Direct Product of `-algebras and Unification.An Application to Residuated Lattices

WOJCIECH DZIK1?, SANDOR RADELECZKI2†

1 Institute of Mathematics, University of Silesia, 40 007 Katowice, Poland2 Institute of Mathematics, University of Miskolc, 3515 Miskolc - Egyetemvaros,

Hungary. Phone: 00(36)(46) 361-951, Fax: 00(36)(46)565-174

We describe classes of `-algebras (which are based on lattices)such that their finitely presented projective algebras are closedunder finite direct product, that is, for which unification is filter-ing. This implies that unification in such classes is either unitaryor nullary. Following ideas of S. Ghilardi [12], [14] we attempt todescribe filtering unification in a variety by means of propertiesof factor-congruences of algebras of the variety. The results sub-sume some previous results, but not those of [14], and open newareas for applications like residuated lattices. In particular weshow that filtering unification depends on the monoid operation,that is, unification is filtering in varieties generated by residuatedlattices without zero divisors. This implies that unification instrict fuzzy logics like SMTL, ΠMTL and many others is unitaryor nullary.

Key words: unification, `-algebras, finitely presented algebra, projec-tive algebra, 1-regular variety, congruence kernel, central element, fac-tor congruence, residuated lattice, zero divisors.

? email: wojciech.dzik@us.edu.pl† email: matradi@uni-miskolc.hu

2000 Mathematics Subject Classification: 06B05, 06B10, 08A05

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1 INTRODUCTION

In this paper we provide algebraic conditions for filtering unification in a classof lattice-based algebras, and hence, for the fact that unification type in theclass is either unitary or nullary.

Unification in equational theories, called E-unification, is an importanttool in equational reasoning applied in automated theorem proving, in par-ticular in Automated Deduction and Term Rewriting, see [2], as well as inrecognizing admissibility of inference rules in logic.

Given an equational theory E, a pair of terms (t1, t2) is called a “unifi-cation problem” and a solution to it, or a unifier for it in E, is a substitutionσ such that `E σ(t1) = σ(t2). The terms t1, t2 are called unifiable if thereis a unifier for them. A substitution σ is more general than a substitution τ ,τ � σ, if there is a substitution θ such that `E θ ◦ σ ≈ τ .? .

A most general unifier, in short a mgu, for (t1, t2), is a unifier that is moregeneral than any unifier for (t1, t2). A mgu can be interpreted as “the best”solution to the unification problem. An equational theory E has unitary uni-fication, or type 1 unification, if for every unifiable terms t1, t2 there is a mgufor t1, t2 in E. Unification types can be unitary, finitary, infinitary or nullarydepending on number (in the ’worst’ case of pairs of terms) of maximal withrespect to� unifiers, cf. [2], [12]. A theory E has nullary unification, or type0, if there are unifiable terms t1, t2), such that for any unifier τ there is an-other unifier which is (essentially) more general then τ , i.e. a maximal unifierfor unifiable terms t1, t2 does not exist. Instead of an equational theoryE oneconsiders, equivalently, a corresponding equational class, a variety, V of al-gebras. For example, unification in Boolean algebras is unitary. Establishingthe unification type of a given variety or a logic is often a hard problem andthere are few general results like, for example S. Burris [3]: all discrimina-tor varieties have unitary unification. Also characterizing theories, or logics,with unitary (or finitary), unification is quite hard (see e.g. [1]). However S.Ghilardi and L. Sacchetti [14] managed to characterize modal logics in whichunification is filtering.

Unification in a variety V is called filtering if, for any pair of unifiers forgiven terms t, s there is a unifier for t, s which is more general than both ofthem, cf. [14]. If unification in V is filtering then it is either unitary or nullary.S. Ghilardi and L. Sacchetti [14] characterized normal modal logics extendingK4 with filtering unification by showing that for a variety V of K4-algebras,the following conditions are equivalent:

? Some authors use the reverse order and write σ � τ in this case.

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(i) unification in V is filtering,(ii) finitely presented projective V-algebras are closed under binary direct

products,(iii) ♦+�+x→ �+♦+x = 1 holds in V .Equivalence of (i) and (ii) is a more general algebraic fact that does not

depend on modal logic. We will investigate varieties of `-algebras in whichthe condition (ii) holds † . More exactly, we follow some ideas from S. Ghi-lardi and L. Sacchetti [14], but instead of modal K4-algebras which are not`-algebras we search for classes of lattice-based `-algebras in which the abovecondition (ii) holds. Our aim is to find conditions on factor-congruencesof algebras from a variety V of lattice based algebras of the form A =

(A,∧,∨, 0, 1, F ), whereF is a set of compatible operations, and (A,∧,∨, 0, 1)

is a bounded lattice, such that finitely presented projective V-algebras areclosed under binary direct product. As a corollary we get that unification inV is filtering, which means that unification is either unitary or nullary. Note,however, that K4-algebras are not `-algebras, hence S. Ghilardi and L. Sac-chetti result [14] does not follow from the results of this paper.

As an application we get that if the Stone identity ¬¬x ∨ ¬x = 1 holdsin a variety of residuated lattices, then unification in it is filtering. This al-gebraic description of lattice based algebras with filtering unification givessome purely algebraic insight into reasons for a particular unification type.

After preliminaries in Section 2 that include notions of the center of alattice, `-algebras, 1-regular algebras etc., the main result of the paper isachieved in two steps. The first step, in Section 3, is concerned with findingalgebraic conditions for a variety V of `-algebras such that the direct productof two finitely presented V-algebras is finitely presented. In the second step,in Section 4, extra conditions are added on a variety V such that the directproduct of two finitely presented projective V-algebras is again projective.

Since the main result holds for any variety V of lattice based `-algebras thepresent approach opens new areas of applications. In Section 5, we show for avariety V of bounded commutative integral residuated (even non-distributiveor non-modular) lattices, if subdirectly irreducible residuated lattices gener-ating the variety V have no zero divisors, or the Stone identity ¬¬x∨¬x = 1

holds in V , then unification in V is filtering. As a partial converse, if unifi-cation in a variety V of such residuated lattices is filtering and the identity¬x� ¬x = ¬x holds in V , then the Stone identity holds in V ‡ . Hence, uni-

† In the context of (ii) note that closure of finitely generated free algebras under binary prod-ucts is sufficient for filtering unification but is not necessary for it, see [14]

‡ Example A shows that ¬x � ¬x = ¬x does not imply the Stone identity, hence filtering

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fication in varieties of such residuated lattices validating the Stone identity isunitary or nullary. This helps to find the unification type of several fuzzy log-ics. In particular this applies to varieties of SMTL-algebras, ΠMTL-algebras(and include Product algebras and Godel algebras), to some Archimedeanclasses of residuated chains, see [16], and others.

2 PRELIMINARIES

Finding the unification type of a given equational theory, or a variety, is oftena difficult task. S. Ghilardi in [12] introduced algebraic and categorial ap-proach to unification that can make this task much easier. The main featureof Ghilardi’s approach is that unification depends only on finitely presentedand projective objects. We recall basic notions of this approach.

Given an equational theory E, let VE be the variety corresponding to E(we often write V). An algebra B ∈ VE is finitely presented if it is isomorphicto a finitely generated free algebra factorized by a finitely generated (or a com-pact) congruence; i.e. if there is a finite set of variables, {x1, . . . , xk} = x

and a finite set S of equations of terms with variables in x such that B is iso-morphic to a quotient algebra FE(x)/∼, where ∼ is a (compact) congruencedefined as follows:

t1 ∼ t2 iff S `E t1 ≈ t2

Thus ∼ is the congruence generated by the set of pairs in S. The pair (x, S)

is then called a presentation for B. Note that `E is substitution invariant butit is not allowed to produce substitution consequences.

An algebra P in a variety V is projective in V if for every A and B of Vand homomorphisms f : P→ B, g : A→ B (where g is regular epi; here inVE it is a surjective homomorphism) there is a homomorphism h : P → A

such that the following diagram commutes:

P

A P

h

g

f

It is known, see e.g. [12], [14], that for a finitely presented algebra P ∈ VEthe following are equivalent:

unification is essential here

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(i) P is projective in V(ii) P is a retract of a free algebra FE(x) for a finite set x in V , i.e. there

are morphisms q and m and a free algebra FV (x) such that q ◦m = 1 (q isonto, m is one–to–one).

P FV (x) Pm q

In the approach of Ghilardi [12] an E-unification problem corresponds toa finitely presented algebra A ∈ VE . A unifier (a solution) for A is a pairgiven by: a projective algebra P and a homomorphism u : A→ P .

A finitely presented algebra A is unifiable if it has a unifier. Given twounifiers u1 and u2 for A, u1 : A → P1 is a more general algebraic unifierthan u2 : A → P2, u2 � u1, if there is a homomorphism g such that thefollowing diagram commutes:

A

P2 P1

u1

g

u2

After finding, in Section 3, conditions for a variety V implying that finitelypresented projective V-algebras are closed under (binary) direct product wemake use of the following purely algebraic result of Ghilardi and Sacchetti[14], Theorem 3.2.

Theorem 1 (see [14]). Unification in V is filtering iff finitely presented pro-jective V-algebras are closed under (binary) direct product.

To give an idea of the proof, assume that finitely presented projective V-algebras are closed under finite direct product. Then, for any two unifiers forA, u : A → P1 and v : A → P2, the unifier given by the direct productf : A → P1 × P2, is more general than u, v, as shown in the followingdiagram

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A

P1 P1 × P2 P2

u f v

π1 π2

Let L be a bounded lattice and let 0 and 1 stand for the least and thegreatest element of L. We denote the principal filter of an element a ∈ L

by [a). An element a ∈ L is called a central element of L if a is comple-mented and for all x, y ∈ L the sublattice generated by {a, x, y} is distribu-tive. The central elements of L form a Boolean sublattice of the lattice Ldenoted by Cen(L). Hence any element c ∈ Cen(L) has a single complementc ∈ Cen(L); the pair {c, c} is also called a central pair ofL. Any c ∈ Cen(L),induces a congruence

θc = {(x, y) | x ∧ c = y ∧ c}.

of L. Notice, that [1]θc = [c). It is well-known (see e.g. [15]) that for any c ∈Cen(L), θc and θc form a factor congruence pair ofL and conversely, if θ1 andθ2 are factor congruences of a bounded lattice L, that is, L ∼= L/θ1 × L/θ2,then there exists a c ∈ Cen(L) such that θ1 = θc, θ2 = θc. Moreover,L/θ1 ∼= [c) and L/θ2 ∼= [c).

An algebra A = (L,∧,∨, 0, 1, F ) is called an `-algebra if (L,∧,∨, 0, 1)

is a bounded lattice, and any n-ary term f : Ln → L of it is center-preserving,that is, for every c ∈ Cen(L),

(xi, yi) ∈ θc, i = 1, ..., n implies (f(x1, ..., xn), f(y1, ..., yn)) ∈ θc.

The best known examples of `-algebras are bounded lattices, p-algebras, or-tholattices and Heyting algebras (see e.g. [18] and [22]). On the other hand,S4-modal and K4-modal algebras (A,∧,∨,→,�,♦, 0, 1) are not `-algebras.Since `-algebras have a lattice reduct, they are congruence distributive, andthe factor congruences of an `-algebra and of its underlying latticeL coincide.An `-algebra will be called distributive, if its underlying lattice is distributive.

Let A = (A,F ) be an algebra and B ⊆ A a nonempty set. The congru-ence θ(B) =

⋂{θ ∈ ConA|B2 ⊆ θ} is the least congruence θ ∈ ConA with

the property B2 ⊆ θ, and it is called the congruence generated by the set B.It is well-known that

θ(B) =∨{θ(a, b)|a, b ∈ B, a 6= b}. (1)

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A congruence θ of ConA is said to be compact if for every (nonempty) subsetΦ of ConA, θ ≤

∨Φ implies θ ≤

∨F for some finite nonempty F ⊆ Φ.

Observe, that this definition and (1) yields θ = θ(a1, b1)∨ ...∨ θ(an, bn), forsome n ∈ N and (a1, b1), ..., (an, bn) ∈ θ.

An algebra A = (A,F ) with a constant 1 is called 1-regular (or weaklyregular) (see [4] or [6]), if [1]ϕ = [1]θ implies ϕ = θ, for each ϕ, θ ∈ ConA.A variety V with a constant 1 is 1-regular if eachA ∈ V is 1-regular. Clearly,any congruence ϕ of a 1-regular algebra A is generated by its congruenceclass [1]ϕ, i.e. ϕ = θ([1]ϕ). From B. Csakany characterization of regularvarieties [4] it follows that

Theorem 2. A variety V with a constant 1 is 1-regular if and only if thereexist binary terms d1(x, y), ..., dn(x, y) such thatdi(x, x) = 1, for i = 1, ..., n, andd1(x, y) = 1,...,dn(x, y) = 1 imply x = y.

Hence in a 1-regular variety we haved1(x, y) = 1,...,dn(x, y) = 1⇐⇒ x = y.

Corollary 3. Let V be a variety with a constant 1 which is 1-regular. Then

(1) Every equation on terms t1 = t2 in V can be replaced by an equivalentfinite set of equations of the form t = 1

(2) For any finitely presented algebra in V any equation in the set S of itspresentations (x, S) can be taken to be an equation of the form t = 1.

(3) If V has a meet-operation ∧, then all finitely generated congruences ofits algebras A are principal of a specific form: θ(s, 1).

Proof. (1) and (2) are clear. In case of (3) by defining a binary term d(x, y) =

d1(x, y) ∧ ... ∧ dn(x, y), in view of Theorem 2 we obtain d(x, y) = 1 ⇐⇒x = y. As any finitely generated congruence θ ∈ ConA has the form θ =

θ({s1 = t1, ..., sn = tn}), we obtain θ = θ(d(s1, t1) ∧ ... ∧ d(sn, tn), 1). �

It is also clear, that by adding new operations to the operations of an algebraA from a 1-regular variety, we obtain again a 1-regular algebra A′.

Next, let (L,∧,∨, 1) be a lattice with the greatest element 1, and A =

(L,∧,∨, 1, F ) be an algebra. Since any θ ∈ ConA is also a congruence of(L,∧,∨), it follows that the congruence class [1]θ is a filter of L, called thekernel of θ. We will denote it by ker(θ). If A is 1-regular, then obviously,θ1 ≤ θ2⇔ ker(θ1) ⊆ ker(θ2), for any θ1, θ2 ∈ ConA.

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Lemma 4. Let A = (L,∧,∨, 0, 1, F ) be a 1-regular algebra. Then anycompact congruence θ ∈ ConA is generated by a principal filter of L andfor any elements x, a, b ∈ L we have

θ([x)) = θ(x, 1), (2)

θ(a, 1) ∨ θ(b, 1) = θ(a ∧ b, 1). (3)

Proof. θ(x, 1) ≤ θ([x)) is clear. If y, z ∈ [x), then x ≤ y, z imply (y, 1) =

(x ∨ y, 1 ∨ y) ∈ θ(x, 1) and (z, 1) = (x ∨ z, 1 ∨ z) ∈ θ(x, 1), whence(y, z) ∈ θ(x, 1). Thus we get θ([x)) ≤ θ(x, 1), by the definition of θ([x)).This proves (2). As any compact congruence θ ofA by Corollary 3(3) has theform θ = θ(s, 1), we get that θ = θ([s)).

Since a, b ∈ [a∧ b), we obtain θ(a, 1)∨ θ(b, 1) ≤ θ([a∧ b)) = θ(a∧ b, 1).Because (a ∧ b, 1) = (a, 1) ∧ (b, 1), (a, 1), (b, 1) ∈ θ(a, 1) ∨ θ(b, 1) imply(a∧b, 1) ∈ θ(a, 1)∨θ(b, 1), whence we get also θ(a∧b, 1) ≤ θ(a, 1)∨θ(b, 1).Thus (3) holds true. �

In view of Lemma 4, if a lattice based variety V with a constant 1 is 1-regular, then the compact congruences in every free algebra FV (x) ∈ V aregenerated by the principal filters [t(x)) corresponding to a single term t(x).Hence, denoting the congruence induced by the filter [t) by θ[t), we obtain

Corollary 5. Let V be a variety of algebras of the formA = (L,∧,∨, 0, 1, F )

which is 1-regular. Then a finitely presented algebra of V can be represented(up to isomorphism) by a quotient FV (x)/θ[t), where t = 1 corresponds toequations in the presentation S that define the congruence ∼ in FV (x).

3 CLOSURE OF FINITELY PRESENTED ALGEBRAS UNDERDIRECT PRODUCTS

In this Section we will consider 1-regular varieties V of `-algebras of the formA = (L,∧,∨, 0, 1, F ) satisfying the following properties:

(A) Each (L,∧,∨, 0, 1) is a bounded lattice where every complemented ele-ment a ∈ L is central.

(B) Each algebra A has a unary term g such that for any v ∈ Lv ∧ g(v) = 0 and g(0) = 1,

and we will prove that the direct product of two finitely presented algebrasfrom this variety V is also a finitely presented algebra. Observe, that condition

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(A) is related only with the underlying lattices of these algebras and it issatisfied obviously if these lattices are distributive ones.

We will need the following known facts in our proofs. Let ϕ ∈ ConA.Then to any congruence θ ⊇ ϕ corresponds a factor congruence

θ/ϕ = {([a]ϕ, [b]ϕ) ∈ (A/ϕ)2 | (a, b) ∈ θ}

of the factor algebra A/ϕ; moreover, the mapping

α : [ϕ,OA] −→ ConA/ϕ, where α(θ) = θ/ϕ

is a lattice isomorphism. Clearly, α maps ϕ into 4ϕ = ϕ/ϕ, the 0-elementof ConA/ϕ. It is easy to check that for any a, b ∈ A with ϕ ≤ θ(a, b),θ(a, b)/ϕ = θ([a]ϕ, [b]ϕ) in ConA/ϕ. First, we prove the following:

Lemma 6. If V is a 1-regular variety of `-algebras satisfying condition (A),then for any a, b ∈ L with a ∧ b = 0 we have

θ(a, 1) ∧ θ(b, 1) = θ(a ∨ b, 1),

and the congruences θ(a, 1)/θ(a∨ b, 1) and θ(b, 1)/θ(a∨ b, 1) form a factorcongruence pair of the factor algebra A/θ(a ∨ b, 1).

Proof. Let ψ = θ(a ∨ b, 1). Since a ∨ b ∈ [a) ∩ [b), and θ(a, 1) = θ[a),θ(b, 1) = θ[b) by (2), we get (a ∨ b, 1) ∈ θ(a, 1) ∧ θ(b, 1), i.e. ψ ≤ θ(a, 1) ∧θ(b, 1). Then ψ = OA implies θ(a, 1)∧θ(b, 1) = OA, and we are done. Thuswe may assume ψ 6= OA. Now, consider the factor lattice L/ψ. We obtain:

[a]ψ ∧ [b]ψ = [0]ψ and [a]ψ ∨ [b]ψ = [a ∨ b]ψ = [1]ψ .

Hence [a]ψ and [b]ψ are the complements of each other in L/ψ, therefore,they are central elements in L/ψ and define a factor congruence pair

%1 = {([x]ψ, [y]ψ) ∈ (L/ψ)2 | [x]ψ ∧ [a]ψ = [y]ψ ∧ [a]ψ},

%2 = {([x]ψ, [y]ψ) ∈ (L/ψ)2 | [x]ψ ∧ [b]ψ = [y]ψ ∧ [b]ψ}.

of L/ψ. Since A/ψ = (L/ψ,∧,∨, [0]ψ, [1]ψ, F ) ∈ V is an `-algebra, %1, %2are also factor congruences of the algebra A/ψ. As ker(%1) = [[a]ψ) andA/ψ is 1-regular, by using (2) we get %1 = θ([[a]ψ)) = θ([a]ψ, [1]ψ) =

θ(a, 1)/ψ. Similarly, using ker(%2) = [[b]ψ) we get %2 = θ([b]ψ, [1]ψ) =

θ(b, 1)/ψ. Then

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(θ(a, 1) ∧ θ(b, 1))/ψ = θ(a, 1)/ψ ∧ θ(b, 1)/ψ = %1 ∧ %2 = 4ψ = ψ/ψ.Since the mapping α(θ) = θ/ψ is one to one, we deduceθ(a, 1) ∧ θ(b, 1) = ψ = θ(a ∨ b, 1),

and that %1 = θ(a, 1)/θ(a ∨ b, 1) and %2 = θ(b, 1)/θ(a ∨ b, 1) form a factorcongruence pair of the algebra A/ψ = A/θ(a ∨ b, 1). �

Theorem 7. Let V be a 1-regular variety of `-algebras satisfying the con-ditions (A) and (B). If A,B ∈ V are finitely presented algebras, then theirdirect product A× B is also finitely presented.

Proof. By definition, A and B can be represented in the form F(m)/θ1and F(n)/θ2, where θ1, θ2 are two compact congruences on free algebrasF(m) = F(x) ∈ V and F(n) = F(y) ∈ V , respectively. We are going toprove that there is a compact congruence ϕ on a free algebra F(m+n+1) =

F(x, y, v) such that:

F(m)/θ1 ×F(n)/θ2 ∼= F(m+ n+ 1)/ϕ

We can present the two free algebras, given at the beginning, as follows:F(m)/θ1 ∼= F(m+ n)/ϕ1 and F(n)/θ2 ∼= F(m+ n)/ϕ2,

where ϕ1, ϕ2 ∈ ConF(m + n). Clearly, ϕ1 = θ1 ∨ θ({(y1, 1), ..., (yn, 1)})and ϕ2 = θ2∨θ({(x1, 1), ..., (xm, 1)}). Since θ1 and θ2 are compact congru-ences, by Corollary 3(3) they have the form θ1 = θ(s1, 1) and θ2 = θ(s2, 1),where s1 ∈ F(x) and s2 ∈ F(y). By using Lemma 4 we obtain:

ϕ1 = θ(s1 ∧ y1 ∧ ... ∧ yn, 1) and ϕ2 = θ(s2 ∧ x1 ∧ ... ∧ xm, 1).

Hence ϕ1 and ϕ2 are compact congruences. Let t1(x, y) = s1 ∧ y1 ∧ ...∧ ynand t2(x, y) = s2 ∧ x1 ∧ ... ∧ xm. Then t1(x, y), t2(x, y) ∈ F(m+ n), andby (2) we get ϕ1 = θ[t1(x, y)) and ϕ2 = θ[t2(x, y)). By using (2) and (3)we obtain θ[t1(x, y) ∧ v) = θ(t1(x, y), 1) ∨ θ(v, 1) = θ(v, 1) ∨ θ([t1(x, y)).Now let g be the unary term from condition (B). Using (2) and (3) we get:θ[t2(x, y)∧g(v)) = θ(t2(x, y), 1)∨θ(g(v), 1) = θ(g(v), 1)∨θ([t2(x, y)). Asin view of (B), v ∧ g(v) = 0, we get (0, v) = (v ∧ g(v), v ∧ 1) ∈ θ(g(v), 1)).Therefore, one can deduce:

Claim 8.(i) F(m+ n+ 1)/θ[t1(x, y) ∧ v) ∼= F(m+ n)/θ[t1(x, y))

and(ii) F(m+ n+ 1)/θ[t2(x, y) ∧ g(v)) ∼= F(m+ n)/θ[t2(x, y)).

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As a consequence of Claim 8. we deduce

F(m+ n+ 1)/θ[t1(x, y) ∧ v) ∼= F(m+ n)/ϕ1∼= F(n)/θ1; (6)

F(m+ n+ 1)/θ[t2(x, y) ∧ g(v)) ∼= F(m+ n)/ϕ2∼= F(n)/θ2. (7)

Let us consider now the term t =(t1(x, y) ∧ v

)∨(t2(x, y) ∧ g(v)

)and

the principal congruence ψ = θ(t, 1) = θ [t) of the free algebraF(m+n+1).Now, we are going to prove

F(m)/θ1 ×F(n)/θ2 ∼= F(m+ n+ 1)/ψ. (∗)

Indeed, let us consider the elements t1(x, y) ∧ v and t2(x, y) ∧ g(v) inF(m + n + 1) ∈ V . By using condition (B), we get

(t1(x, y) ∧ v

)∧(

t2(x, y) ∧ g(v))

= t1(x, y)∧t2(x, y)∧(v∧g(v)) = 0. Then the assumptionand Lemma 6 imply thatθ(t1(x, y) ∧ v, 1) ∧ θ(t2(x, y) ∧ g(v), 1) =

= θ((t1(x, y) ∧ v

)∨(t2(x, y) ∧ g(v)

), 1)

= θ(t, 1) = ψ,and that F(m+ n+ 1)/ψ is isomorphic to(F(m+n+1)/ψ)/(θ(t1(x, y)∧v, 1)/ψ)×(F(m+n+1)/ψ)/(θ(t2(x, y)∧g(v), 1)/ψ).In view of the second isomorphism theorem of algebras we get(F(m+n+1)/ψ)/(θ(t1(x, y)∧v, 1)/ψ) ∼= F(m+n+1)/θ(t1(x, y)∧v, 1),(F(m+ n+ 1)/ψ)/(θ(t2(x, y) ∧ g(v), 1)/ψ) ∼=∼= F(m+ n+ 1)/θ(t2(x, y) ∧ g(v), 1).Since F(m+ n+ 1)/θ(t1(x, y) ∧ v, 1) = F(m+ n+ 1)/θ[t1(x, y) ∧ v) ∼=F(m)/θ1 andF(m+n+1)/θ(t2(x, y)∧g(v), 1) = F(m+n+1)/θ[t2(x, y)∧g(v)) ∼= F(n)/θ2 according to (6) and (7), we obtain F(m + n + 1)/ψ ∼=F(m)/θ1 ×F(n)/θ2, thus (∗) holds.

Hence F(m+ n+ 1)/ψ ∼= A× B. Since the congruence ψ is defined bya single term t ∈ F(m + n + 1), the algebra F(m + n + 1)/ψ is finitelypresented, and this completes the proof of our theorem. �

4 CLOSURE OF FINITELY PRESENTED PROJECTIVE ALGEBRASUNDER DIRECT PRODUCTS

Let V be a variety of `-algebras, and consider the following condition:

(C) Each algebra A = (L,∧,∨, 0, 1, F ) ∈ V has two unary terms h and h′,such that for every v ∈ L, h(v) and h′(v) are the complements of eachother in the lattice L, and h(0) = h′(1) = 1.

11

Clearly from condition (C) it follows also:h(1) = h′(0) = 0.

Our aim is to prove the following

Theorem 9. Let V be a 1-regular variety of `-algebras satisfying the condi-tions (A), (B) and (C). If A,B ∈ V are finitely presented projective algebras,then A× B is also a finitely presented projective algebra of V .

In order to simplify our proof¶ we introduce the notations:

x− = x∧h′(v), x+ = x∧h(v) and a∗b = a−∨b+ = (a∧h′(v))∨(b∧h(v))

for arbitrary a, b, v, x ∈ L. Then clearly, x−− = x− and x++ = x+.

First, we will prove

Lemma 10. Let V be a variety of `-algebras satisfying the conditions (A)and (C). Then for any k-ary polynomial f the identity

f(x1, ..., xk) ∗ f(y1, ..., yk) = f(x1 ∗ y1, ..., xk ∗ yk)

is satisfied in V .

Proof. The assumptions (A) and (C) imply that in any algebra (L,∧,∨, 0, 1, F )

from V the elements h(v) and h′(v) form a central pair of the underlying lat-tice L, and hence they define a factor congruence pair ≡(1), ≡(2) of L asfollows:x ≡(1) y ⇔ x ∧ h(v) = y ∧ h(v)⇔ x+ = y+,x ≡(2) y ⇔ x ∧ h′(v) = y ∧ h′(v)⇔ x− = y−.

Since A = (L,∧,∨, 0, 1, F ) is an `-algebra, by definition any factor congru-ence of L is preserved by any polynomial of A, therefore, fromx1 ≡(1) x1

+, ..., xk ≡(1) xk+ it follows f(x1, ...xk) ≡(1) f(x1

+, ..., xk+),

that is,f(x1, ...xk)

+= f(x1

+, ..., xk

+)+. (8)

Similarly we prove:

f(x1, ...xk)−

= f(x1−, ..., xk

−)−. (9)

Because h(v) and h′(v) form a central pair of L, for any a, b ∈ L we obtain:

(a∧h′(v))∨(b∧h(v)) ≡(1) b∧h(v) and (a∧h′(v))∨(b∧h(v)) ≡(2) a∧h′(v).

¶ We wish to thank the referee for suggesting the simplification in the proof that follow.

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In other words, we get:

a ∗ b ≡(1) b+ and a ∗ b ≡(2) a

−. (10)

These relations imply:

(a ∗ b)+ = b++ = b+ and (a ∗ b)− = a−− = a−. (11)

Hence in view of (11) we have

(f(x1, ..., xk) ∗ f(y1, ..., yk))+ = (f(y1, ..., yk))+, (12)

(f(x1, ..., xk) ∗ f(y1, ..., yk))− = (f(x1, ..., xk))−. (13)

(10) yields also that the relations

xi ∗ yi ≡(1) yi+ and xi ∗ yi ≡(2) xi

− hold for all i ∈ {1, ..., k}.Since the factor congruences ≡(1) and ≡(2) are preserved by f , we get thatf(x1 ∗ y1, ..., xk ∗ yk) ≡(1) f(y1

+, ..., yk+), and

f(x1 ∗ y1, ..., xk ∗ yk) ≡(2) f(x1−, ..., xk

−),whence by using (8) and (9) we obtain

f(x1 ∗ y1, ..., xk ∗ yk)+ = f(y1+, ..., yk

+)+ = f(y1, ..., yk)+, (14)

f(x1 ∗ y1, ..., xk ∗ yk)− = f(x1−, ..., xk

−)− = f(x1, ..., xk)−. (15)

Now, from (12) – (15) we infer:

f(x1 ∗ y1, ..., xk ∗ yk)+ = (f(x1, ..., xk) ∗ f(y1, ..., yk))+, (16)

f(x1 ∗ y1, ..., xk ∗ yk)− = (f(x1, ..., xk) ∗ f(y1, ..., yk))−. (17)

Finally, observe that the maps x 7→ x−, x ∈ L and x 7→ x+, x ∈ L areenough to recover an element x ∈ L. Indeed, since h(v), h′(v) is a centralpair in L, for each x ∈ L the equality x = (x∧h′(v))∨(x∧h(v)) = x−∨x+holds. Hence for any a, b ∈ L, if a− = b− and a+ = b+, then

a = a− ∨ a+ = b− ∨ b+ = b.

Therefore, in view of this observation (16) and (17) imply:

f(x1 ∗ y1, ..., xk ∗ yk) = f(x1, ..., xk) ∗ f(y1, ..., yk),

and this is exactly the identity that we intended to prove. �

Proof of Theorem 9. Let A,B ∈ V two finitely presented projective algebras.Then, by Theorem 7, D = A × B is finitely presented. Therefore, as we

13

pointed out in Section 2, to prove that D is projective, it suffices to show thatit is a retract of a free algebra of V . Because A and B are retracts of somefinitely generated free algebras F(m) and F(n), respectively, D is a retractof a direct product F(m)×F(n). Clearly, by showing that F(m)×F(n) isa retract of a free algebra F ∈ V , it follows that D is also a retract of F , andour proof is finished.

Hence, in what follows, we will show that F(m)×F(n) is a retract of thefree algebra F(m + n + 1), that is, in case of free algebras F(m) = F(x),F(n) = F(y) and F(m+n+1) = F(x, y,v) there are two homomorphismsm and q such that

m : F(m)×F(n) −→ F(m+ n+ 1),q : F(m+ n+ 1) −→ F(m)×F(n),

and q ◦m = id (on F(m)×F(n)).

In order to prove this, let us define for arbitrary t1(x) ∈ F(x), t2(y) ∈ F(y):m((t1(x), t2(y))

)= t1(x) ∗ t2(y),

and consider the mappingsτ1(xi) = xi, 1 ≤ i ≤ m; τ1(yj) = 1, 1 ≤ j ≤ n; τ1(v) = 1 andτ2(xi) = 0, 1 ≤ i ≤ m; τ1(yj) = yj , 1 ≤ j ≤ n; τ1(v) = 0.

Then there are some (unique) homomorphisms σ1 : F(m+ n+ 1)→ F(m)

and σ2 : F(m+ n+ 1)→ F(n), such that σ1 is an extension of τ1 and σ2 isan extension of τ2, respectively. Now, we defineq(t(x, y, v)) :=

(σ1(t(x, y, v)), σ2(t(x, y, v))

).

Clearly, q : F(x, y, v) −→ F(x)×F(y) is a homomorphism, andq(t(x, y, v)) = (t(τ1(x), τ1(y), τ1(v)), t(τ2(x), τ2(y), τ2(v)).

According to the definition of τ1 and τ2 we can use the shortened notation:q(t(x, y, v)) = (t(x, y/1,v/1), t(x/0, y,v/0)).

First, let us prove that q ◦m is equal to the identity map on F(m)×F(n).Take (t1(x), t2(y)) ∈ F(x)×F(y) arbitrary. Then (q ◦m)((t1(x), t2(y)) =

= q(t1(x) ∗ t2(y)) = q((t1(x) ∧ h′(v)) ∨ (t2(y) ∧ h(v))

)=

= ((t1(x)∧h′(1))∨(t2(y/1)∧h(1)), (t1(x/0)∧h′(0))∨(t2(y)∧h(0))).Since h′(1) = 1, h(1) = 0 and h′(0) = 0, h(0) = 1, we obtain

(q ◦m)((t1(x), t2(y)) = (t1(x), t2(y)).

Now, it remains to show only that m is a homomorphism, that is, any k-arypolynomial f of F(m)×F(n), commutes with m. Let (t

(1)1 (x), t

(1)2 (y)),...,

(t(k)1 (x), t

(k)2 (y)) ∈ F(m)×F(n). Then by definition

f((t(1)1 (x), t

(1)2 (y)),..., (t(k)1 (x), t

(k)2 (y))) =

= (f(t(1)1 (x), ..., t

(k)1 (x)), f(t

(1)2 (y), ..., t

(k)2 (y)), so we have to prove that

14

m(f(t(1)1 (x), ..., t

(k)1 (x)), f(t

(1)2 (y), ..., t

(k)2 (y)) =

= f(m(t(1)1 (x), t

(1)2 (y)), ...,m(t

(k)1 (x), t

(k)2 (y))), that is,

f(t(1)1 (x), ..., t

(k)1 (x)) ∗ f(t

(1)2 (y), ..., t

(k)2 (y)) =

= f(t(1)1 (x) ∗ t(1)2 (y), ..., t

(k)1 (x) ∗ t(k)2 (y)). (18)

Since V satisfies conditions (C) and (A), we can apply Lemma 10 with x1 =

t(1)1 (x), ..., xk = t

(k)1 (x), y1 = t

(1)2 (y), ..., yk = t

(k)2 (y), and this yields (18).

Hence m is a homomorphism, and this completes our proof. �

Since any bounded distributive lattices satisfies condition (A), the follow-ing consequence is immediate:

Corollary 11. Let V be a variety of 1-regular distributive `-algebras sat-isfying conditions (B) and (C). If A,B ∈ V are finitely presented projectivealgebras, then A× B is also a finitely presented projective algebra.

Finally, by applying the result of Ghilardi and Sacchetti [14], i.e. Theorem 1,we obtain:

Theorem 12. Let V be a variety of 1-regular `-algebras. If the conditions(A), (B) and (C) are satisfied byV , then unification in V is filtering.

An existing application for Heyting algebras. We observe that the existingresult for Heyting algebras, see [6], [7], is a special case of the above Theorem12. As we pointed out in the preliminaries, Heyting algebras are `-algebras.All Heyting algebras contain the constant 1 and the operation→ satisfies thecondition

x ≤ y ⇔ x→ y = 1,

thus they are also 1-regular. Since Heyting algebras are based on boundeddistributive lattices, they clearly satisfy condition (A). Hence, to apply ourresults, it remains to check conditions (B) and (C). Consider the variety H ofHeyting algebras (A,∧,∨,→, 0, 1). Since ¬v = v → 0, for all v ∈ A, Hclearly satisfies condition (B) with g(v) = ¬v (a pseudocomplement). Nowlet S be a subvariety of H which satisfies the Stone identity:

¬¬x ∨ ¬x = 1.

Then condition (C) is satisfied by setingh(v) = ¬v, h′(v) = ¬¬v.Hence, the unification in S is filtering.

15

Moreover, since (Th(¬¬x ∨ ¬x = 1), Th(22 ⊕ 1)) is the splitting pairfor the lattice of theories of Heyting algebras one gets that: unification in avariety V ⊆ H is filtering iff the Stone identity holds in V , see [6].

The advantages of the present results are that they are preserved by expan-sions of algebras with compatible operations (note that any compatible op-eration is center-preserving), they can be applied to algebras without EDPCand to algebras based on non-distributive lattices, as shown below.

5 FILTERING UNIFICATION IN SOME VARIETIES OF RESIDU-ATED LATTICES

An algebra L = (L,∧,∨,�,→, 0, 1) is called an integral bounded commu-tative residuated lattice, IBCRL or simply bounded residuated lattice, if

(R1) (L,∧,∨, 0, 1) is a bounded lattice;(R2) (L,�) is a commutative monoid with unit element 1;(R3) x� y ≤ z ⇔ x ≤ y → z, for all x, y, z.

It is known that the following distributivity rule holds, see e.g. [11]:(x ∨ y)� z = x� z ∨ y � z,

moreover, we define ¬x = x→ 0. The variety IBCRL is 1-regular, and it wasproved in [20] that in bounded residuated lattices, although the underlyinglattices need not to be distributive, the complements are unique. In whatfollows, we are going to prove that any bounded residuated lattice is an `-algebra satisfying condition (A). First, we present some known properties ofthese algebras.The results of Kowalski and Ono [20]; Lemma 1.3. and [20]; Lemma 1.6.,see also [11], can be summarized as the following Kowalski - Ono Lemma:

Lemma 13. Let (L,∧,∨,�,→, 0, 1) be a bounded residuated lattice and leta ∈ L have a complement b ∈ L. Then the following hold:

(i) If c is a complement of a in L then c = b;(ii) ¬a = b and ¬b = a;(iii) a2 = a;(iv) a� x = a ∧ x, for any x ∈ L.

As an immediate consequence we obtain:

Corollary 14. If a ∈ L is a complemented element of a residuated lattice(L,∧,∨,�,→, 0, 1), then a∧(x∨y) = (a ∧ x)∨(a∧y), for every x, y ∈ L.

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Proof. Take any x, y ∈ L. In view of Lemma 13(iv) we obtain:a ∧ (x ∨ y) = a� (x ∨ y) = a� x ∨ a� y = (a ∧ x) ∨ (a ∧ y). �

An i-filter (implicative filter) of an (integral commutative) residuated latticeL = (L,∧,∨,�,→, 1) is a nonempty subset F ⊆ L which satisfies thefollowing conditions:

(a) F is an order filter of the lattice L;(b) If x, y ∈ F then x� y ∈ F .

In view of [5] an equivalence θ ⊆ L × L is a congruence of the algebra L ifand only if the θ-equivalence class θ[1] is an i-filter, and this is equivalent to

(x, y) ∈ θ ⇔ (x→ y) ∧ (y → x) ∈ θ[1].

Now let z, z ∈ L be the complements of each other in a bounded lattice L. Inview of [21] (see also [23]), z, z ∈ Cen(L) if and only if the following hold:

(c) x = (x ∧ z) ∨ (x ∧ z), for any x ∈ L;(d) z ∧ (u ∨ z) = u, for any u ≤ z and z ∧ (v ∨ z) = v, for any v ≤ z.

Theorem 15. Any bounded residuated lattice is an `-algebra satisfying con-dition (A).

Proof. Let L = (L,∧,∨,�,→, 0, 1) be a bounded residuated lattice andc ∈ L a central element of the lattice L. We are going to prove that thecongruence θc = {(x, y) ∈ L2 | x ∧ c = y ∧ c} of L is also a congruenceof the algebra L. First, observe that θc[1] = {x ∈ L | x ∧ c = c} = [c),that is, θc[1] is principal filter of L. Now take x, y ∈ θc[1]. Then c ≤ x, y.Since c is a complemented element of L, by using Lemma 13(iv) we getc = c ∧ y = c� y ≤ x� y, and hence x� y ∈ [c) = θc[1]. Thus θc[1] is ani-filter in L. Now let (x, y) ∈ θc. Then x ∧ c = y ∧ c implies x� c = y � c.We claim that

(x→ y) ∧ (y → x) ≥ c.

Indeed, c� x = y � c = y ∧ c ≤ y, yields c ≤ x→ y and c� y = c� x =

c ∧ x ≤ x gives c ≤ y → x. Thus (x→ y) ∧ (y → x) ≥ c.Conversely, suppose (x → y) ∧ (y → x) ≥ c. Then c ≤ x → y andc ≤ y → x imply c ∧ x = c � x ≤ y and c ∧ y = c � y ≤ x. Hencex ∧ c ≤ y ∧ c and y ∧ c ≤ x ∧ c, thus we get x ∧ c = y ∧ c, i.e. (x, y) ∈ θc.

As we proved that (x, y) ∈ θc ⇔ (x → y) ∧ (y → x) ∈ [c) = θc[1], thismeans that θc is a congruence of the algebra L. Therefore, L is an `-algebra.

17

Further, let a, a be the complements of each other in the lattice L. To provethat L satisfies condition (A), we have to show only that a, a ∈ Cen(L). Wewill show this by verifying that conditions (d) and (c) hold.Indeed, let u ≤ a and v ≤ a. Then, in view of Corollary 14, we obtain:a ∧ (u ∨ a) = (a ∧ u) ∨ (a ∧ a) = u anda ∧ (v ∨ a) = (a ∧ v) ∨ (a ∧ a) = v,

and hence (d) is satisfied.Finally, to prove (c) take any x ∈ L. Since both a and a are complemented

elements, in view Lemma 13(iv), we have x� a = x ∧ a and x� a = x ∧ a.Thus we can write:x = x� 1 = x� (a ∨ a) = x� a ∨ x� a = (x ∧ a) ∨ (x ∧ a).

Since this proves (c), in view of [21] we obtain a, a ∈ Cen(L), and hence Lsatisfies condition (A). �

Now let L = (L,∧,∨,�,→, 0, 1) be a residuated lattice satisfying

¬x ∧ x = 0, for all x ∈ L. (P)

Such a residuated lattice is always pseudocomplemented, i.e. for any x, y ∈ L

y ∧ x = 0⇔ y ≤ ¬x

see e.g. [5] . Therefore, a bounded residuated lattice is called pseudocomple-mented if it satisfies condition (P). Observe, that defining g(v) = ¬v, for allv ∈ L condition (B) holds for L and hence we obtain:

Corollary 16. If A,B are finitely presented pseudocomplemented residuatedlattices then A× B is also finitely presented.

A residuated lattice L (or a variety) is called Stonean if it satisfies the Stoneidentity

¬¬x ∨ ¬x = 1, for all x ∈ L. (S)

Since x ∧ ¬x ≤ ¬¬x ∧ ¬x = ¬(¬x ∨ ¬¬x) = ¬1 = 0, each Stoneanresiduated lattice is pseudocomplemented and, obviously, bounded. Hencewe have:

Theorem 17. Let V be a variety of Stonean (integral commutative) residuatedlattices. Then the unification in V is filtering.

Proof. By Theorem 15 any algebra L ∈ V is a 1-regular `-algebra which sat-isfies condition (A). If axiom (S) is satisfied then L is pseudocomplemented,and hence condition (B) is satisfied with g(v) = ¬v.

18

Now, let h(v) = ¬v, h′(v) = ¬¬v. Since h(v) ∧ h′(v) = 0 and h(v) ∨h′(v) = 1, h(v) and h′(v) are the complements of each other, and hencecondition (C) is also satisfied by L. Then, in view of Theorem 12, unificationin V is filtering. �

In view of Theorem 9, from the above proof it follows also

Corollary 18. Let V be a variety of integral commutative residuated latticeswhich satisfies axiom (S). If A,B ∈ V are finitely presented projective alge-bras, then A× B is also a finitely presented projective algebra.

Corollary 19. Unification in any variety of (integral commutative) residuatedlattices satisfying the Stone identity (S) is either unitary or nullary.

Now we will show a partial converse of Theorem 17, that is, some resid-uated lattices A with filtering unification are Stonean. We use the followingknown properties of residuated lattices:

Lemma 20. Let V be a variety of integral commutative residuated lattices.Then the following conditions hold in V .

(i) x→ (y → z) = x� y → z = y → (x→ z) (com),

(ii) x� y ≤ x ∧ y, x� (x→ y) ≤ y, ¬x� x = 0.

(iii) x→ (y ∧ z) = (x→ y) ∧ (x→ z), x→ y = 1 ⇐⇒ x ≤ y,

(iv) if x ≤ y, then ¬y ≤ ¬x;

(v) x ≤ ¬¬x,

(vi) if ¬x � ¬x = ¬x, i.e. ¬x is idempotent, then the principal filter [¬x)

is an i-filter and θ([¬x)) is a congruence given by an i-filter.

(vii) if A is Stonean, then ¬x� ¬x = ¬x holds in A, for any A ∈ V .

For (vi), observe that for any a, b ∈ [¬x) we have a, b ≥ ¬x and thisimplies a� b ≥ ¬x� ¬x = ¬x, hence a� b ∈ [¬x).

We show (vii): ¬x = ¬x� 1 = ¬x� (¬x ∨ ¬¬x) =

= (¬x� ¬x) ∨ (¬x� ¬¬x) = ¬x� ¬x.

We prove a partial converse of Theorem 17.

19

Theorem 21. Let V be a variety of (commutative integral bounded) residu-ated lattices. Then the following are equivalent:

(i) unification in V is filtering and the identity ¬x�¬x = ¬x holds in thevariety V ,

(ii) V is Stonean.

Proof. (i)⇒ (ii) is clear by Theorem 17 and Lemma 20 (vii).(i) ⇒ (ii). Let V be a variety of residuated lattices. All equations writtenbelow are related to V . We use the well known fact that any mapping onvariables {x1, . . . , xk} has a unique extension to a homomorphism on thefree algebra F(x1, . . . , xk) generated by {x1, . . . , xk}.Let us consider a unification problem (x ∨ ¬x, 1), where x is an arbitraryvariable. The problem has two unifiers: λ0(x) = 0 and λ1(x) = 1.

Since unification in V is filtering there is a unifier µ for it: µ(x∨¬x) = 1,and the unifier µ is more general then λ0 and λ1. This means that there aresubstitutions σ0, σ1 such that for any variable z, σ1(µ(z)) = λ1(z) = 1 andσ0(µ(z)) = λ0(z) = 0, and hence ¬σ0µ(z) = 1.

Let {x1, . . . , xn} = x be the set of all variables in the term µ(x), let y bea fresh variable and let F(x, y) be the free lattice generated by x and y.

Since, by assumption, the elements ¬¬y and ¬y are idempotent, they gen-erate two i-filters [¬¬y), [¬y) i.e. two congruences θ([¬¬y)) and θ([¬y))

in F(x, y). Later in the proof we will define a substitution τ on the set{x1, . . . , xn} such that the following conditions hold:(D) (τ(xj), σ1(xj)) ∈ θ([¬¬y)) and(E) (τ(xj), σ0(xj) ∈ θ([¬y)).Since the congruences θ([¬¬y)) and θ([¬y)) preserve operations, we get:(D’) (τ(µ(x)), σ1(µ(x))) ∈ θ([¬¬y)) and(E’) (τ(µ(x)), σ0(µ(x)) ∈ θ([¬y)) i.e. (¬τ(µ(x)),¬σ0(µ(x)) ∈ θ([¬y)),hence(D’) ¬¬y ≤ (τ(µ(x))↔ σ1(µ(x))) and(E’) ¬y ≤ (¬τ(µ(x))↔ ¬σ0(µ(x))).By the fact that σ1(µ(x)) = 1 and ¬σ0(µ(x)) = 1, we get(D’) ¬¬y ≤ τ(µ(x)) and(E’) ¬y ≤ ¬τ(µ(x)). By (20) (iv),(v) we can write (D’) as follows:(D’) ¬τ(µ(x)) ≤ ¬¬¬y = ¬y. By (v), it follows from (E’) that1 = ¬y → ¬τ(µ(x)) = τ(µ(x))→ ¬¬y, thus(E’) τ(µ(x)) ≤ ¬¬y.Now, by taking joins on both sides of (D’) and (E’) we get1 = τ(µ(x ∨ ¬x)) = ¬τ(µ(x)) ∨ τ(µ(x)) ≤ ¬y ∨ ¬¬y, as required.

20

Consider the following substitution:

τ(xj) := (¬¬y → σ1(xj)) ∧ (¬y → σ0(xj)).

For simplicity of the notation we set: t = τ(xj), s0 = σ0(xj) and s1 =

σ1(xj). Moreover, we have t = (¬¬y → s1) ∧ (¬y → s0). It remains onlyto show that (D) and (E) hold (using Lemma 20). By the definition of thecongruences θ([¬¬y)) and θ([¬y)) we get the equivalent forms of(D) ¬¬y ≤ (s1 ↔ t) , and of (E) ¬y ≤ (s0 ↔ t).

For the proof of (D), using 20 (iii): a ≤ b iff 1 = a→ b, we show that(?) 1 =¬¬y → (s1 ↔ t) = [¬¬y → (s1 → t)] ∧ [¬¬y → (t→ s1)].

Consider the first ’conjunct’ of (?): [¬¬y → (s1 → t)]. By the definitionof t as ”∧” of two parts, we have, by (iii), two cases:1) ¬¬y → (s1 → (¬¬y → s1)) = ¬¬y → (¬¬y → (s1 → s1)) = 1, and2) by (i), (ii) and (iii) of Lemma 20¬¬y → (s1 → (¬y → s0)) = ¬¬y → (¬y → (s1 → s0)) =

= ¬¬y � ¬y → (s1 → s0) = 1.Now consider the second ’conjunct’ of (?): [¬¬y → (t→ s1)]; we show it isequal to 1. By the definition of t we show¬¬y → (t→ s1) = ¬¬y → ([(¬¬y → s1) ∧ (¬y → s0)]→ s1) =

= [(¬¬y → s1) ∧ (¬y → s0)]→ (¬¬y → s1) = 1. Hence (D) is proved.For the proof of (E), similarly we show that:

(??) 1 = ¬y → (s0 ↔ t) = [¬y → (s0 → t)]∧ [¬y → (t→ s0)] . Considerthe first ’conjunct’ of (??): [¬y → (s0 → t)]. Again, by the definition of twe have, by (iii), two cases:1) ¬y → (s0 → (¬¬y → s1)) = ¬y → (¬¬y → (s0 → s1) = 1, and2) ¬y → (s0 → (¬y → s0) = ¬y → (¬y → (s0 → s0)) = 1.Now consider the second ’conjunct’ of (??): [¬y → (t → s0)]. By thedefinition of t we obtain¬y → (t→ s0) = ¬y → ([(¬¬y → s1) ∧ (¬y → s0)]→ s0) =

= [(¬¬y → s1)∧ (¬y → s0)]→ (¬y → s0) = 1. Hence (E) is proved. Thisends the proof of the theorem. �

Remark. Without assuming that the identity ¬x � ¬x = ¬x holds inthe variety, filtering unification does not imply that the variety is Stonean.Consider a three-element MV-algebra {0, 1/2, 1}. Unification in the varietyV3 generated by it is unitary, see [8], hence it is filtering, but V3 is not Stoneanand the condition ¬x� ¬x = ¬x does not hold in V3 (put x = 1/2).

Note that both unification types 1 and 0 actually occur in some varietiesspecified in Corollary 18. Heyting algebras satisfying (S) will be called here

21

Heyting – de Morgan algebras; the variety of Heyting – de Morgan algebrashas unitary unification. Some of its subvarieties e.g. all the varieties of Godelalgebras have unitary unification, Ghilardi [13] showed that there are subva-rieties of Heyting – de Morgan algebras in which unification is nullary.

M. Kondo [19] studied strict residuated lattices, i.e. such lattices L, that

¬(x� y) = ¬x ∨ ¬y, for all x, y ∈ L. (Str)

which generalize well known SMTL-algebras and SBL-algebras, i.e. MTL-algebras and BL-algebras, respectively, satisfying (Str); a t-norm� satisfying(Str) is called strict. By [19] Prop.3, a residuated lattice satisfies (Str) iff it isStonean.An operation ¬ is called a Godel negation if ¬x = 0, for x 6= 0, and ¬0 = 1.

FACT 1. In any residuated lattice L the following conditions are equivalent:(a) L has no (non-trivial) zero divisors: x� y = 0⇒ (x = 0 or y = 0),(b) negation ¬x = x→ 0 in L is Godel negation.For (a) → (b) see [19]. For the converse, assume that 0 has non-trivial

divisors, say z � x = 0 and z 6= 0 6= x; then by z � x = 0 ⇐⇒ z ≤ ¬x,we have 0 6= z ≤ ¬x, i.e. ¬ is not Godel negation, a contradiction.

FACT 2. For any subdirectly irreducible (s.i.) bounded residuated lattice Lthe following conditions are equivalent§ :

(i) L satisfies (Str), or L is Stonean,(ii) L has no zero divisors,(iii) the negation ¬x = x→ 0 in L is a Godel negation,

We show (i) → (ii). Suppose that x � y = 0. Then 1 = ¬(x � y) =

¬x∨¬y, but since L is subdirectly irreducible, Prop.1.4 in [20] gives ¬x = 1

or ¬y = 1, hence x = 0 or y = 0. (ii)→ (iii) and (iii)→ (i) are similar toProposition 4. in [19].

Corollary 22. Let V be a variety of (commutative integral bounded) residu-ated lattices. Then the following are equivalent:

(i) L satisfies (Str), i.e. L is Stonean, for every L ∈ V ,(ii) every subdirectly irreducible (s.i.) member of V has no zero divisors,(iii) the negation ¬x = x→ 0 in every s.i. L ∈ V is the Godel negation.

Hence lack of zero divisors in s.i. residuated lattices that generate thevariety V , determine filtering unification in V . In general, since x� ¬x = 0,the free algebra in V has zero divisors, it generates V , but it is not s.i.

§ In [19] FACT 2 is proved for a chain L.

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Corollary 23. Let V be a variety of (commutative integral bounded) residu-ated lattices. Then unification in V is filtering and the identity ¬x�¬x = ¬xholds in V iff the subdirectly irreducible members of V have no zero divisors.

Residuated lattices that satisfy prelinearity (x → y) ∨ (y → x) = 1 arecalled MTL-algebras. They form an algebraic semantics for the monoidal t-norm based logic, MTL, the fuzzy logics of all left-continuous t-norms, see[9]. BL-algebras are MTL-algebras satisfying the axiom (x∧ y) = x� (x→y). They form an algebraic semantics for P. Hajek’s Basic Fuzzy Logic BL.ΠMTL-algebras (Π-algebras) are MTL-algebras (BL-algebras) that satisfy¬x∨ ((x→ x� y)→ y) = 1. BL-algebras with � = ∧ are called Godel al-gebras. SMTL-algebras (SBL-algebras) additionally satisfy (Str). For variousvarieties of residuated lattices see e.g. [11], [20].

It is known, see e.g. [5], 1.9, 1.8, that MTL-algebras are subdirect productsof bounded residuated chains, and a bounded residuated chain is Stonean ifand only if it is pseudocomplemented. Thus we obtain

Corollary 24. Unification is either unitary or nullary in the following va-rieties: all varieties of SMTL-algebras, MTL-algebras with Godel negation,SBL-algebras, ΠMTL-algebras, etc.

Many examples of such varieties can be found in [16] and [17].

Example A. Let L1 be a residuated lattice which is a five-element Heytingalgebra, that is the square 2× 2 with an additional top 1, i.e. 0 < a, b <

v < 1, a, b incomparable, ¬a = b, ¬b = a and let V1 ⊆ H be a variety ofHeyting algebras containing L1. Since in Heyting algebras � = ∧, we have¬x � ¬x = ¬x, but (S) fails in L1 (and in V1): ¬a ∨ ¬¬a = v 6= 1, L1

has zero divisors: a � b = 0. L1 is the particular splitting of the lattice ofsubvarieties of H, hence unification in V1 is not filtering, see [6], [7].

Example B. LetL2 be a residuated lattice of seven-elements || which consistsof the non-distributive diamond M3 (with the bottom u), plus additional top1 and bottom 0 added, and x � 1 = x, x � 0 = 0; x � y = u otherwise;i.e. L2 has no zero divisors. By FACT 2 L2 is Stonean, hence HSP(L2) is aStonean variety, thus unification in HSP(L2) is filtering (unitary or nullary).Distributivity fails in L2.

Example C. Let L3 be a residuated lattice of seven-elements which consistsof the non-modular pentagon N5 (with the bottom u), plus additional top 1

|| This is Example 2 of [10].

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and bottom 0 added, and x � 1 = x, x � 0 = 0; x � y = u otherwise; i.e.L3 has no zero divisors. By FACT 2 L3 is Stonean, hence unification in thevariety HSP(L3) is filtering (unitary or nullary), however modularity fails inL3

# .

6 CONCLUSIONS

We proved that unification is filtering, i.e. unitary or nullary, in any varietyof `-algebras, which satisfy some algebraic conditions (A), (B) and (C). Thisis true even if the underlaying lattices are not modular. Here `-algebras arebounded lattice-based algebras in which every term preserves congruencesinduced by central elements in the underlying lattice. Condition (A) con-cerns central elements of the underlying lattice, (B) demands a kind of weakpseudocomplement and (C) requires generation of pairs of complemented el-ements.

This general result is then applied to a particular case of bounded, com-mutative, integral residuated lattices. It is shown that unification is filteringin Stonean residuated lattices (or equivalently, in varieties generated by s.i.residuated lattices without zero divisors). These conditions are sufficient, butin particular subvarieties they are also necessary. The result on filtering uni-fication apply also to a wide range of fuzzy logics, especially to fuzzy logicsin which negation is represented by the pseudocomplement, in particular tologics with Godel negation.

ACKNOWLEDGEMENT

The authors thank the anonymous referee for his/her insightful comments andsuggestions that improved the paper significantly. They also wish to thankMichiro Kondo for his remarks on Theorem 15 and Peter Jipsen for providingnon-modular examples.

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# This example was suggested to us by Peter Jipsen in a private communication.

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