KnG is an element of C∈KnG C satisfying the prefix property, that is, if C, C  are chains in KnG with a common prefix W0 < · · · < Wh and C = Wi for some 0 i h, then also  C = Wi . Proof. By induction on the structure of . If  = xi (resp.  = 0) then C = Wi if and only if Wi is the unique element of C containing xi (resp. 0) if and  only if Wi is the unique element of C  containing xi (resp. 0) if and only if C = Wi . C C C C C Let  = 1 ∧ 2 and suppose theorem holds for 1 and 2 . Either 1 < C 2 or 2 1 . Suppose 1 < 2     C C C C hence C = C 1 = Wi , then by induction 1 = Wi . If 2 < 1 then 2 would be equal to some Wj < Wi hence  C C C C C C C 2 = Wj < Wi in contradiction with our hypothesis  = 1 . So 2 1 and  = 1 = Wi . The other case is analogous. C C C C C C C C Let  = 1 → 2 . Then either C 1 2 and  = 1 or 1 > 2 and  = 2 . In the first case,  = Wi C C  C implies Wi = {1} hence C = C . In the other case,  = 2 = Wi implies, by induction hypothesis, 2 = Wi .  C C C C C C C If it were C 1 2 then 1 = Wj with 0 j i and so 1 = Wj and  = 1 . So it must be 1 > 2 hence   C = C 2 = Wi .  Definition 5.3. If C is a set of chains, we denote by (C) the poset uniquely determined by the following prescriptions (see [4] for an algorithmic construction of (C)): • If C1  = C2 are chains of C and C1 = P ⊕ C1 , C2 = P ⊕ C2 with P their longest common prefix, then P ⊕ (C1 C2 ) is a subposet of (C). • Each branch of (C) is a copy of a unique chain in C and for each chain C there is a unique branch of (C) that is a copy of C. Let Gn = (KnG ) and let S(Gn ) be the poset of sections over Gn . By the bijective correspondence between chains in KnG and branches in B(Gn ), the algebraic structure of each chain is preserved in the corresponding branch. The componentwise defined operations ∧ and → over sections make the set S(Gn ) into a Gödel algebra. By construction: Lemma 5.4. The map (C )C∈KnG → [B ]B∈B(Gn ) is a monomorphism from C∈KnG C to S(Gn ). Hence the free algebra Fn (G) can be embedded in the algebra S(Gn ) of sections over Gn .

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Define connectives x  y := (y → x) → y

and

x  y := (x ↔ y) → x.

Note that interpretation in each Gödel chain is such that x  y = 1 iff either x < y or x = y = 1 and x  y = y iff x y, while x  y = 1 iff x  = y and x  y = x iff x = y. Consider a branch B = {W0 < W1 < · · · < Ww < {1}}. Let z0 = 0 and for each i ∈ {1, . . . , w} let zi be an arbitrarily fixed element of Wi . Define the formulas  B,i := (zi ↔ x) x∈Wi ,x=zi

and for each i ∈ {0, . . . , w − 1}

B,i := zi  zi+1 . Then set for each p ∈∈ B, p ∈ / W0 , B,p := p ∧ Empty

w 

B,i ∧



i=0

w−1 

B,i .

i=0

-conjunctions are safely omitted when writing B,p .

Theorem 5.5. For any B = {W0 < · · · < Wk < {1}} ∈ B(Gn ) and p ∈∈ B, B,p is a syntactical minterm for Gn . B  Proof. It is straightforward to see that B B,p = p . Let B = {W0 < · · · < Wk−1 < Vk < · · · < Vv } ∈ B(Gn ) with 

Vk  = Wk , (possibly k = 0) and denote by Wi = ziB the element of B  containing zi . By Definition 3.4 and Lemma  5.2, we have to prove that B B,p = Vk . Note that it may happen Wi < Wj even when i > j . It is easy to see that  {1} if Wi ⊆ Wi , B B,i = W ∈ B  such that minB  {x|x ∈ Wi } ∈ W otherwise,   ,  {1} if Wi < Wi+1 =

B  B,i otherwise. Wi+1  w B  B  Observe that either w i=0 B,i = Vh where h is the minimum index such that Vh  = Wh or i=0 B,i = {1} if such h does not exist. Now suppose that Wk Vk . Then there exists x ∈ Vk and x ∈ / Wk , hence x ∈ Wh for some h > k. There are two     cases: if zh ∈ Vk then (zh−1  zh )B = zhB = Vk ; if zh ∈ / Vk then (zh ↔ x)B = x B = Vk .   / Vk hence x ∈ Vh with h > k. If zk ∈ Vk then (zk ↔ x)B = zkB = Vk ; If Wk Vk then there exists x ∈ Wk and x ∈   / Vk then Vk  = Wk hence w

B = Vk . if zk ∈ w w−1i=0 B,iB  In all cases, ( i=0 B,i ∧ i=0 B,i ) = Vk and this settles the claim.  In order to describe Gödel maxterms we proceed as follows. Definition 5.6. For each branch B = {W0 < W1 < · · · < Ww < Ww+1 = {1}} ∈ B(Gn ) let z0 = 0 and zi be an arbitrarily fixed element of Wi , for i ∈ {1, . . . , w}. Define the formulas  B,i := (zi  y) y∈Wi ,y=zi

and for each i ∈ {0, . . . , w − 1} and each j ∈ {0, . . . , w}, k ∈ {j + 1, . . . , w + 1}  (y → zj ). B,i := zi+1 → zi , kB,j := y∈Wk

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Note that w+1 B,j ≡G zj . Then for each element p ∈∈ B, with p ∈ Wj for some j ∈ {0, . . . , w}, we set B,p := p ∨

j  i=0

Empty

B,i ∨

j −1 i=0

B,i ∨

w 

iB,j .

i=j +1

-disjunctions are safely omitted when writing B,p .

Theorem 5.7. For each branch B = {W0 < W1 < · · · < Ww < Ww+1 = {1}} ∈ B(Gn ) and each element p ∈∈ B, the formula B,p is a syntactical maxterm for Gn .  Proof. Assume p ∈ Wj . It is immediate to see that B B,p = Wj . Consider a branch B = {W0 < W1 < · · · < Wk−1 < Vk < · · · < Vv } ∈ B(Gn ) with Vk  = Wk for some k 0, and let j k. By Definition 3.4, we have to show that  B B,p = {1}. Let h be the unique index such that zj ∈ Vh . Set LB = {W0 < W1 < · · · < Wj } and LB  = {W0 < W1 < · · · < Vh }.     We may assume that z0B < z1B < z2B < · · · < zjB (which implies h j ), otherwise there is an index 0 t < j such



  j −1 that (zt+1 → zt )B = {1} and hence ( i=0 B,i )B = {1}. Moreover, we may assume LB  ⊆ LB , otherwise there

 i B is an x ∈ LB  and an index t > j with x ∈ Wt ; this would imply (x → zj )B = {1}, and ( w i=j

+1 B,j ) = {1}.

For any t ∈ {0, . . . , w}, let Wt be the unique element of B  containing zt . Note that LB  ⊆ LB together with Wk  = Vk implies that it is impossible that Wt ⊆ Wt for all t ∈ {0, . . . , j }, whence there is a t ∈ {0, . . . , j } such   / Wt : this yields (zt  x)B = {1} and B that Wt Wt . Then there exists x ∈ Wt and x ∈ t,B = {1}. This concludes the proof. 

In writing minterms and maxterms, simplifications can be applied: for instance note that x ↔ 0 ≡G ¬x, 0  x ≡G ¬¬x and x  0 ≡G ¬¬x. Example 5.8. We give some examples of minterms and maxterms for G2 . • Let B = {{0, x} < {y} < {1}} ∈ B(G2 ). Then B,0 = 0 ↔ x, B,0 = 0  y. Hence B,y = y ∧ (0 ↔ x) ∧ (0  y) ≡G y ∧ ¬x ∧ ¬¬y ≡G y ∧ ¬x. On the other hand, B,0 = 0  x, B,0 = y → 0, 2B,1 = 1 → y ≡G y. Then B,y = y ∨ (0  x) ∨ (y → 0) ≡G y ∨ ¬¬x ∨ ¬y. • Let B = {{0} < {x} < {y} < {1}} ∈ B(G2 ). Then B,0 = 0  x, B,1 = x  y. Hence B,x = x ∧ (0  x) ∧ (x  y) ≡G x ∧ ¬¬x ∧ (x  y) ≡G x ∧ (x  y), while B,y = y ∧ (0  x) ∧ (x  y) ≡G y ∧ ¬¬x ∧ (x  y). On the other hand, B,0 = x → 0, B,1 = 2B,1 = y → x, 3B,1 = 1 → x, 3B,2 = 1 → y. Hence B,x = x ∨ (x → 0) ∨ (y → x) ≡G x ∨ ¬x ∨ (y → x), while B,y = y ∨ (x → 0) ∨ (y → x) ≡G y ∨ ¬x ∨ (y → x). • Let B = {{0} < {x, y} < {1}} ∈ B(G2 ). Then B,1 = x ↔ y, B,0 = 0  x. Hence B,y = y∧(x ↔ y)∧(0  x) ≡G y ∧ (x ↔ y) ∧ ¬¬x. On the other hand, B,1 = x  y, B,0 = x → 0. Hence B,y = y ∨ (x  y) ∨ (x → 0) ≡G y ∨ (x  y) ∨ ¬x. • Let B = {{0, x, y} < {1}} ∈ B(G2 ). Then B,1 = B,1 = (0 ↔ x) ∧ (0 ↔ y) ≡G ¬x ∧ ¬y. On the other hand, B,0 = B,0 = (0  x) ∨ (0  y) ≡G ¬¬x ∨ ¬¬y. Let NF() and NF() be defined as in Definition 4.5. From Corollary 4.4 and Theorem 4.6 we have NF() ≡G  ≡G NF(). A detailed analysis of similar constructions can be found in [2]. Analogous kinds of normal forms for Gödel logic are found in [6,17,8]. Theorem 5.9. The free algebra Fn (G) is isomorphic to the algebra of sections S(Gn ). Proof. By Lemmas 4.1 and 5.4, Fn (G) can be embedded in S(Gn ). We have to prove that for any section [pB ]B∈B(Gn )  S(G ) there is a term  in n variables such that [B ]B∈B(Gn ) = [pB ]B∈B(Gn ) . By Theorem 5.5, B,pn = [B B,p ]B  ∈B(Gn ) is the smallest section taking value p over B. Then, for any section [pB ]B∈B(Gn ) over Gn , by Theorem 3.5, ⎛ ⎞S(Gn ) ⎡ ⎤    B ⎝ B,pB ⎠ =⎣ B,pB ⎦ = [pB ]B∈B(Gn ) .  B∈B(Gn )

B∈B(Gn )

B  ∈B(Gn )

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The order type of the nice poset Gn is a forest of upward growing trees: Theorem 5.10. Gn is nice, its type being o(Gn ) = Hn (Hn )⊥ with H0 = 1 and Hn =

n−1  i=0

 n (Hi )⊥ . i

Proof. Recall that Gn = (KnG ), hence we must compute the order type of the poset obtained by merging at common prefixes the set of all chains in KnG , that is, the set of all ordered partitions W0 < · · · < Wk of n + 2 elements 0, x1 , . . . , xn , 1 such that 0 is in the first block W0 and Wk = {1}. Observe that to any ordered partition {V0 < · · · < Vh } of {x1 , . . . , xn } there correspond exactly two distinct chains in KnG , namely {{0} ∪ V0 < · · · < Vh < {1}} and {{0} < V0 < · · · < Vh < {1}}. We can then reduce the computation of o(Gn ) to the computation of the type Hn of the poset Pn = Qn ⊕ {1}, where Qn is the poset obtained by merging at common prefixes the set of all ordered partitions of n symbols {x1 , . . . , xn }. It is then straightforward to verify that o(Gn ) = Hn (Hn )⊥ . We proceed by induction on n. If n = 1, then Q1 = {x1 } and the order type of P1 is H1 = 2. For sake of convenience we set H0 = 1 (hence H1 = (H0 )⊥ ). Since Pn is obtained merging chains at common prefixes, Pn is a forest of upward growing trees. It is clear  nthe   that set of roots of all trees in the forest is the set of all subsets of {x1 , . . . , xn }. In particular, Pn contains exactly nk = n−k trees whose root is a set of k elements. Each tree T with a root RT consisting of a set of k elements is obtained as the merging of the set CT of all chains having common prefix RT . Let CT = {C \{RT , 1C } | C ∈ CT }. Then T = (CT ) = RT ⊕ ({C ⊕ {1}|C ∈ CT }). Observe that CT is the set of all ordered partitions over the set of n − k many symbols {x1 , . . . , xn }\ RT . Hence, by induction hypothesis, o(({C ⊕ {1}|C ∈ CT })) = Hn−k and o(T ) = (Hn−k )⊥ . We can conclude that n−1  n (Hk )⊥ .  Hn = o(Pn ) = k k=0

Notice that the case n = 0 yields o(G0 ) = 1 2 and hence o(S(G0 )) = 2. The following first appeared in [17]: (ni) Corollary 5.11. |Fn (G)| = |S(Hn )|2 + |S(Hn )|, for |S(Hn )| = n−1 i=0 (|S(Hi )| + 1) . Example 5.12. o(G1 ) = 2 3. o(G2 ) = 2 2(3) (1 ⊕ (2 2(3))). 5.1. Gödel functions In this subsection, and in the analogous Sections 6.1 and 7.1, we consider each xi as the ith projection function xi (t1 , . . . , tn ) = ti and 0 and 1 as constant functions. Let DnG = {DC |C ∈ KnG } as in Definition 4.3. Definition 5.13. A function f : [0, 1]n → [0, 1] is a Gödel function over n variables if it satisfies the following properties: (1) f is a piecewise linear function and f coincides with a function in {0, 1, x1 , . . . , xn } over each domain D ∈ DnG ; (2) Prefix: Let U, V ∈ KnG and let W = {W1 < · · · < Wk } be their longest common prefix. Then f DU ∈ Wi if and only if f DV ∈ Wi , for each i ∈ {1, . . . , k}. Lemma 5.14. The free Gödel algebra Fn (G) is isomorphic to the algebra of Gödel functions over n variables, equipped with the pointwise operations of the standard Gödel algebra. Proof. Recall that the set of branches of Gn is in bijective correspondence with the set KnG . Let I be the map sending each Gödel function f into the section defined as follows. For any branch B ∈ B(Gn ), we have either f DB = xi for some xi or f DB = 0 or f DB = 1. Then let I (f ) = [pB ]B∈B(Gn ) where pB is the block of B containing f DB .

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Property 5.13.2 assures that I (f ) = [pB ]B∈B(Gn ) is a section. Vice versa, let J be the function mapping each section [pB ]B∈B(Gn ) into the function f such that f DB coincides with each element in pB . It is easy to check that f is well defined and satisfies properties in 5.13. As the algebra generated by applying standard operations to f (t1 , . . . , tn ) is isomorphic to a chain in CnG , the function I can be easily checked to be a monomorphisms having J as inverse. Hence the claim follows from Theorem 5.9.  Corollary 5.15. The variety V(G) is generated by the standard Gödel algebra. Proof. Suppose (x1 , . . . , xn ) is not valid in logic G. Then Fn (G) < 1 and by Lemma 5.14 there exist t1 , . . . , tn ∈ [0, 1] such that [0,1] (t1 , . . . , tn ) < 1.  Remark 5.16. The logic GH is the 0-free fragment of G [12]. Chains of KnGH are 0-free subreducts of chains in KnG . Corollaries of the preceding results are that Fn (GH) is isomorphic to S(Hn ), and to the restriction to (0, 1]n of the algebra of Gödel functions over n variables. Moreover, V(GH) is generated by the 0-free subreduct of the standard Gödel algebra. 6. Poset representation for NM Due to the involutiveness of negation, in NM chains double negation does not bring new information. As a matter of fact NM chains generated by x1 , . . . , xn can be represented as C = {¬Wk < · · · < ¬W1 ∝ W1 < · · · < Wk },

(4)

where {Wi , ¬Wi }ki=1 is an ordered partition of {0, x1 , . . . , xn , ¬x1 , . . . , ¬xn , 1}, ∝∈ {∪, <} and: • 1 ∈ Wk ; • if xi ∈ Wj then ¬xi ∈ ¬Wj and if ¬xi ∈ Wj then xi ∈ ¬Wj . There is only one way to add double negations to C, respecting NM axioms, i.e. adding ¬¬xi to the block containing xi . Operations are defined in accordance with Eqs. (1), (2). We let CnNM be the set of NM chains as described in (4). Lemma 6.1. NM chains in KnNM have the form C = {¬Wk < · · · < ¬W1 ∝ W1 < · · · < Wk }, where Wk = {1} (hence ¬Wk = {0}). Proof. The claim follows from Lemma 4.2, since given a congruence R on a NM chain C  , either R is the identity  congruence or the top element 1C of C  is congruent to some other block of C  . C is a proper quotient of C  if and only  if the top element 1C of C contains a block of C  different from 1C if and only if 1C  = {1}.  Definition 6.2. Let C = {¬Wk < ¬Wk−1 < · · · < ¬W1 ∝ W1 < · · · < Wk } ∈ CnNM . If ∝ is ∪, then we denote by C ∗ the set {¬W1 ∪ W1 }. If ∝ is <, then by C ∗ we mean the empty set. Subchains C↓ and C ↑ of C are defined univocally by C = C↓ ⊕ C ∗ ⊕ C ↑ and are called lower and upper semichains, respectively. It is straightforward to see that operations in upper semichains C ↑ of NM-chains coincide with Gödel operations. Definition 6.3. The sign of a chain C = C↓ ⊕ C ∗ ⊕ C ↑ is the sequence (C) = (1 , . . . , n ) ∈ {0, −1, 1}n such that ⎧ ⎨ −1 if xi ∈ W for some W ∈ C↓ , if xi ∈ W for C ∗ = {W }, i = 0 ⎩ 1 if xi ∈ W for some W ∈ C ↑ . The length of (C), denoted by  (C), is the number of components of (C) different from 0. Note that if U, V ∈ KnNM are such that (U ) = (V ) then U ∗ = V ∗ .

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Definition 6.4. Let ∼ be the relation on KnNM given by U ∼ V if and only if (U ) = (V ). It can be easily shown that ∼ is an equivalence relation. Let C/ ∼ denote the equivalence class of C and (C/ ∼)↓ = {B↓ |B ∼ C} and (C/∼)↑ = {B ↑ |B ∼ C}. Lemma 6.5. For any formula (x1 , . . . , xn ) of NM, (C )C∈KnNM is an element of property and the sign property, that is,



C∈KnNM

C satisfying the prefix

• Sign: Let U, V ∈ KnNM with U ∼ V : if U < ¬U then V < ¬V , if U = ¬U then V = ¬V and if U > ¬U then V > ¬V . • Prefix: Let U, V ∈ KnNM with U ∼ V and let W = {W1 < · · · < Wk } be the longest common prefix of U ↑ and V ↑ . Then U = Wi iff V = Wi and U = ¬Wi iff V = ¬Wi , for each i ∈ {1, . . . , k}. Proof. Proofs are by structural induction. By interdefinability of connectives, it is sufficient to deal with negation and conjunction. If  = 0 the properties are trivially valid. Let  = xi . The proof of the prefix property is trivial. Sign property follows from Definition 6.3. Let  = ¬. The sign property easily follows from induction hypothesis since ¬¬ = . Regarding the prefix property, if U = ¬U = Wi then U = ¬Wi hence V = ¬Wi and V = Wi . The other case is analogous. Let  = 1  2 and let R ∈ { , >}. U U U V V V V U U Sign. If U 1 R¬1 and 2 R¬2 then  R¬ and, by induction hypothesis, 1 R¬1 and 2 R¬2 and this V V U U U V V V V V V V implies  R¬ , since if 1 > ¬1 then  = min(1 , 2 ), otherwise  = 0 . If 1 R¬1 and not U 2 R¬2

U U U U U U U then without loss of generality we can suppose U 1 > ¬1 and 2 ¬2 . Then  = min(1 , 2 ) = 2 if U U U U U U V V 1 > ¬2 , and  = 0 otherwise. In both cases  ¬ and by induction hypothesis,  ¬ . U U U Prefix. The case U 1 R¬1 and 2 R¬2 easily follows from the sign property. U U U U U U U If 1 R¬1 and not 2 R¬2 then without loss of generality we can suppose U 1 > ¬1 and 2 ¬2 . Then U U U U U U U U U U  = min(1 , 2 ) = 2 if 1 > ¬2 , and  = 0 otherwise. In the first case  and 2 evaluates to the same block ¬Wi , and by inductive hypothesis V2 = ¬Wi , too. By sign property, V1 evaluates to some block W of V ↑ ; if W V lies in the common prefix, then V1 > ¬V2 by induction hypothesis, otherwise U 1 and 1 are greater than any element V V V in the common prefix, and, once again, 1 > ¬2 . This case is settled, since  = min(V1 , V2 ) = V2 = ¬Wi . In the second case U = 0U and hence U = V . 

By j (C) we mean the order dual of (C). Definition 6.6. For every C/∼ ∈ KnNM /∼, we set j

NC/∼ := j ((C/ ∼)↓ ) ⊕ ((C/∼)↑ ) if  (C) = n, j

NC/∼ := j ((C/∼)↓ ) ⊕ C ∗ ⊕ ((C/∼)↑ ) Let NMn =



if  (C) < n.

NC/∼ ,

C/∼∈KnNM /∼

and let S(NMn ) be the poset of sections over NMn . Remark 6.7. Each branch B ∈ B(NMn ) can be naturally described as B = B↓ ⊕ B ∗ ⊕ B ↑ . In particular if B is  a branch of NC/∼ , then B ∗ is a copy of C ∗ . For any chain C ∈ KnNM there are uniquely determined B↓ , B ↑ , with B ∼ B  ∈ B(NMn ) such that C is a copy of B↓ ⊕ B ∗ ⊕ B ↑ . Vice versa, for all branches B ∈ B(NMn ) there are 

uniquely determined C↓ , C ↑ , with C ∼ C  ∈ KnNM such that C ↑ is a copy of B ↑ and C↓ is a copy of B↓ . Notice that C and C  can be distinct chains, that is, there are branches B ∈ B(NMn ) that are not copies of any chain in KnNM . We can cope with this problem as follows. Let B  be the subset of B(NMn ) containing all branches which

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are copies of some chain in KnNM . It is then easy to see that each section in S(NMn ) can be written as [pB  ]B  ∈B . We can hence define operations on sections componentwise: [pB  ]B  ∈B  [qB  ]B  ∈B = [(p  q)B  ]B  ∈B and similarly for other connectives. By construction: Lemma 6.8. The map (C )C∈KnNM → [B ]B∈B(NMn )

is a monomorphism from C∈KnNM C to S(NMn ). Hence the free algebra Fn (NM) can be embedded in the algebra S(NMn ) of sections over NMn . Define connectives ◦ x := (x ↔ ¬x)  (x ↔ ¬x), +x := ¬((¬(x  x))  (¬(x  x))) and

− x := +(¬x).

Observe that ◦ x = 1 iff x = ¬x, ◦ x = 0 otherwise. Further, note that +x = 1 iff x > ¬x, +x = 0 otherwise. We call the formulas +x, ◦ x, −x the sign witnesses for x. Let  ∈ {−1, 0, 1}n be a sign and for each j ∈ {−1, 0, 1} let Tj = {i|i = j }. Define    −xi ∧ ◦ xi ∧ +xi .  := i∈T−1

i∈T0

i∈T1

We have  (x1 , . . . , xn ) = 1 iff for each i ∈ {1, . . . , n}: xi < ¬xi iff i = −1, xi = ¬xi iff i = 0, xi > ¬xi iff i = 1. Otherwise we have  (x1 , . . . , xn ) = 0. Since DeMorgan laws hold for ∧ and ∨, we have    ¬ ≡ ¬ − xi ∨ ¬ ◦ xi ∨ ¬ + xi . i∈T−1

i∈T0

i∈T1

We get minterms and maxterms for NM by playing with sign witnesses and by modifying Gödel minterms and maxterms through suitable substitutions: Let B ∈ B(NMn ) and let VB ↑ ⊆ {xi , ¬xi |1 i n} be the set of 0-negs and 1-negs occurring in elements of B ↑ and let B : VB ↑ → {xi |1i n}|B ∗ be the substitution map fixing each xi and sending each ¬xj to xj . Note B is bijective. By abuse of notation, we apply B to branches and formulas, with the obvious meaning of performing the prescribed substitution to every occurrence of an element of VB ↑ . We apply the same notation to the inverse of B , too. Let B0 = {0}⊕B (B ↑ ). Observe that B0 is a copy of a branch in B(Gn ). Let us denote by B,p and B,p , respectively, the minterm and the maxterm in Gödel logic for B (a copy of) a branch in B(Gn ) and p ∈∈ B. ˜ B,p := −1 Let ˜ B,p := −1 B (B0 ,B (p) ) and  B (B0 ,B (p) ). We set ⎧ if p ∈∈ B ∗ , ⎨ p ∧  (B) if p ∈∈ B ↑ , B,p := ˜ B,p ∧  (B) ⎩ ¬˜B,¬p ∧  (B) if p ∈∈ B↓ , ⎧ if p ∈∈ B ∗ , ⎨ p ∨ ¬ (B) if p ∈∈ B ↑ , B,p := ˜ B,p ∨ ¬ (B) ⎩ ¬˜ B,¬p ∨ ¬ (B) if p ∈∈ B↓ . Theorem 6.9. For any B ∈ B(NMn ) and p ∈∈ B, B,p is a syntactical minterm for NMn and B,p is a syntactical maxterm for NMn .

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B B ∗ ↑ Proof. To check that B B,p = p we first observe that  (B) = {1}. The case p ∈∈ B is trivial. If p ∈∈ B we recall that NM operations and G operations coincide on upper semichains of KnNM . If p ∈∈ B↓ , we identify B (B, ¬p) with B B B its copy in B(Gn ). Since B00 ,B (¬p) evaluates to B (¬p)B0 we have that (¬−1 B (B0 ,B (¬p) )) = p . 

B  Let B   = B be a distinct branch of B(NMn ). If B   ∼ B then  (B) = {0} and we are done. If B ∼ B and p ∈∈ B ∗ , since B ∗ is both a subset of B and B  there is nothing more to prove. If B  ∼ B and p ∈∈ B ↑ then let B0 = {0}⊕B (B ↑ ), and identify B0 with its copy in B(Gn ). Since B0 ,B (p) is a minterm in Gödel logic for (B0 , B (p)), B

we have that B00 ,B (p) evaluates in B(Gn ) to the smallest possible value Q ∈ B0 . Observe Q  = {0} since {0} is in the common prefix of B0 and B0 . B0 Since NM operations and G operations coincide on upper semichains of KnNM we have that (−1 B (B0 ,B (p) )) evaluates in B(NMn ) to the smallest possible value of B  , and this case is settled. If B  ∼ B and p ∈∈ B↓ we reason analogously, with the only differences of using the appropriate Gödel logic maxterm for (B, ¬p). Negation of this maxterm gives the desired minterm. Due to the involutiveness of negation, the argument for maxterms is analogous.  Let NF() and NF() be defined as in Definition 4.5. From Corollary 4.4 and Theorem 4.6 we have NF() ≡NM  ≡NM NF(). Theorem 6.10. The free algebra Fn (NM) is isomorphic to the NM algebra S(NMn ) of sections over NMn . Proof. By Lemmas 4.1 and 6.8, Fn (NM) can be embedded in S(NMn ). We have to prove that for any section [pB ]B∈B(NMn ) there is a term  in n variables such that [B ]B∈B(NMn ) = [pB ]B∈B(NMn ) .  By Theorem 6.9, [B B,p ]B  ∈B(NMn ) is the smallest section taking value p over B. Then, for any section [pB ]B∈B(NMn ) over NMn , by Theorem 3.5, ⎡ ⎤   ⎣ ⎦ B = [pB ]B∈B(NMn ) .  B,p(B) B∈B(NMn )

B  ∈B(NMn )

Theorem 6.11. NMn is nice, its type o(NMn ) being n−1     j j i n (Hi ⊕ 1 ⊕ Hi ) 2n (Hn ⊕ Hn ). 2 i i=0

 Proof. First recall that NMn = C/∼∈KnNM /∼ NC/∼ . Then NMn is the horizontal sum of as many components as the cardinality KnNM /∼. It is straightforward to check that this number is 3n . In particular for each 0 k n there are n of k exactly k 2 distinct signs of length k. j

Given an equivalence class C/∼, with k =  (C) < n, recall that NC/∼ = j ((C/∼)↓ ) ⊕ C ∗ ⊕ ((C/∼)↑ ). Observe that by the proof of Theorem 5.10, the order type of ((C/∼)↑ ) is Hk since it is the set of all ordered partitions j j j of k symbols merged at common prefixes. It is trivial to see that o(j ((C/∼)↓ )) = Hk , hence o(NC/∼ ) = Hk ⊕1⊕Hk . j

The case  (C) = n now immediately gives o(NC/∼ ) = Hn ⊕ Hn . This concludes the proof, since the horizontal sum over all components yields the desired formula.  Corollary 6.12. n

|Fn (NM)| = (2|S(Hn )|)2 ·

n−1 

i n

(2|S(Hi )| + 1)2 ( i ) .

i=0

Example 6.13. o(NM1 ) = 4 3 4. o(NM2 ) = 3 4(5) 4((3 2 3) ⊕ (3 2 3)).

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6.1. NM functions Consider the standard NM algebra given by x  y = 0 if x 1 − y and x  y = x ∧ y otherwise. Let DnNM = {DC |C ∈ KnNM }. Definition 6.14. A function f : [0, 1]n → [0, 1] is a NM function over n-variables if satisfies the following conditions: (1) f is piecewise linear and it coincides with one of the functions in {0, x1 , . . . , xn , ¬x1 , . . . , ¬xn , 1} over each domain D ∈ DnNM . (2) Sign: Let U, V ∈ KnNM with U ∼ V and let  be any of {<, =, >}. If f DU  21 then f DV  21 . (3) Prefix: Let U and V be two chains in KnNM such that U ∼ V and let W = {W1 < · · · < Wk } be the longest common prefix of U ↑ and V ↑ . Then f DU ∈ Wi iff f DV ∈ Wi and f DU ∈ ¬Wi iff f DV ∈ ¬Wi , for each i ∈ {1, . . . , k}. Lemma 6.15. The free algebra Fn (NM) is isomorphic to the algebra of NM functions over n variables equipped with the pointwise operations of the standard NM algebra. Proof. With any NM function f we associate the section [pB ]B∈B(NMn ) such that for any B ∈ B(NMn ), if B ↑ is the copy of the upper semichain C ↑ and f DC  21 then we set pB = f DC ; while if B↓ is the copy of C↓ and f DC  < 21 we put pB = f DC  . Vice versa, write any section [pB ]B∈B(NMn ) equivalently as [pB  ]B  ∈B , for B  as defined in Remark 6.7. We associate with [pB  ]B  ∈B the function f such that for any C ∈ KnNM , f DC = pBC , where BC is the copy of C in B  . Then the argument runs as in the proof of Lemma 5.14, using Theorem 6.10 instead of Theorem 5.9.  Corollary 6.16. The variety V(NM) is generated by the standard NM algebra. 7. Poset representation for NMG We say that a chain {W0 < · · · < Wk } ∈ CnNMG is order compatible with c1 c2 (c1 , c2 ∈ Fn ) if c1 ∈ Wi , c2 ∈ Wj and Wi Wj . Observe that in any NMG chain C we have either ¬¬xiC = xiC or ¬¬xiC = 1C . Then, in any NMG chain each occurrence of a 2-neg ¬¬xi must appear either in the block containing xi or in the top element of the chain. The set KnNMG of all NMGn chains generated by x1 , . . . , xn that are not proper quotients of elements of CnNMG can be described in terms of KnNM and KnG , as follows. An NM(G)n -chain is an element of CnNMG obtained from an element of KnNM by adding ¬¬xi to each node containing xi (for i ∈ {1, . . . , n}). Analogously, a (NM)Gn -chain is an element of CnNMG obtained from an element of KnG by adding for i ∈ {1, . . . , n}, ¬xi and ¬¬xi in the only possible consistent way, that is: if the bottom does not contain xi , then ¬xi is added to the bottom and ¬¬xi is added to the top, while if the bottom does contain xi , then ¬xi is added to the top and ¬¬xi to the bottom. For any C ∈ CnNMG , define its ith cycle as the one variable subchain generated by the element containing xi , that is: {W ∈ C|{0, xi , ¬xi , ¬¬xi , 1} ∩ W  = ∅}. Lemma 7.1. The set KnNMG is the set of all ordered partitions C of Fn satisfying the following properties: (i) For any i ∈ {1, . . . , n}, the ith cycle of C is either an NM(G)1 chain or a (NM)G1 chain. (ii) If the ith cycle is an NM(G)1 chain and the jth cycle is a (NM)G1 chain then either xj occurs in the bottom element of C or the order of C is compatible with xi < xj . (iii) If the set of cycles that are (NM)G1 chains is not empty then there is at least one such cycle, say the jth cycle, such that xj does not occur in the bottom element of C. Proof. Suppose C ∈ KnNMG . Condition (i) holds since C is an NMG chain and by axioms of NMG it must be either xi = ¬¬xi or xi = xi  xi . In particular, no 0-negs appear in the top element of C.

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Condition (ii) follows from properties of negation. Indeed let xi and xj be as in (ii): if 0 < xj xi < 1 then ¬xi ¬xj = 0 hence ¬xi = 0 and xi = 1. A contradiction. Regarding (iii), suppose that {xj }j ∈J are all the variables generating Gödel cycles and that they are all in the bottom element of C. Then the chain C would be the quotient of a suitable NM(G) chain C  by a congruence R such that xj is    not in the bottom 0C of C  and xjC R 0C for all j ∈ J . Consider now a chain C satisfying conditions (i), (ii) and (iii). By directly checking axioms of NMG algebra we see that C ∈ CnNMG . Let xi be the greatest 0-neg in C generating a (NM)G1 chain. By condition (iii), xi is not in the bottom element of C. By condition (i) the top element 1C of C cannot contain any 0-neg, so xi is not in 1C . By (ii) each 0-neg generating a NM(G)1 chain is smaller than xi and all 1-negs are either equal to 1 or smaller than xi . So the block xiC containing xi is a co-atom of C.   Suppose C is a quotient of C  through a congruence R and denote by 1C and xiC the top element of C  and the block containing xi , respectively. Since in 1C there are no 0-negs, then blocks of C  containing 0-negs are not R-congruent     to 1C . If (¬xj )C is congruent to (but different from) the top element of C  , since xiC is a coatom of C, (¬xj )C < xiC  hence by Lemma 4.2, we have xiC = [(¬xj )C ]R = 1C , which is a contradiction. Again by Lemma 4.2, R can only be the diagonal congruence and the claim is proved.  As in Definitions 6.2 and 6.3 we can write any NMG chain C as C↓ ⊕ C ∗ ⊕ C ↑ , where C↓ = {W0 < · · · < Wk } for

k C C C C ∗ i=0 Wi = {0} ∪ {xj , ¬¬xj |xj < ¬xj } ∪ {¬xj |¬xj < xj } and C is either the empty set or the only block of C containing {xi , ¬xi , ¬¬xi }, for some i. We then define signs accordingly. Definition 7.2. Let xi be the smallest 0-neg in a chain C ∈ KnNMG not belonging to the bottom of C and generating a (NM)G1 cycle. Then ¬¬xi is in the top element of C. Let H = {h|xh ∈ xiC } and K = {k|¬xk ∈ 1C }. We obtain a chain D ∈ KnNMG by modifying C as follows: • Move every element of the set {¬¬xi , ¬¬xh , ¬xk |h ∈ H, k ∈ K} into the block containing xi . • Create a new block W by taking {¬xi , ¬xh , xk , ¬¬xk |h ∈ H, k ∈ K} away from 0C , and insert W as an atom of D. We say that the chain D arises from C by the involutive promotion of xi . Notice that if D arises from C, then the top element of D does not contain any occurrences of 1-negs. Example 7.3. Let C = {{0, ¬x2 , ¬x3 , x4 , ¬¬x4 , ¬x5 } < {¬x1 } < {x1 , ¬¬x1 } < {x2 , x5 } < {x3 } < {1, ¬¬x2 , ¬¬x3 , ¬x4 , ¬¬x5 }}. Then the involutive promotion of x2 is D = {{0, ¬x3 } < {¬x2 , x4 , ¬¬x4 , ¬x5 } < {¬x1 } < {x1 , ¬¬x1 } < {x2 , x5 , ¬¬x2 , ¬x4 , ¬¬x5 } < {x3 } < {1, ¬¬x3 }}. C is equivalent to D (in symbols, C ≡ D) if either C arises from D or D arises from C by a sequence of involutive promotions. The relation ≡ is easily seen to be an equivalence over CnNMG refining the sign equivalence relation ∼ given in Definition 6.4. Each equivalence class [C] ∈ KnNMG / ≡ contains exactly one NM(G)n chain, denoted by I(C). This is the chain in which there are no 0-negs generating (NM)G1 chains. All other chains in [C] have at least one ith cycle that is a (NM)G1 chain. Example 7.4. In Example 7.3 the involutive promotion of x3 in D is the NM(G)5 chain E = {{0} < {¬x3 } < {¬x2 , x4 , ¬¬x4 , ¬x5 } < {¬x1 } < {x1 , ¬¬x1 } < {x2 , x5 , ¬¬x2 , ¬x4 , ¬¬x5 } < {x3 , ¬¬x3 } < {1}}. In this example [C] = {C, D, E} and I(C) = E. The class [C] can be constructed as a descending sequence of chains Ch , Ch−1 , . . . , C0 , where C0 = I(C) and each Ci arises by involutive promotion from Ci+1 . We can then describe the lower parts Ci↓ of chains in [C] as follows. The negative prefix I(C)¬ of I(C) is the prefix {{0} < W1 < · · · < Wh } of I(C) such that for each 1 i < h the element Wi contains only 1-negs, Wh contains at least one 1-neg and: either Wh also contains some 0-negs, or the

S. Aguzzoli, B. Gerla / Fuzzy Sets and Systems 159 (2008) 1131 – 1152

successor of Wh in I(C) contains no 1-negs. For each i ∈ {0, . . . , h} let Wi◦ = {0} ∪ Wi+1 < · · · < Wh } ⊕ (I(C)↓ \ I(C)¬ ). Let [C]↓ = {B↓ |B ∈ [C]} and [C]↑ = {B ↑ |B ∈ [C]}. Lemma 7.5. For any formula (x1 , . . . , xn ), (C )C∈KnNMG is an element of properties:



1147

i

j =1 Wj .

C∈KnNMG

Then Ci↓ = {Wi◦ <

C satisfying the following

• Sign: Let U, V ∈ KnNMG with U ∼ V . For any  ∈ {<, >, =}, if U ¬U then V ¬V . • Prefix: Let U, V ∈ KnNMG with U ∼ V . Let W = {W1 < · · · < Wk } be the longest common prefix of U ↑ and V ↑ . Then U = Wi iff V = Wi for each i ∈ {1, . . . , k} and U = ¬Wi iff V = ¬Wi , for each i ∈ {1, . . . , h}. • Lower unification: Let I(C) be the NM(G)-chain in [C] and I(C)↓ = {{0} = W0 < W1 < · · · < Wk }. Display [C] as {C0 , . . . , Ch }, where Ci↓ = {Wi◦ < Wi+1 < · · · < Wh < · · · < Wk }. Then: For 0 i h and for all j ∈ {i, . . . , h}, I (C) = Wi implies Cj = Wj◦ . Proof. Sign and prefix properties are proved as in Lemma 6.5. We just notice that if C is obtained from D by an j j involutive promotion, then the longest common prefix of C↓ and D↓ is the negation of the longest common prefix of C ↑ and D ↑ . Lower unification follows from the other two properties. As a matter of fact, if Cj ∈ Cj ↓ \ {Wj◦ } for some j ∈ {i, . . . h}, say Cj = Wt for some t > j i, it follows by sign and prefix properties that I (C) = Wt  = Wi .  Remark 7.6. Observe that the lower unification property implies that, as far as lower semichains are concerned, all semichains not of the form I(C)↓ bear redundant information. Just as in the case of Gödel and NM, we are going to find a more suitable representation of elements of C∈KnNMG C satisfying the properties of Lemma 7.5. In complete analogy with NMn , the poset NMGn is a horizontal sum of a finite number of components which in turn are vertical sums of a lower part, a central part and an upper part. In order to describe the upper part of a component, we shall introduce the splitting construction, which takes the upper part ((C/∼)↑ ) of a component of NMn and iteratively duplicates pieces of it to reflect the structure of the set of sections over {[D]↑ |D ∈ C/∼}. Further, we shall show that set of sections over the lower part of a component coincides with the set of sections over all lower semichains of NM(G)n chains in the same component. We start with the splitting construction. • Let T be a tree whose branches are ordered partitions of Fn . Call an element W of T involutive iff either it is equal to {1} or 1 ∈ / W , W contains at least one 0-neg and xi ∈ W implies ¬¬xi ∈ W . An element W  = {1} is splittable iff all elements comparable with W are involutive and every element W  > W contains no occurrences of 1-negs. • If W ∈ T is splittable, the splitting of W in T is the poset T  obtained by replacing in T the set {W  |W  W } with {W  |W  W } {W  |W  W }G where AG is the poset obtained from A by moving all the 1-negs and 2-negs from A to its top element. Note that T  is a tree iff W is not the root of T. Otherwise T  is a forest of two trees. • To split a tree T, perform the splitting of each splittable element exactly once, proceeding upwards from the root. Example 7.7. Consider the NM(G)3 -chain C whose upper part is given by C ↑ = {{x1 , ¬¬x1 } < {x2 , ¬¬x2 } < {x3 , ¬¬x3 } < {1}}. Since each node of C ↑ \{1} is splittable, the forest obtained by splitting C ↑ has type 4 (1 ⊕ (3 (1 ⊕ (2 2)))). The four element chain is {{x1 } < {x2 } < {x3 } < {1, ¬¬x1 , ¬¬x2 , ¬¬x3 }}, the root of the largest tree is {x1 , ¬¬x1 }, the three element chain is {{x2 } < {x3 } < {1, ¬¬x2 , ¬¬x3 }}, the structure of the remaining part of the poset is {x2 , ¬¬x2 } ⊕ ({{x3 } < {1, ¬¬x3 }} {{x3 , ¬¬x3 } < {1}}). Definition 7.8. Given a set C of upper semichains of NM(G)n chains, we denote by (C) the splitting of (C). Lemma 7.9. ([C]↑ ) = (I(C)↑ ) and ((C/∼)↑ ) = ({I(D)↑ |D ∈ C/∼}). Proof. By Definition 7.8, since each branch of (I(C)↑ ) is a copy of the upper semichain of a chain in [C].



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S. Aguzzoli, B. Gerla / Fuzzy Sets and Systems 159 (2008) 1131 – 1152 j

j

Lemma 7.10. The order type of S(j ([C]↓ )) coincides with the order type of I(C)↓ . Moreover o(S(j ((C/∼)↓ ))) = j

o(S(j ({I(D)↓ |D ∈ C/∼}))).

j

Proof. Let [C] = {C0 , C1 , . . . , Ch }. By lower unification property of 7.5, the poset ([C]↓ ) is of the form D ⊕ T (h), where D is a chain of length |I(C)↓ | − h − 1 and T (h) = 1 (1 ⊕ T (h − 1)), with T (0) = 1. An easy induction on h shows that S((D ⊕ T (h))j ) is order isomorphic to D ⊕ 1 ⊕ h, a chain of |I(C)↓ | many elements. The second part of the lemma follows immediately.  Example 7.11. Let C be the NM(G)4 chain such that C↓ = {{0} < {¬x1 } < {¬x2 , x3 , ¬¬x3 } < {x4 , ¬¬x4 }}. Then C ¬ = {{0} < {¬x1 } < {¬x2 , x3 , ¬¬x3 }} and [C]↓ = {C0↓ , C1↓ , C2↓ } where C0 = C, C1↓ = {{0, ¬x1 } < j {¬x2 , x3 , ¬¬x3 } < {x4 , ¬¬x4 }}, C2↓ = {{0, ¬x1 , ¬x2 , x3 , ¬¬x3 } < {x4 , ¬¬x4 }}. Hence the poset S(j ([C]↓ )) is given by the chain { ({0}, {0, ¬x1 }, {0, ¬x1 , ¬x2 , x3 , ¬¬x3 }) < ({¬x1 }, {0, ¬x1 }, {0, ¬x1 , ¬x2 , x3 , ¬¬x3 }) < ({¬x2 , x3 , ¬¬x3 }, {0, ¬x1 , ¬x2 , x3 , ¬¬x3 }) < ({x4 , ¬¬x4 }) }. Definition 7.12. For each sign  ∈ {−1, 0, 1}n let A be an arbitrarily chosen NMGn chain with (A ) = . We set NMGn = ∈{−1,0,1}n P , for j

P = j ({I(C)↓ | (C) = }) ⊕ A∗ ⊕ ({I(C)↑ | (C) = }). To turn S(NMGn ) into an NMG algebra we reason as follows. Consider the poset N = Q is the poset



∈{−1,0,1}n

Q , where each

j

j ((C/∼)↓ )) ⊕ A∗ ⊕ ({I(C)↑ | (C) = }). By Lemma 7.10, S(N ) is order-isomorphic to S(NMGn ). We observe that for any chain C ∈ KnNMG there are uniquely  determined B↓ , B ↑ , with B ∼ B  ∈ B(N ) such that C is a copy of B↓ ⊕B ∗ ⊕B ↑ . Vice versa, for all branches B ∈ B(N ) 

there are uniquely determined C↓ , C ↑ , with C ∼ C  ∈ KnNM such that C ↑ is a copy of B ↑ and C↓ is a copy of B↓ . As in Remark 6.7, we define operations over sections in S(N ) componentwise on the set of branches that are copies of chains in KnNMG . Now, by Remark 7.6 to Lemma 7.5, and Lemma 7.10, S(NMGn ) results equipped with the structure of NMG algebra. Lemma 7.13. The map (C )C∈KnNMG → [B ]B∈B(NMGn ) is a monomorphism from



C∈KnNMG

C to S(NMGn ).

Let B ∈ B(NMGn ) be a branch whose upper part B ↑ = {W1 < W2 < · · · < Wu } is a branch of ({I(C)↑ | (C) = }). / H }. Further, for each i ∈ H ∪ I let h(i) be the unique index Let H = {i|¬¬xi ∈ 1B } and let I = {i|xi ∈∈ B ↑ , i ∈ such that xi ∈ Wh(i) . Note that for each i ∈ H , the formula ¬¬xi evaluates to 1B over B, while ¬¬xi → xi evaluates to Wh(i) . On the contrary, for each i ∈ I , the formula ¬¬xi evaluates to Wh(i) over B, while ¬¬xi → xi evaluates to 1B . We call the formulas ¬¬xi and ¬¬xi → xi the splitting witnesses for xi . Let B,p and B,p , respectively, denote the formulas giving the minterm I (B),p and the maxterm I (B),p in the logic NM. By Remark 7.6 and Lemma 7.10, we can set B,p := B,p and B,p := B,p for any p ∈∈ B \ B ↑ . If, on the other hand, p ∈∈ B ↑ , let j ∈ {1, . . . , u − 1} be the unique index such that p ∈ Wj . Then   B,p := B,p ∧ ¬¬xi ∧ (¬¬xi → xi ) i∈H

and B,p := B,p ∨

i∈I

 i∈H,h(i)  j

(¬¬xi → xi ) ∨

 i∈I,h(i)  j

¬¬xi .

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Theorem 7.14. For any B ∈ B(NMGn ) and p ∈∈ B, B,p is a syntactical minterm and B,p is a syntactical maxterm of S(NMGn ).   Proof. It is immediate to see that i∈H ¬¬xi ∧ i∈I (¬¬xi → xi ) = 1B over B. Let B   = B be a branch in B(NMGn ) such that B  ≡ B. Then there is a uniquely determined index t ∈ I such that xtB is the least block of B which is not in the common prefix of B ↑ and B ↑ . If B is obtained from B  by a sequence of involutive promotions, then ¬¬xt ∈ xtB    and hence t ∈ I , while ¬¬xt ∈ 1B . In this case we have (¬¬xt → xt )B = xtB and we are done. In the case B  is obtained from B by a sequence of involutive promotions, then ¬¬xt belongs to the top element of B, and hence t ∈ H ,    while ¬¬xt ∈ xtB . It follows that (¬¬xt )B evaluates to xtB . This settles the proof for minterms. The argument for maxterms is analogous.  Let NF() and NF() be defined as in Definition 4.5. From Corollary 4.4 and Theorem 4.6 we have NF() ≡NMG  ≡NMG NF(). Theorem 7.15. The free algebra Fn (NMG) is isomorphic to the algebra of sections S(NMGn ). Proof. By Lemmas 4.1 and 7.13, it is enough to prove that for every section [pB ] ∈ S(NMGn ) there exists a term  such that (B )B∈B(NMGn ) = [pB ]B∈B(NMGn ) . By Theorems 7.14 and 3.5 ⎡ ⎤   ⎣ ⎦ B = [pB ]B∈B(NMGn ) .  B,p(B) B∈B(NMGn )

B  ∈B(NMGn )

Remark 7.16. The logic NMG is properly weaker than the intersection of NM and G. As a matter of fact, consider the following branch B ∈ B(NMG2 ): B = {{0, ¬y} < {x, ¬x, ¬¬x} < {y} < {1, ¬¬y}}. Then the maxterm B,y = y ∨ (y → 0) ∨ (¬ ◦ x) ∨ (¬ + y) ∨ (¬¬y → y) is not a NMG tautology, while it is both a G and an NM tautology. A moment’s reflection then shows that the simpler formula (¬¬y → y) ∨ ¬(x ↔ ¬x) would do as well. Theorem 7.17. The poset NMGn is nice and its type o(NMGn ) is n−1 i  i=0 k=0

   n    n i n j j (Hi ⊕ 1 ⊕ Lki ) (Hn ⊕ Lkn ) i k k k=0

for L0i = Hi and Lki

= Hk

j i−1  j =0 h=0

   k i−k (Lhj )⊥ . h j −h

Proof. By Theorem 6.11 and Definition 7.12 both NMGn and NMn are horizontal sums of sets of 3n many components, and the two sets containing the respective components are in bijection, through the sign relation. j j By Lemma 7.10, the lower part j ({I(C)↓ | (C) = }) of each component has the same structure Hi of the lower part of the corresponding component in NMn . It remains to analyse the structure of the posets ({I(C)↑ | (C) = }). We say that a 0-neg xi occurs in a sign  iff i = 1. It is immediate to see that if 1  = 2 are signs of the same length and with the same number of occurrences of 0-negs, then the order type of ({I(C)↑ | (C) = 1 }) is the same as the type of ({I(C)↑ | (C) = 2 }), since a renaming of variables carries one poset to the other. Let  be a sign of length i and with k many occurrences of 0-negs. We denote by Lki the order type of a poset ({I(C)↑ | (C) = }).

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First we notice that for any 0 k i, there are exactly 0-negs. Hence i n−1  i=0 k=0

i  k

signs of length i that have exactly k many occurrences of

   n    n i n j j k (Hi ⊕ 1 ⊕ Li ) (Hn ⊕ Lkn ). i k k k=0

To describe the type Lki of ({I(C)↑ | (C) = }), we first observe that this poset is a forest of upwards growing trees, and by Definition 7.8 each one of its trees is obtained by splitting a poset (CR ) for CR being the set of all upper semichains I(D)↑ with (D) = , that have as common prefix their bottom element R. We reason by induction on k. Clearly L0i = Hi , since a tree containing no occurrences of 0-negs does not split. Consider now any tree T in the forest ({I(C)↑ | (C) = }), for C a chain whose sign has length i and containing k > 0 many occurrences of 0-negs. We can display T as RT ⊕ T  for RT the root of T. Let CT be the set of all chains in {I(C)↑ | (C) = } having RT as bottom element, and let CT = {C \{RT }|C ∈ CT }. Then T  is the forest obtained by splitting (CT ). Further, observe that if in the splitting of (CT ) the root RT does not split, then the splitting of (CT ) is T. If RT does split, then the splitting is T (CT )G . Since the root RT is not empty, T  contains a total of h many occurrences of 0-negs and j − h many occurrences of 1-negs, for some 0 hj < i, h k. By induction hypothesis o(T  ) = Lhj . Hence o(T ) = (Lhj )⊥ .    In ({I(C)↑ | (C) = }) there are hk ji−k −h trees whose root has h many occurrences of 0-negs and j −h occurrences of 1-negs. To complete the construction we just have to take into account the cases the splitting of the tree (CT ) splits its root RT . This happens if and only if T  contains only occurrences of 0-negs and RT contains at least one 0-neg. In j those cases h = j < i, h < k, and the type of the splitting T (CT )G of (CT ) is a forest of two trees: (Lj )⊥

(Hj )⊥ . Then, if k > 0, the poset Lki has the form Lki =

k−1  j =0

  j   i−1  k k i−k (Hj )⊥ (Lhj )⊥ . j h j −h j =0 h=0

The proof is complete, for Hk =

k−1 k  j =0 j

(Hj )⊥ .



Corollary 7.18. The cardinality of Fn (NMG) is  n  n−1 i    n n i k k ( ( ) )( ) k i k (|S(Hn )| + |S(Ln )|) (|S(Hi )| + |S(Li )| + 1) · k=0

i=0 k=0

for |S(L0i )| = |S(Hi )| and |S(Lki )| = |S(Hk )|

j i−1  

k

i−k

(|S(Lhj )| + 1)(h)(j −h) .

j =0 h=0

j

Example 7.19. o(NMG1 ) = 4 3 (2 ⊕ (2 2)) (see Fig. 1). o(NMG2 ) = 3 2(5) 2(3 ⊕ (2 2)) (H2 ⊕ H2 ) j

j

2(H2 ⊕L12 ) (H2 ⊕L22 ), where H2 = 3 2 3, L12 = 3 2(2) (1⊕(2 2)) and L22 = (1⊕(2 2)) 3 2(2) 3 (1⊕ (2 2)). 7.1. NMG functions Consider the standard NMG algebra given by x  y = 0 if x  21 − y and x  y = x ∧ y otherwise. In this algebra ¬ 41 = 41 . Let DnG = {DC |C ∈ KnNMG }.

S. Aguzzoli, B. Gerla / Fuzzy Sets and Systems 159 (2008) 1131 – 1152

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Fig. 1. NMG1 .

Definition 7.20. A function f : [0, 1]n → [0, 1] is a NMG function over n-variables if satisfies the following conditions: (1) f is piecewise linear, and f coincides with one of the functions in Fn over each domain D ∈ DnNMG . (2) Sign: Let U, V ∈ KnNMG with U ∼ V and let  be any of {<, =, >}. If f DU  41 then f DV  41 . (3) Prefix: Let U, V ∈ KnNMG with U ∼ V and let W = {W1 < · · · < Wk } be the longest common prefix of U ↑ and V ↑ . Then f DU ∈ Wi iff f DV ∈ Wi and f DU ∈ ¬Wi iff f DV ∈ ¬Wi , for each i ∈ {1, . . . , k}. (4) Lower unification: Let I(C) be the NM(G)-chain in [C] and I(C)↓ = {{0} = W0 < W1 < · · · < Wk }. Display [C] as {C0 , . . . , Ch }, where Ci↓ = {Wi◦ < Wi+1 < · · · < Wh < · · · < Wk }. Then: For 0 i h and for all j ∈ {i, . . . , h}, f DI (C) ∈ Wi implies f DCj ∈ Wj◦ . Lemma 7.21. The free NMG algebra Fn (NMG) is isomorphic to the algebra of NMG functions over n variables equipped with the pointwise operations of the standard NMG algebra. Proof. With any NMG function f we associate the section [pB ]B∈B(NMGn ) such that for any B ∈ B(NMGn ), if B ↑ is the copy of an upper semichain C ↑ and f DC  41 then we set pB = f DC ; while if f DC < 41 by lower unification property it is enough to put pB = f DI (C) for C↓ the copy of B↓ . Vice versa, we write any section [pB ]B∈B(NMGn ) equivalently as [pB  ]B  ∈B , for B  the subset of B(NMGn ) containing all the branches that are copies of chains in KnNMG . We associate with [pB  ]B  ∈B the function f such that for any C ∈ KnNMG , f DC = pBC , where BC is the copy of C in B  . Then the argument runs as in the proof of Lemma 5.14, using Theorem 7.15 instead of Theorem 5.9.  Corollary 7.22. The variety V(NMG) is generated by the standard NMG algebra. Conclusions: To conclude the paper we include a table which reports for each logic L considered in the paper, the cardinality of its set Fn (L) of classes of logically equivalent formulas and the cardinality of the underlying nice poset Ln , for n = 1, 2, 3, where approximations are from below (it is easy to compute the exact values with any software for unbounded length arithmetic). n=1

n=2

n=3

|Fn (B)| |Bn |

4 4

16 8

256 16

|Fn (G)| |Gn |

6 5

342 17

137 186 159 382 77

|Fn (NM)| |NMn |

48 11

3 149 280 000 87

approx. 2.79 × 1070 845

|Fn (NMG)| |NMGn |

72 13

738 910 317 600 111

approx. 1.02 × 10104 1123

References [1] S. Aguzzoli, M. Busaniche, V. Marra, Spectral duality for finitely generated nilpotent minimum algebras with applications, J. Logic Comput. 17 (2007) 749–766.

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