quasi-discriminator varieties

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International Journal of Algebra and ComputationVol. 24, No. 3 (2014) 375–411c© World Scientific Publishing CompanyDOI: 10.1142/S0218196714500179

Quasi-discriminator varieties

Francesco Paoli∗ and Antonio Ledda†

Dipartimento di Pedagogia, Psicologia, FilosofiaUniversita di Cagliari, Via Is Mirrionis 1

Cagliari 09123, Italy∗[email protected]

†[email protected]

Tomasz Kowalski

Department of Mathematics and StatisticsLa Trobe University, Bundoora

VIC 3086, [email protected]

Matthew Spinks

Dipartimento di Pedagogia, Psicologia, FilosofiaUniversita di Cagliari, Via Is Mirrionis 1

Cagliari 09123, [email protected]

Received 10 October 2012Accepted 3 April 2014Published 14 May 2014

Communicated by K. A. Kearnes

We generalize the notion of discriminator variety in such a way as to capture severalvarieties of algebras arising mainly from fuzzy logic. After investigating the extent towhich this more general concept retains the basic properties of discriminator varieties,we give both an equational and a purely algebraic characterization of quasi-discriminatorvarieties. Finally, we completely describe the lattice of subvarieties of the pure pointedquasi-discriminator variety, providing an explicit equational base for each of its members.

Keywords: Discriminator varieties; quasi-subtractive varieties; subvarieties lattice.

Mathematics Subject Classification: 03C05, 08A40, 08B15

1. Introduction

A discriminator variety [60] is a variety V with a ternary term t(x, y, z) that realizesthe ternary discriminator function

t(a, b, c) ={c if a = b,

a otherwise

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on any subdirectly irreducible member of V . In their influential textbook on univer-sal algebra [12, p. 186], Burris and Sankappanavar call discriminator varieties “themost successful generalization of Boolean algebras to date”. This remark of Burrisand Sankappanavar can be justified in at least two ways. On the one hand, manyimportant classes of algebras arising in algebra and logic form discriminator vari-eties, including the varieties of Boolean algebras, monadic algebras, n-dimensionalcylindric algebras, Post algebras, n-valued MV algebras, and basic logic algebraswith Baaz Delta [58]. On the other hand, discriminator varieties turn out to bevery easy and tractable to work with in general. In particular, algebras in discrim-inator varieties admit an extremely useful Boolean product representation, whichtypically gives a deep insight into the algebraic and logical properties of the classin question. For a very readable survey of discriminator varieties, see [36].

Historically, the study of discriminator varieties arose from the theory of primalalgebras. A finite algebra A is said to be primal if every n-ary function on A isrepresentable by a term; recall that the two-element Boolean algebra is primal.Primality is the mathematical basis of the algebraic theory of switching circuitsand also plays an important role in mathematical logic. It was Foster [27, 28] whoinitiated the study of primal algebras from the perspective of general algebra inthe 1950s. Subsequently, he and his collaborators wrote a series of papers exploringmany generalizations of primality [47, 20, 30]. Of these various generalizations, themost important has turned out to be quasi-primality. Recall from Werner [59] thata finite algebra A is quasi-primal if and only if the ternary discriminator is a termfunction on A; the generalization to discriminator varieties follows immediately.For an overview of primality and its generalizations, see [32, Appendix 5] or [37].

An attractive characterization of discriminator varieties [29] states that a varietyV is a discriminator variety if and only if it is congruence permutable and semisimplewith equationally definable principal congruences (EDPC). Independently, in theearly 1980s Blok and Pigozzi observed a close connection between the deduction-detachment theorem in logic and EDPC in varieties of algebras. This, together withPigozzi’s observation in the late 1970s that varieties with EDPC are congruencedistributive [38] turned the study of varieties with EDPC into a major topic, andnumerous further generalizations of the ternary discriminator such as the ternarydeductive term quickly followed [5, 6, 8, 9]. However, unlike early developments inthe theory of discriminator varieties, which were algebraically driven, these newerdevelopments were inspired by applications to logic.

Whether algebraically or logically motivated, what all generalizations of theternary discriminator have in common is the desire to preserve as many of theattractive properties of discriminator varieties as possible — such as the avail-ability of (weak) Boolean product representations with directly indecomposablefactors, or the reducibility of universal first-order formulas to identities — whilesimultaneously enlarging the class of structures encompassed by the generalizationin question. Following suit, and urged by considerations from logic, we introducethe class of quasi-discriminator varieties in this paper. The precise definition of

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Quasi-discriminator varieties 377

quasi-discriminator variety that we adopt is motivated in part by its naturalness, inpart by the fact that it encompasses numerous varieties of algebras (like Godel alge-bras or product algebras) that have been extensively investigated in the literature,especially in fuzzy logic, and in part because quasi-discriminator varieties turn outto be quasi-subtractive. Quasi-subtractive varieties are a modalized generalizationof Ursini’s subtractive varieties [1, 34, 53] that were introduced recently by three ofthe present authors in connection with the study of varieties enjoying a satisfactorytheory of open filters [40]. The present paper consists in an exploratory study ofquasi-discriminator varieties with an eye to investigating their potentialities.

After dispatching the necessary preliminaries in Sec. 2, we study quasi-subtractive varieties where the modal operator � is an endomorphism on everymember of the class (Sec. 3). Then, we further specialize this concept and proceedto define quasi-discriminator varieties, pointing to some noteworthy examples aboveand beyond discriminator varieties (Sec. 4). Basic properties of quasi-discriminatorvarieties (including on the one hand a description of the mutual relationshipsbetween quasi-discriminator algebras, simple members and s.i. members; and onthe other hand, a weak Boolean product representation with directly indecompos-able factors in the double pointed case), are investigated in Sec. 5. Two differentcharacterizations of quasi-discriminator varieties are discussed next: an equationaldescription along the lines of the one given by Vaggione [56] for discriminator vari-eties (Sec. 6), and a purely algebraic one following in the footsteps of Fried et al.’sresult [29], given in Sec. 8. In between these sections, we insert some ancillary mate-rial on the equational definability of principal open congruences (Sec. 7). Section 9is devoted to the pure pointed quasi-discriminator variety: we completely describethe lattice of subvarieties of this variety, providing explicit equational bases for allthe members of this lattice.

Our terminological and notational conventions are the usual ones in universalalgebra. Deviations from standard usage will be explicitly noted in what follows. Inparticular, if K is a class of similar algebras, we let K+ denote the class K augmentedby a trivial algebra of the same type.

2. Preliminaries

2.1. Abstract algebraic logic

In this section, we briefly go over some notions from abstract algebraic logic to beused in the present paper. A standard reference for the material that follows is [7].

A (propositional) logic is a pair L = (Fm,�), where Fm is the formula algebraof some given language L and � is a substitution-invariant consequence relationover Fm. Logical languages and algebraic similarity types will be hereafter identi-fied; therefore, “formula” and “term” should be understood as synonymous, whileequations are viewed as pairs of formulas. If L = (Fm,�L) is a logic in the languageL, a set τ = {δi(x) ≈ εi(x) : i ≤ n} of equations in a single variable of L can also beregarded as a function which maps formulas in Fm to sets of equations of the same

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language; this mapping will be called formula-equation translation, or sometimessimply translation.

A class K of similar algebras is an algebraic semantics for L if, for some trans-lation τ , the following condition holds for all Γ ∪ {t} ⊆ Fm:

Γ �L t if and only if for every A ∈ K and every −→a ∈ An,

if τ(s)A(−→a ) for all s ∈ Γ, then τ(t)A(−→a ),

a condition which can be rewritten as

Γ �L t if and only if {τ(s) : s ∈ Γ} �K τ(t).

If K is given, we also say that the logic L having the same language as K, whoseconsequence relation �L is defined as in the previous display, is the τ -assertionallogic of K.

In their paper on assertionally equivalent quasivarieties [11], Blok and Rafteryintroduce a notion of τ -class that relativizes the usual notion of congruence classto a given translation (which we assume henceforth to be finite). Although thatpaper considers the more general setting of quasivarieties, in the present discussionwe shall restrict ourselves to varieties: if V is a variety, A ∈ V , θ ⊆ A2 and τ ={δi(x) ≈ εi(x) : i ≤ n} is a translation in the similarity type of V , the τ -class of θin A — in symbols τA/θ — is defined as

τA/θ = {a ∈ A : δAi (a)θεAi (a) for every i ≤ n}.

A variety V is τ -regular if and only if for any congruences θ, ϕ on anyA∈V , τA/θ = τA/ϕ implies θ = ϕ; if τ = {x ≈ 1}, we get as a specialcase the standard notion of 1-regularity [22]. Blok and Raftery also consider aproperty of τ -permutability obtained by appropriately generalizing the notion of1-permutability to varieties that need not be pointed: a variety V is τ -permutableif and only if for any congruences θ, ϕ on any A ∈ V , τA/θ ◦ ϕ = τA/ϕ ◦ θ.

Given an equation t ≈ s and a set of formulas in two variables ρ = {tj(x, y)}j∈J ,we use the abbreviation

ρ(t, s) = {tj(x/t, y/s)}j∈J .

ρ will be also regarded as a function mapping equations to sets of formulas, andcalled an equation-formula translation.

A logic L = (Fm,�L) is said to be algebraizable with equivalent algebraic seman-tics K (where K is a class of algebras of the same language as Fm) if and only ifthere exist a formula-equation translation τ , and an equation-formula translationρ, such that the following conditionsa hold for any t, s ∈ Fm, for any Γ ⊆ Fm and

aIf Γ, ∆ are sets of formulas, Γ �L ∆ means Γ �L t for all t ∈ ∆; if E, E′ are sets of equations,E �Eq(K) E′ means E �L ε for all ε ∈ E′.

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for any E ⊆ Fm2:

AL1: Γ �L t if and only if τ(Γ) �K τ(t);AL2: E �K t ≈ s if and only if ρ(E) �L ρ(t, s);AL3: t �L ρ(τ(t));AL4: t ≈ s �K τ(ρ(t, s)).

The mappings τ(x) and ρ(x, y) are respectively called a system of defining equa-tions and a system of equivalence formulas for L and K. A logic L is algebraizable(tout court) if and only if, for some K, it is algebraizable with equivalent algebraicsemantics K. By virtue of [7, Corollary 2.11], K may be identified with the quasiva-riety generated by K; in other words, K is an equivalent quasivariety semantics. IfK is a variety, then K is an equivalent variety semantics, and L is said to be stronglyalgebraizable. Also, L is algebraizable with equivalent algebraic semantics K if andonly if it satisfies either AL1 and AL4, or else AL2 and AL3.

Let A be an algebra of language L, and F a subset of A. An L-matrix (orsimply a matrix, when L is understood) is a tuple 〈A, F 〉. For any matrix 〈A, F 〉and Γ ∪ {t} ⊆ Fm, let |=〈A,F 〉 be the relation defined by:

Γ |=〈A,F 〉 t if and only if for any −→a ∈ An,

s(−→a ) ∈ F for any s ∈ Γ implies t(−→a ) ∈ F.

If M is a class of matrices, then Γ |=M t if Γ |=〈A,F 〉 t, for every 〈A, F 〉 ∈M . If Lis a logic of language L and A is an algebra of the same language, F ⊆ A is calleda deductive filter on A of L, or just a deductive filter on A, when L is understood,if Γ �L t implies Γ |=〈A,F 〉 t, for all Γ ∪ {t} ⊆ Fm.

2.2. Quasi-subtractive varieties

Recall that a variety V with (at least) an equationally definable constant 1 in itstype is 1-subtractive (or simply subtractive when no ambiguity is possible) if thereis a binary term, denoted by → and written in infix notation, such that V satisfiesthe following equations:

S1 x→ x ≈ 1,S2 1 → x ≈ x.

In their paper [34], Gumm and Ursini essentially observe that a variety V with1 is 1-permutable if and only if it is 1-subtractive.

The following generalization of the preceding concept was proposed in [40].Unless otherwise specified, all the results stated without proof in the present sectionare proved in detail in this paper.

Definition 2.1. A variety V whose type ν includes an equationally definablenullary term 1 and an equationally definable unary term � is called quasi-subtractivewith respect to 1 and � if and only if there is a binary term→ (x, y) (hereafter

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written in infix notation) of type ν such that V satisfies the equations

Q1 �x→ x ≈ 1,Q2 1 → x ≈ �x,Q3 �(x→ y) ≈ x→ y,Q4 �(x→ y) → (�x→ �y) ≈ 1.

Whenever no risk of confusion is present, reference to 1 and � will be left implicitand the qualification “equationally definable” will be omitted. When discussingparticular examples, we will sometimes say that “the terms →,� and 1 witnessquasi-subtractivity for V”, meaning that V is quasi-subtractive with respect to1 and � and that → is the binary term referred to in Definition 2.1. On otheroccasions, we will say instead that “→ witnesses quasi-subtractivity with respectto 1 and � for V”. These expressions, as well as slight stylistical variants thereof,should be understood as synonymous. Members of quasi-subtractive varieties willbe called, by extension, quasi-subtractive as well.

Every quasi-subtractive variety is {�x ≈ 1}-permutable (or, as we will sayfor short, (�x, 1)-permutable; a similar convention will be applied to “ (�x, 1)-regular”, “(�x, 1)A/θ”, “(�x, 1)-class” and similar expressions): indeed, it can beshown that a variety of the appropriate type is (�x, 1)-permutable if and onlyif it satisfies Q1 and Q2 for some binary term →. However, there are (�x, 1)-permutable varieties that fail to be quasi-subtractive, and Theorem 2.8 will providean appropriate justification for the more specialized concept we chose to focus on.

An element a ∈ A ∈ V is called open if �a = a. The set {a ∈ A : �a = a} ofopen elements of A is denoted A�.

Every 1-subtractive variety with witness term → is automatically quasi-subtractive with witness terms →, 1, and the identity term as �. The following tablelists some other examples of quasi-subtractive varieties. Observe that some of thesevarieties are indeed subtractive but can be viewed as properly quasi-subtractivewith a different choice of witness terms.

Variety Ref. x → y �x 1-Subtr.?

Residuated lattices [31] (x\y) ∧ 1 x ∧ 1 YesSubresiduated lattices [25] x ⇒ y 1 ⇒ x YesQuasi-MV algebras [41] x′⊕y x ⊕ 0 NoVar. with a comm. TD term [9] p(x, p(x, y, x), 1) p(x, 1, 1) NoPseudointerior algebras [9] x → y x◦ NoInterior algebras [4] �(¬x ∨ y) �x YesIntegral k-potent res. lattices [31] (x\y)k xk Yes

The following terminology and notation will be repeatedly used in what follows:

Definition 2.2. (1) An algebra A in a quasi-subtractive variety V is called openin case it satisfies the equation x ≈ �x, i.e. if all its elements are open.

(2) A congruence θ on A is said to be open if and only if A/θ is open.

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We ambiguously denote by Conσ(A) the lattice of open congruences of an alge-bra A in a quasi-subtractive variety V , as well as its universe. For X ⊆ A, θσ(X)will denote the smallest open congruence collapsing all members of X ; as usual,outer brackets will be omitted whenever X is a pair or an n-tuple.

The following concept of open filter is as central for the investigation of quasi-subtractive varieties as the Gumm–Ursini concept of ideal [34] is for the investiga-tion of subtractive varieties.

Definition 2.3. Let V be a variety whose type ν includes a nullary term 1 anda unary term �. A V-open filter term in the variables −→x is an n + m-ary termp(−→x ,−→y ) of type ν such that

{�xi ≈ 1 : i ≤ n} �V �p(−→x ,−→y ) ≈ 1.

The wording “V-open filter term” will be simplified to “open filter term”whenever this replacement is unambiguous. The same applies to “V-open filter”below.

Definition 2.4. Let V be as in Definition 2.3. A V-open filter of A ∈ V is a subsetF ⊆ A with the following properties:

(i) F is closed with respect to all V-open filter terms p: if a1, . . . , an ∈ F andb1, . . . , bm ∈ A, then p(−→a ,−→b ) ∈ F ;

(ii) for every a ∈ A, we have that a ∈ F if and only if �a ∈ F .

Observe that 1 is a member of any open filter since the constant term 1 is anopen filter term. OpfilV(A) will denote the set (or, with a slight notational abuse,the lattice) of V-open filters of A; again, all subscripts in sight will be droppedwhenever no ambiguity is possible.

As we already mentioned, all the results below are proved in detail in [40].The next theorem is an analog of a classical result from the theory of subtractivevarieties [53, Proposition 1.4].

Theorem 2.5. Every quasi-subtractive variety V has normal open filters: for allA ∈ V , every V-open filter of A is a (�x, 1)-class of some θ ∈ Con(A).

Its importance is highlighted by the following noteworthy consequence. If S(V)denotes the (�x, 1)-assertional logic of V , then we have the following theorem.

Theorem 2.6. If V is a variety with normal open filters and A ∈ V , then theV-open filters of A coincide with the deductive filters on A of S(V). Therefore, ifV is a quasi-subtractive variety and A ∈ V , the V-open filters of A coincide withthe deductive filters on A of S(V).

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In the theory of subtractive varieties, ideal generation can be nicely described(in fact, this is the key reason why ideals in subtractive varieties are so useful towork with). A similar result holds for open filters. If A is any algebra in a varietyV of the appropriate type, and we define for X ⊆ A:

↑ X = X ∪ {a : �a ∈ X};

ΓX = {pA(−→a ,−→b ) : −→a ∈ X,−→b ∈ A, p an open filter term},

we get the following theorem.

Theorem 2.7. Let V be a quasi-subtractive variety, A ∈ V and X ⊆ A. TheV-open filter [X) generated by X is precisely ↑ ΓX.

Among its consequences, the preceding theorem yields a characterization ofjoins of open filters and the following welcome property, an analog of the propertyof modularity for the lattices of ideals in subtractive varieties [53, Proposition 1.5].

Theorem 2.8. Let V be a quasi-subtractive variety. Then the lattice of open filtersof any A ∈ V is modular.

Let V1 and V2 denote the varieties with equational bases {Q1, Q2, Q3} and{Q1, Q2, Q4}, respectively. Then for i = 1, 2, there exists A in Vi such that thelattice of open filters on A is not modular. This circumstance provides some justi-fication for the choice of Definition 2.1 among the many possible specializations ofthe notion of (�x, 1)-permutable variety that still count as generalizations of theconcept of subtractive variety.

In every algebra A belonging to a 1-ideal determined (i.e. 1-subtractive and1-regular) variety V the lattice of congruences on A is isomorphic to the lattice ofV-ideals of A. Some isomorphism results along the same lines, available for examplein the theories of residuated lattices, of pseudointerior algebras, or of quasi-MValgebras, do not lend themselves to be viewed as special cases of this theorem,either because the variety at issue fails to be ideal determined, or else because thelattice proved isomorphic to Con(A) is not the lattice of V-ideals of A. A suitablyweakened version of the property of τ -regularity is defined hereafter.

Definition 2.9. A variety V is called weakly τ -regular if and only if the τ -assertionallogic S(V , τ) of V is strongly and finitely algebraizable.

Observe that a weakly τ -regular variety V is τ -regular if and only if, in addition,V is the equivalent variety semantics of S(V , τ).

Leaving once again the general scenario of an arbitrary translation τ so as tofocus on translations of the form {�x ≈ 1}, we make a note of the next result; here,a congruence θ on an algebra A ∈ V is said to be a V-V ′ congruence if A/θ ∈ V ′.

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Theorem 2.10 ([7]). If V is weakly (�x, 1)-regular and V ′ is the equivalent varietysemantics of S(V), then for any A ∈ V there is a lattice isomorphism between thelattice of V-V ′ congruences on A and the lattice of deductive filters on A of S(V).

By Theorems 2.6 and 2.10, we get the following corollary.

Corollary 2.11. If V is quasi-subtractive and weakly (�x, 1)-regular and V ′ isthe equivalent variety semantics of S(V), then for any A ∈ V there is a latticeisomorphism between the lattice of V-V ′ congruences on A and the lattice of V-open filters on A.

Besides generalizing similar results (see e.g. [7, 10, 23]), Corollary 2.11 subsumesthe lattice isomorphism theorems mentioned above.

2.3. t-arithmetical varieties

The notions in this section were in part introduced in [39] and are currently underthorough scrutiny in the work in progress [15], which contains full proofs of all theresults we mention hereafter.

Definition 2.12. If V is a variety of type ν, a term t of the same type is a V-compatible term if and only if the operation tA is an endomorphism on every algebraA ∈ V . Moreover, t is a V-idempotent term if and only if V satisfies the identityt(t(x)) ≈ t(x).

Definition 2.13. Let V be a variety, and let t, s be V-compatible and V-idempotentterms. V is (t, s)-distributive if and only if for every a, b ∈ A ∈ V and for everyθ, ϕ, ψ ∈ Con(A), we have

(tA(a), sA(b)) ∈ θ ∧ (ϕ ∨ ψ) if and only if (tA(a), sA(b)) ∈ (θ ∧ ϕ) ∨ (θ ∧ ψ).

V is (t, s)-permutable if and only if for every a, b ∈ A ∈ V and for every θ, ϕ ∈Con(A), we have

(tA(a), sA(b)) ∈ θ ◦ ϕ if and only if (tA(a), sA(b)) ∈ ϕ ◦ θ.

V is (t, s)-arithmetical if and only if it is (t, s)-distributive and (t, s)-permutable.

It is easy to see that (t, s)-permutability simultaneously generalizes both thestandard notion of congruence permutability and the already mentioned conceptof τ -permutability: the former by letting t, s be the identity, and the latter byletting a and b in the above definition be the same element; mutatis mutandis,the same applies to (t, s)-distributivity. If a variety V is (t, s)-distributive (respec-tively, (t, s)-permutable, (t, s)-arithmetical) and t = s, we say simply that V is

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t-distributive (respectively, t-permutable, t-arithmetical). Finally, by V1 we meanthe variety axiomatized with respect to V by the equation x ≈ t(x).

True to form, all the previous properties are Mal’cev properties.

Theorem 2.14. V is (t, s)-distributive if and only if there exists a set of m + 1ternary terms {pj(x, y, z)}j≤m such that V satisfies the following equations:

p0(x, y, z) ≈ t(x),

pm(x, y, z) ≈ s(z),

pj(s(x), s(y), t(x)) ≈ s(t(x)) ≈ pj(s(x), t(y), t(x)) for all j ≤ m,

pj(x, t(x), y) ≈ pj+1(x, t(x), y) for even j,

pj(x, s(y), y) ≈ pj+1(x, s(y), y) for odd j.

Theorem 2.15. A variety V is (t, s)-permutable if and only if there exists a ternaryterm p(x, y, z) such that

V � p(x, s(y), y) ≈ t(x),

V � p(x, t(x), y) ≈ s(y).

Theorem 2.16. For a variety V the following are equivalent:

(1) V is t-arithmetical;(2) There exist terms p and M, where p is as in Theorem 2.15, such that

V � M(x, y, t(x)) ≈M(x, t(x), y) ≈M(y, t(x), x) ≈ t(x);

(3) There exists a ternary term m such that

V � m(x, y, t(x)) ≈ m(x, t(y), y) ≈ m(y, t(y), x) ≈ t(x).

A version of Jonsson’s Lemma holds for t-distributive varieties.

Theorem 2.17. Let V = V (K) be t-distributive. Then all s.i. members of V1 arein HSPU (K).

2.4. Church varieties

The key observation motivating the introduction of Church algebras [42] is thatmany algebras arising in completely different fields of mathematics — includingHeyting algebras, rings with unit, or combinatory algebras — have a term operationq satisfying the fundamental properties of the if-then-else connective: q(1, x, y) ≈ x

and q(0, x, y) ≈ y. As simple as they may appear, these properties are enough toyield rather strong results. This motivates the next definition.

Definition 2.18. An algebra A of type ν is a Church algebra if there are termdefinable elements 0A, 1A ∈ A and a term operation qA such that, for all a, b ∈ A,

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qA(1A, a, b) = a and qA(0A, a, b) = b. A variety V of type ν is a Church variety ifevery member of V is a Church algebra with respect to the same term q(x, y, z) andthe same constants 0, 1.

Examples of Church algebras include FLew-algebras (commutative, integral anddouble-pointed residuated lattices, for which see [31]) and, in particular, Heytingalgebras and thus also Boolean algebras; ortholattices; rings with unit; combinatoryalgebras. Expanding on an idea due to Vaggione [54], we also define the following.

Definition 2.19. An element e of a Church algebra A is called central if the pair(θ(e, 0), θ(e, 1)) is a pair of complementary factor congruences on A. A centralelement e is nontrivial if e /∈ {0, 1}. By Ce(A) we denote the center of A, i.e. theset of central elements of the algebra A.

By defining x ∧ y = q(x, y, 0), x ∨ y = q(x, 1, y) and x′ = q(x, 0, 1), we get thefollowing theorem.

Theorem 2.20 ([49]). Let A be a Church algebra. Then Ce(A) = (Ce(A);∨,∧,′ ,0, 1) is a Boolean algebra which is isomorphic to the Boolean algebra of factorcongruences of A.

This result, together with theorems by Comer [21] and Vaggione [54], impliesthe following theorem.

Theorem 2.21 ([49]). Let A be a Church algebra, S be the Boolean space ofmaximal ideals of Ce(A) and f : A→ ΠI∈SA/θI be the map defined by

f(a) = (a/θI : I ∈ S),

where θI =⋃

e∈I θ(0, e). Then we have:

(1) f gives a weak Boolean representation of A.(2) f provides a Boolean representation of A if and only if, for all a �= b ∈ A, there

exists a least central element e such that q(e, a, b) = a (i.e. (a, b) ∈ θ(0, e)).

Theorem 2.22 ([49]). Let V be a Church variety of type ν. Then, the followingconditions are equivalent:

(i) For all A ∈ V , the stalks A/θI (I ∈ S a maximal ideal) are directly indecom-posable.

(ii) The class VDI of directly indecomposable members of V is a universal class.

2.5. Semi-Boolean-like varieties

In a generic Church algebra, of course, there is no need for the set of central ele-ments to comprise all elements of the algebra — not any more than an arbitraryortholattice needs to be a Boolean algebra, or a ring with unit a Boolean ring. In[49], Church algebras where the set of central elements comprises all elements of thealgebra were introduced and investigated under the label of Boolean-like algebras,

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while the name of semi-Boolean-like algebras was reserved for the concept definedbelow.

Definition 2.23. We say that a Church algebra A of type ν is a semi-Boolean-like algebra (or a SBlA, for short) if it satisfies the following equations, for alle, a, a1, a2, b, c ∈ A:

Ax0. q(1, a, b) = a = q(0, b, a),Ax1. q(e, a, a) = a,Ax2. q(e, q(e, a1, a2), a) = q(e, a1, a) = q(e, a1, q(e, a2, a)),Ax3. q(e, g(b), g(c)) = g(q(e, b1, c1), . . . , q(e, bn, cn)), for every g ∈ ν.

If A satisfies Ax0–Ax3 plus

Ax4. q(a, 1, 0) = a

then we say that A is a Boolean-like algebra (or a BlA, for short).

Definition 2.24. A variety V of type ν is a (semi-)Boolean-like variety if everymember of V is a (semi-)Boolean-like algebra with respect to the same term q(x, y, z)and the same constants 0, 1.

It turns out that, if we define c(x) = q(x, 1, 0), an element a in a semi-Boolean-like algebra is central just in case c(a) = a. By Ax4, therefore, Boolean-like algebrasare precisely those semi-Boolean-like algebras where every element is central.

The “pure semi-Boolean-like” variety SBlA0, consisting of all the term reductsof the form (A, q, 0, 1) of semi-Boolean-like algebras, and axiomatized by Ax0–Ax3

above, is of independent interest. The term c is SBlA0-compatible and SBlA0-idempotent and thus, if A is a member of SBlA0, c[A] is a retract of A. Examplesof members of SBlA0 are the following.

Example 2.25. Let 3 = ({0, 1, 2}; q, 0, 1) be the Church algebra completely spec-ified by the stipulation that q(0, a, b) = q(2, a, b) for all a, b ∈ {0, 1, 2}. It can bechecked that 3 is semi-Boolean-like. However, c(2) = q(2, 1, 0) = 0 �= 2. Moreover,3 is a nonsimple subdirectly irreducible algebra, with the middle congruence cor-responding to the partition {{1}, {0, 2}}. Therefore, V (3) is not a discriminatorvariety, although it is a binary 1-discriminator variety in the sense of [14] withbinary 1-discriminator term y′ ∨ x.

Example 2.26. Let 3′ = ({0, 1, 2}; q, 0, 1) be the Church algebra completely spec-ified by the stipulation that q(1, a, b) = q(2, a, b) for all a, b ∈ {0, 1, 2}. It can bechecked that 3′ is semi-Boolean-like. However, c(2) = q(2, 1, 0) = 1 �= 2. Moreover,3′ is a nonsimple subdirectly irreducible algebra, with the middle congruence cor-responding to the partition {{0}, {1, 2}}. Therefore, V (3′) is not a discriminatorvariety, although it is a binary 0-discriminator variety with binary 0-discriminatorterm y′ ∧ x.

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The algebras we just introduced are actually more than workaday examples ofpure semi-Boolean-like algebras. In fact, let 4 be the fibered product 3×23′, i.e. thealgebra whose universe is {(0, 0), (2, 0), (1, 2), (1, 1)} and whose algebraic structureis described by the self-explanatory diagram below.

We have the following theorem.

Theorem 2.27. SBlA0 = V ({3,3′}) = V (4).

SBlA0 has three proper nontrivial subvarieties:

• ISBlA0, the subvariety generated by 3′, whose equational basis relative to SBlA0

is given by the single identity x ∧ x ≈ x;• JSBlA0, the subvariety generated by 3, whose equational basis relative to SBlA0

is given by the single identity x ∨ x ≈ x;• BlA0, the variety consisting of all the term reducts of the form (A, q, 0, 1) of

Boolean-like algebras, generated by the two-element Boolean-like algebra, whoseequational basis relative to SBlA0 is given either by the single identity x ∧ y ≈y∧x, or by the two idempotency identities x∧x ≈ x, x∨x ≈ x, or by the identityc(x) ≈ x.

More generally, a semi-Boolean-like variety is said to be meet (join) idempotentif it satisfies the identity x ∧ x ≈ x (x ∨ x ≈ x). Although an arbitrary semi-Boolean-like variety needs not be e-subtractive or e-regular (e ∈ {0, 1}), a meet(join) idempotent semi-Boolean-like variety is always 0- (1-)subtractive with witness

Fig. 1. Inclusion relationships.

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term y′∧x (y′ ∨ x). Adding 0- (1-)regularity constraints to the preceding conceptssuffices to deliver double pointed discriminator varieties, according to Fig. 1.

3. Quasi-Subtractive Varieties with Compatible Box

Before defining the main concept of the present paper, we explore some middleground between it and the notion of a generic quasi-subtractive variety — namely,we consider quasi-subtractive varieties V , with respect to →,� and 1, where � is aV-compatible term. We call such varieties “quasi-subtractive with compatible box”.

Lemma 3.1. Let V be quasi-subtractive with compatible box, and let A be a memberof V. Then:

(1) A/ker� � �A =(A�, {f � A� : f ∈ ν}) is a maximal open retract of A.(2) The class {A/ker� : A ∈ V} generates a variety which coincides with VO, the

open subvariety of V defined, relative to V , by the equation �x ≈ x.(3) Conσ(A) is isomorphic to Con(A/ker�) via the mapping θ → θ/ker�.

Proof. All the items in the lemma follow easily once we observe that, owing tothe V-compatibility and V-idempotency of �, ker� = {(a, b) ∈ A2 : �a = �b} isa congruence on every algebra A ∈ V — indeed, the smallest open congruence inCon(A).

Let V be a quasi-subtractive variety and A ∈ V . From [40] we briefly recall thefollowing concepts. For any term t(x1, . . . , xk), the open translation t� of t is:

• x� = x, for a variable x,• f�(t1, . . . , tk) = �f(t�1 , . . . , t

�k ), for a k-ary basic operation f and terms

t1, . . . , tk.

The open contraction A� of A is the algebra (A�, (f�)f∈ν). We put K� ={A� : A ∈ K}. If A� ∈ V for every A ∈ V then V is called contractive. AlthoughA� need not be isomorphic to A/ker� � �A in general, we have the followinglemma.

Lemma 3.2. Let V = V (K) be quasi-subtractive with compatible box. Then theopen contractions of algebras from V are retracts in V. In particular, such varietiesare contractive.

Proof. Let � be K-compatible. If f ∈ ν is a basic operation of A ∈ V (K), itfollows that f� = f � A�, whence A� = �A. Our result is thus a consequence ofLemma 3.1(1).

It follows from [40, Lemma 50] and Lemma 3.1(2) that in such cases V� =VO = V ({A/ker� : A ∈ V}). Indeed, something more holds. In the next lemma,

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recall that a subvariety VF ⊆ V is flat, if Mod(Eq(VF ) ∪ {�x ≈ x}) is the trivialvariety.

Lemma 3.3. Let V = V (K) be quasi-subtractive with compatible box. Then thefollowing hold:

(1) For the open subvariety VO of V , we have VO = V (K�).(2) For any flat subvariety VF of V , we have VF ∩ K� ⊆ T , where T is the trivial

variety of an appropriate type.

Proof. Clearly, any member of V (K�) belongs to VO. To show the converse, takeA ∈ VO. Then, A ∈ HSP (K), but since A� = A, by Lemma 3.2 we get A ∈HSP (K�). This proves (1), and then (2) follows immediately by remarking thatVF ∩ VO = T . Observe that the inclusion in (2) can be strengthened to equality ifK� contains the trivial algebra.

Open congruences have pleasant properties in the varieties we are investigating.

Lemma 3.4. θσ(a, b) = θ(a, b) ∨ ker�.

Proof. θσ(a, b) ≤ θ(a, b)∨ker � because the latter congruence is open and collapsesa and b. Conversely, θ(a, b) ≤ θσ(a, b) by definition and ker� ≤ θσ(a, b) becauseker� is the smallest open congruence on A.

Lemma 3.5. θσ(a, b) = θσ(�a,�b).

Proof. Clearly, (�a,�b) ∈ θσ(a, b), whence θσ(�a,�b) ≤ θσ(a, b). For the otherinclusion, by Lemma 3.4 θσ(a, b) = θ(a, b)∨ker�; now, ker� ≤ θσ(�a,�b) becauseit is the smallest open congruence, and θ(a, b) ≤ θσ(�a,�b) as

aθσ(�a,�b)�aθσ(�a,�b)�bθσ(�a,�b)b.

Corollary 3.6. Every principal congruence in Conσ(A) has the form θσ(a, b),where a, b are open elements.

4. Quasi-Discriminator Varieties: Definition and Examples

We are now ready to introduce the notion of a quasi-discriminator variety.

Definition 4.1. Let A be an algebra whose type ν includes a unary term �. Afunction n : A4 → A defined by:

n(a, b, c, d) =

{c if �a = �bd if �a �= �b

,

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is a (quaternary) quasi-discriminator function on A. A quaternary term n(x, y, z, w)of type ν that realizes the quasi-discriminator function on A (respectively, on theclass K) is called a quasi-discriminator term for A (respectively, for K). An alge-bra A is called a quasi-discriminator algebra if there exists a quasi-discriminatorterm for A. A class K is called a quasi-discriminator class if there exists a quasi-discriminator term for K and � is V (K)- idempotent and V (K)-compatible. If K isa quasi-discriminator class, then we say that V (K) is a quasi-discriminator variety.

Discriminator varieties can be equivalently defined in terms of the quaternarydiscriminator or of the ternary discriminator. The same option is also available inour setting, although it takes a little more work to establish the equivalence. Infact, let A be an arbitrary algebra of type ν. Suppose there exists a function t suchthat

t(a, b, c) =

{c if �a = �ba if �a �= �b

,

hold for all a, b, c ∈ A. We will call such a function t a ternary quasi-discriminatorfunction, and a ternary term that realizes it on the algebra A a ternary quasi-discriminator term for A. Observe that if t is a ternary quasi-discriminator functionon A, then t(a, b, a) = a holds for any a, b ∈ A.

Lemma 4.2. Given a ternary term t, define quaternary terms p and q by

p(x, y, u, v) = t(v, t(x, y, v), t(x, y, u)),

q(x, y, u, v) = t(v, t(y, x, v), t(y, x, u)).

If t is a ternary quasi-discriminator term for an algebra A, then the following holdfor any a, b, c, d ∈ A:

pA(a, b, c, d) =

c if �a = �b,d if �d �= �a �= �b,a if �d = �a �= �b,

qA(a, b, c, d) =

c if �b = �a,d if �d �= �b �= �a,b if �d = �b �= �a.

Proof. If �a = �b, we have p(a, b, c, d) = t(d, t(a, b, d), t(a, b, c)) = t(d, d, c) = c,and q(a, b, c, d) = t(d, t(b, a, d), t(b, a, c)) = t(d, d, c) = c. If �a �= �b, thenp(a, b, c, d) = t(d, a, a), and q(a, b, c, d) = t(d, b, b), and thus p(a, b, c, d) = d if�d �= �a, and p(a, b, c, d) = a otherwise. Similarly, q(a, b, c, d) = d if �d �= �b,and q(a, b, c, d) = b otherwise.

Lemma 4.3. Let t, p and q be as in Lemma 4.2. Define

n(x, y, u, v) = t(p(x, y, u, v), x, q(x, y, u, v)).

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If t is a ternary quasi-discriminator term on an algebra A, then we have

nA(a, b, c, d) =

{c if �a = �bd if �a �= �b

,

for any a, b, c, d ∈ A.

Proof. If �a = �b, then n(a, b, c, d) = t(p(a, b, c, d), a, q(a, b, c, d)) = t(c, a, c) = c.We now suppose �a �= �b, and go through a case-splitting argument. If �d �= �aand �d �= �b, then n(a, b, c, d) = t(d, a, d) = d. If �d = �a, then �d �= �b, andso n(a, b, c, d) = t(a, a, d) = d. If �d = �b, then �d �= �a, and so n(a, b, c, d) =t(d, a, b) = d. Since the case �d = �a, �d = �b is impossible, this ends theproof.

Theorem 4.4. An algebra A has a ternary quasi-discriminator term if and onlyif it has a quaternary quasi-discriminator term.

Proof. The forward direction follows by Lemma 4.3. The backward direction fol-lows by taking t(x, y, z) = n(x, y, z, x).

All discriminator varieties are clearly quasi-discriminator: it suffices to pick theidentity term as �. The examples that follow show that the concept introduced inDefinition 4.1 is a proper generalization of the notion of discriminator variety.

Example 4.5. MTL algebras [26] are the equivalent variety semantics of monoidalT-norm based logic, one of the fundamental T-norm based fuzzy logics, roughlycorresponding to the substructural logic FLew augmented with the prelinearityaxiom (α → β) ∨ (β → α). In general, MTL algebras fail to be quasi-subtractivewith respect to �x = ¬¬x and x⇒ y = ¬¬(x → y), because the quasi-subtractiveaxiom Q1 is not generally satisfied. An MTL algebra that does satisfy the equation¬¬(¬¬x → x) ≈ 1 is called a Glivenko MTL algebra by Cignoli and Torrens [18].Glivenko MTL algebras that satisfy (¬((¬x)2))2 ≈ ¬(¬(x2))2 are called DL algebrasby the same authors, showing as well that a variety of Glivenko MTL algebras isa variety V of DL algebras just in case it has the Boolean retraction property: eachalgebra in V admits a homomorphism onto its subalgebra of Boolean elements.Examples of DL algebras include: pseudocomplemented MTL algebras, hence inparticular Godel algebras and Product algebras [35], as well as the variety of MValgebras generated by Chang’s algebra [17]. Defining ∆x = ¬(¬x2)2, and observingthat this notation is unambiguous in any DL algebra, it turns out that the varietyDL of DL algebras is not discriminator, but is quasi-discriminator witness the terms

n(x, y, z, w) = ((∆x ↔ ∆y) → z) ∧ ((∆x↔ ∆y) ∨ w),

�x = ∆x. (4.1)

Here, the generating class can be taken to be e.g. the class of all DL chains. In anypseudocomplemented MTL algebra, in particular, choosing �x = ¬¬x will do the

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job. Recently, this investigation has been extended to FLew algebras with Booleanretracts in [19].

Example 4.6. Nelson algebras are the equivalent variety semantics of constructivelogic with strong negation (see e.g. [57]). In [51, 52] it is shown that the varietyof Nelson algebras is term equivalent to a particular variety NRL of 3-potentFLew-algebras. NRL fails to be a discriminator variety. However, its subvarietyNR of regular Nelson residuated lattices (for which see [13] and, for their termequivalent counterpart, [16]) is quasi-discriminator with the same witness terms asDL algebras.

Example 4.7. The variety SBLA0 (Sec. 2.5) fails to be a discriminator variety —indeed, it satisfies no nontrivial congruence lattice identity. However, it is quasi-discriminator witness the terms

n(x, y, z, w) = q((x′ ∨ c(y)) ∧ (y′ ∨ c(x)), z, w),

�x = c(x).

Example 4.8. Orthomodular lattices with states were introduced in [24], wherethe relevance of states to the hidden variable program in quantum mechanics andto the representation theory of orthostructures is thoroughly explained. The varietyOMLσ of Jauch–Piron orthomodular lattices whose states σ are such that if σ(a) =σ(b) = 1, then σ(a ∧ b) = 1, is quasi-discriminator with witness terms

n(x, y, z, w) = (((σ(x) ∧ σ(y)) ∨ (σ(x)′ ∧ σ(y)′)) → z)

∧((σ(x) ∧ σ(y)) ∨ (σ(x)′ ∧ σ(y)′) ∨ w),

�x = σ(x),

where → denotes the Sasaki hook.

Example 4.9. An example of a simple pointed quasi-discriminator algebra whichfails to be discriminator is the two-element algebra F2 = 〈{a, 1}, n,�, 1〉, where nand � are defined as follows, for any b, c, d, e ∈ {a, 1}:

n(b, c, d, e) = d,

�b = 1.

5. Basic Properties

The reason quasi-discriminator varieties are of interest in the context of quasi-subtractive varieties is provided by the following theorem.

Theorem 5.1. Let V be a variety whose type includes the unary term � and theconstant 1. If V is a quasi-discriminator variety, then it is quasi-subtractive withrespect to � and 1.

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Proof. Define x→ y as n(1, x,�y, 1).

From now on, therefore, we can specialize to the present case all the conceptsand results obtained for quasi-subtractive varieties, although none of the exampleslisted in the table preceding Definition 2.2 counts, in general, as quasi-discriminator.Indeed, Definition 4.1 even ensures that every pointed quasi-discriminator varietyis quasi-subtractive with compatible box. Lemma 3.1 assumes a more attractiveform.

Lemma 5.2. (1) The class {A/ker� : A ∈V} generates a discriminator varietythat coincides with VO, the open subvariety of V.

(2) Conσ(A) is a distributive sublattice of permuting congruences of Con(A).

The previous lemma illustrates the extent to which the lattice of congruencesof an algebra in a quasi-discriminator variety is “well-behaved”: it is such as far asopen congruences go (and, in general, not further than that: see Example 4.7). Withreference to the notions introduced in Sec. 2.3, it is natural to guess that althougha quasi-discriminator variety need not be arithmetical, it is at least �-arithmetical.

Lemma 5.3. Every quasi-discriminator variety V is �-arithmetical.

Proof. By Theorem 2.16, it suffices to show that there exists a term m such that

V � m(x, y,�x) ≈ m(x,�y, y) ≈ m(y,�y, x) ≈ �x.It can be checked that m(x, y, z) = t(�x,�y,�z) does the job.

In the light of Theorem 2.17, the following version of Jonsson’s Lemma holdsfor quasi-discriminator varieties.

Corollary 5.4. If V (K) is quasi-discriminator, then all s.i. open members of V (K)belong to HSPU (K).

A discriminator variety can be equivalently defined as a variety that is gener-ated by some class of discriminator algebras, or as a variety whose s.i. membersare discriminator algebras (in both cases, of course, it is necessary to assume theexistence of a common discriminator term). This circumstance, in principle, maynot be guaranteed in our context, although we will observe in Lemma 6.2 that asimilar equivalence is after all true. To see that the matter is less than obvious,recall the following theorem.

Theorem 5.5 ([12, Theorem IV.9.4]). For a member A of a discriminatorvariety V , the following are equivalent:

(1) A is simple;(2) A is s.i.;(3) A is a discriminator algebra;(4) A ∈ SPU (K+), where K is the class of simple algebras in V.

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Fig. 2. Inclusion relationships.

Here, on the other hand, these concepts can be separated. To give a picture ofthe situation which is as precise as possible, we need an auxiliary concept.

Definition 5.6. An algebra A is said to be σ-simple if and only if its only opencongruences are ker� and ∇.

Clearly, whenever box is the identity, an algebra is σ-simple just when it issimple. We now try to determine which of the equivalences in Theorem 5.5 carryover to our new framework.

Theorem 5.7. The inclusion relationships (Fig. 2) among subclasses of a quasi-discriminator variety V (K) hold (where K is any generating class).

The inclusions represented by unidirectional arrows are proper.

Proof. (1) If A is s.i. and open, it is a discriminator algebra, hence simple. As wehave observed in Example 4.9, F2 is a simple quasi-discriminator algebra that failsto be open.

(2) Being simple implies being s.i. The algebra 3 in Example 2.25 is subdirectlyirreducible but not simple.

(3) To show that s.i. implies σ-simple, we prove the stronger result that alldirectly indecomposable members of V (K) are σ-simple. Thus, suppose A ∈ V (K),and let θ be an open congruence on A such that ker� < θ < ∇. Considerthe open algebra A/ker� ∈ V (K)O . θ/ker� is still a nontrivial congruence ofsuch. Therefore, by the theory of discriminator varieties, it is a nontrivial factorcongruence which induces a decomposition operationb on A/ker�. Then A is afortiori directly decomposable. To see that the inclusion is proper, observe that thefour-element generator 4 = ({0, 1, a, b}, q, 0, 1) of the quasi-discriminator varietySBLA0 (Sec. 2.5) is σ-simple but not subdirectly irreducible. In fact, its lattice ofcongruences is shown in Fig. 3.

bThe definition of decomposition operation is briefly recalled below, in Sec. 6.

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Fig. 3. The congruence lattice of 4.

(4) We show that being a quasi-discriminator algebra implies being σ-simple. Ifa, b ∈ A and �a �= �b, then

(c, d) = (n(a, a, c, d), n(a, b, c, d)) ∈ θ(a, b)

i.e. �a �= �b implies θ(a, b) = ∇. However, if θ > ker�, θ identifies at leasttwo distinct open members of A. The converse implication is proved below, inTheorem 8.6.

(5) By a standard ultraproduct argument (see [12]) we show that if n is a qua-ternary quasi-discriminator term for each Ai ∈ K, then it is a quasi-discriminatorterm for

∏i∈I Ai/U (where U is any nonprincipal ultrafilter of I) and thus for

every B ≤∏

i∈I Ai/U . Now, consider the four-element semi-Boolean-like alge-bra B with universe {0, 1, a, b}, unequivocally specified by the stipulation thatq(a, 1, 0) = 1 = q(b, 1, 0). B has a quasi-discriminator because its sole central ele-ments are 0 and 1. However, it does not belong to SPU ({4}+). In fact, 4 satisfiesthe universal formula

c(x) = 1 & c(y) = 1 & x �= 1 & y �= 1 ⇒ x = y

and thus SPU ({4}+), also satisfies it, unlike B.(6) By Corollary 5.4, all open s.i. members of V (K) belong to HSPU (K). If

HS(H) denotes the class of all quotients of members of H modulo an open congru-ence, then we conclude that all open s.i. members of V (K) belong to HSSPU (K), forno open algebra can arise as a quotient modulo a non-open congruence. By items (4)and (5), then, open quotients yield nothing new and all s.i. open members of V (K)belong to SPU (K+). Therefore, all such algebras have n as a quasi-discriminatorterm. Any non-open algebra which generates a quasi-discriminator variety (e.g. thestandard Godel algebra over the [0, 1] interval) is a counterexample to the converseimplication.

Being in SPU (K+) does not imply being s.i., let alone simple (the semi-Boolean-like algebra 4 is a counterexample); we do not know whether being subdirectlyirreducible or simple implies being a member of SPU (K+).

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In the double pointed case, algebras in quasi-discriminator varieties areamenable to weak Boolean product representations. We first prove the followinglemma.

Lemma 5.8. Let A be a nontrivial algebra in a double pointed quasi-discriminatorvariety V. Then A is directly indecomposable if and only if it is σ-simple.

Proof. One direction follows from the proof of Theorem 5.7(3). Now, suppose thatA is σ-simple, and suppose ex absurdo that θ1, θ2 are nontrivial factor congruences.Then θ1 ∨ θ2 = θ1 ◦ θ2 = ∇, whence given a �= b in A, for some c we have thataθ1cθ2b, whence �aθ1�cθ2�b. Therefore, if �a = �c = �b, then ∇ = θ1 ◦ θ2 ⊆ker�, which is impossible because V is double pointed; if without loss of generality�a �= �c, then θ1 collapses two distinct open elements and therefore coincides withthe full relation since A is σ-simple. But this is a contradiction for we had supposedθ1, θ2 to be nontrivial factor congruences.

The next theorem can be profitably compared to the weak Boolean productrepresentation results in [18, Sec. 4] and with Busaniche and Cignoli’s weak Booleanproduct representation of regular Nelson residuated lattices [13, Sec. 6].

Theorem 5.9. Every algebra in a double pointed quasi-discriminator variety V isrepresentable as a weak Boolean product of directly indecomposable (hence σ-simple)algebras.

Proof. To begin, we observe that every double pointed quasi-discriminator alge-bra, with residually distinct constants 0, 1, is a Church algebra witness the termq(x, y, z) = n(x, 0, z, y). Therefore, every double pointed quasi-discriminator vari-ety V is a Church variety. By Theorem 2.22, therefore, every algebra in V is rep-resentable as a weak Boolean product of directly indecomposable algebras, for (inthe light of Lemma 5.8) the latter are defined by the universal formulas

t(x, x, z) = z,

∀xyz (�x �= �y ⇒ t(x, y, z) = x).

Observe that, in general, one cannot hope to do any better than this, becauseVaggione ([55]; cf. also [44]) has shown that every variety with factorable con-gruences whose members have a Boolean product representation with directlyindecomposable factors is a discriminator variety. On the other hand, there areproper quasi-discriminator varieties with factorable congruences (e.g. SBLA0:see [43]).

6. An Equational Characterization

We now provide an equational characterization for quasi-discriminator varieties,along the lines of the corresponding result proved by Vaggione for discriminator

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varieties [56]. Let us briefly recall some notions first. A decomposition operator onan algebra A is a homomorphism D : A × A → A such that, for all a, b, c, d ∈ A:

(1) D(a, a) = a;(2) D(D(a, b), c) = D(a, c) = D(a,D(b, c)).

With every pair of complementary factor congruences ρ, θ on A we can associatea decomposition operator Dρ,θ(a, b), yielding the unique c such that aρc and cθb.Conversely, from a decomposition operator D we obtain a pair of complementaryfactor congruences ρD, θD defined by ρD = {(a, b) : D(a, b) = b} and θD = {(a, b) :D(a, b) = a}. These correspondences are mutually inverse [46].

Theorem 6.1. A variety V of the appropriate type ν is quasi-discriminator if andonly if there exists a term N such that the following identities are satisfied in V ,for any n-ary operation symbol f of type ν, with n ≥ 1:

(QD1) N(x, y, f(�x), f(�y)) ≈ f(N(x, y, x1, y1), . . . , N(x, y, xn, yn));(QD2) N(x, x, y, z) ≈ y;(QD3) N(x, y, z, z) ≈ z;(QD4) N(x, y,N(x, y, r, s), z) ≈ N(x, y, r, z) ≈ N(x, y, r,N(x, y, s, z));(QD5) N(x, y,�x,�y) ≈ �y ≈ N(y, x,�x,�y);(QD6) N(�x,�y, z, w) ≈ N(x, y, z, w).

Proof. (⇒) It is straightforward to verify that, if V is quasi-discriminator, theseidentities are satisfied with N = n.(⇐) Let A be a subdirectly irreducible member of V that satisfies the above iden-tities. Let a, b, c, d ∈ A, with �a �= �b. Conditions (QD3), (QD4), and (QD1)guarantee that D(z, w) = N(x, y, z, w) is a decomposition operator on A. By con-dition (QD5), �aρD�b. But A is directly indecomposable, and then ρD = ∇.Therefore, N(a, b, c, d) = d if �a �= �b. In case �a = �b, then by (QD6) and(QD2) N(a, b, c, d) = N(�a,�b, c, d) = N(�a,�a, c, d) = c, and therefore N is aquasi-discriminator term for A. It follows that V is quasi-discriminator.

Now, from the proof of Theorem 6.1 it follows that there exists a term n whichis a quaternary quasi-discriminator term on each subdirectly irreducible algebra inV . Since the ternary quasi-discriminator term t is defined from the quaternary one,it follows that the same remarks, mutatis mutandis, hold for t. Let us state thisstraightforward observation as a lemma, because it will play an essential role inwhat follows.

Lemma 6.2. Let V = V (K) be a quasi-discriminator variety. Then there existterms n and t such that n is a quaternary quasi-discriminator term and t a ternaryquasi-discriminator term on any subdirectly irreducible algebra A ∈ V.

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If V is a quasi-discriminator variety, then its open subvariety VO is discriminator;at the other end of the spectrum, any of its flat subvarieties is, intuitively, “as non-discriminator as possible” because it is trivialized by the equation �x ≈ x. We arenow trying to make somewhat more precise the intuitive idea that VF and VO areas far apart as possible among the subvarieties of V . By [40, Lemma 44], if V ispointed, for any flat subvariety VF of V there exists a unary term �x such thatVO |= �x ≈ x ≈ �x and VF |= �x ≈ 1. This term is in general not unique, so wewill refer to it as a full flat box of VF .

Lemma 6.3. Let V = V (K) be a pointed quasi-discriminator variety, with ternaryquasi-discriminator term t(x, y, z). Let VF be a flat subvariety of V with a full flatbox �. Then, there exists a unary term � such that VO |= �x ≈ 1 and VF |= �x≈x.

Proof. Define �x = t(1,�x, x). Then, by Lemma 6.2, on any subdirectly irre-ducible A ∈ VO we have for any a ∈ A, t(1,�a, a) = a if ��a = 1 and t(1,�a, a) =1 otherwise. Since �a = �a = a holds in VO, we get � � a = ��a = �a = a, andthus t(1,�a, a) = a if a = 1 and t(1,�a, a) = 1 otherwise, so t(1,�a, a) = 1. Thus,�x ≈ 1 holds on an arbitrary subdirectly irreducible algebra in VO and thereforeVO |= �x ≈ 1 as desired. Now, reasoning similarly in VF , we get that for anya ∈ A ∈ VF , t(1,�a, a) = a if � � a = 1 and t(1,�a, a) = 1 otherwise. However,�a = 1, so the second case never obtains. Hence, VF |= �x ≈ x as needed.

Recall that two similar varieties V1,V2 of type L are said to be independent ifthere exists a term x y of type L, containing at most the indicated variables, suchthat V1 � x y ≈ x and V2 � x y ≈ y. According to a classic result by Gratzeret al. [33], if V1,V2 are independent, then V1 ∨ V2 = V1 × V2.

Lemma 6.4. Let V be a pointed quasi-discriminator variety, with ternary quasi-discriminator term t(x, y, z). Let VF be a flat subvariety of V with a full flat box �.Then, VF and VO are independent.

Proof. Let � be the term proved to exist in Lemma 6.3. Define x y= t(�x, 1,�y).On any subdirectly irreducible algebra in VF we have t(�a, 1,�b) = t(1, 1,�b) =�b = b for arbitrary a, b. Similarly, on any subdirectly irreducible algebra in VO wehave t(�a, 1,�b) = t(a, 1, 1) = 1 if �a = a = 1, and t(�a, 1,�b) = t(a, 1, 1) = a

otherwise. It follows that VO |= x y ≈ x and VF |= x y ≈ y as claimed.

Theorem 6.5. Let V be a pointed quasi-discriminator variety, and VF a flat sub-variety of V. Then VO ∨ VF = VO × VF .

Proof. By Lemma 6.4 and the remark immediately preceding it.

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7. Open Congruences

It is well-known that discriminator varieties have EDPC. In our more general set-ting, principal open congruences are always equationally definable.

Definition 7.1. A variety V of the appropriate type has equationally definableprincipal open congruences (EDPOC) if and only if there exist terms pi, qi (i ≤ n)of the same type such that for all A ∈ V and all a, b, c, d ∈ A,

(c, d) ∈ θσ(a, b) if and only if pAi (a, b, c, d) = qAi (a, b, c, d) (i ≤ n).

In the light of [44], Theorem 6.1 yields the following theorem.

Theorem 7.2. Every quasi-discriminator variety V = V (K) has EDPOC. If n isa quasi-discriminator term for K, then

n(x, y,�z, x) ≈ n(x, y,�w, x)

defines principal open congruences on V.

Proof. Let A ∈ V , and a, b, c, d ∈ A. Set

ψ = {(c, d) ∈ A2 : n(a, b,�c, a) = n(a, b,�d, a)}.

Evidently, ψ is an equivalence relation, and an (open) congruence at that by (QD1).In the light of Lemma 3.5 and Corollary 3.6, we can assume without loss of generalitythat a, b are open. It can be seen that (a, b) ∈ ψ, for

n(a, b, a, a) =(QD3) a

=(hyp.) �a

=(QD5) n(a, b,�b,�a)

=(hyp.) n(a, b, b, a).

On the other hand, if (c, d) ∈ ψ, then

�c = n(a, a,�c, a)θσ(a, b)n(a, b,�c, a)

= n(a, b,�d, a)θσ(a, b)n(a, a,�d, a) = �d.

So, (�c,�d) ∈ θσ(a, b). Thus, by definition of θσ(a, b), (c, d) ∈ θ(a, b). Therefore,ψ = θ(a, b).

Observe that this lemma yields something more: principal congruences gen-erated by open elements are equationally definable. This result is stronger:in fact, consider the five-element pure semi-Boolean-like algebra with universe{0, a, 1, a′, u}, whose center is the four-element Boolean-like algebra, and whosesole non-central element is u such that �u = 1. The congruence θ whose cosets are

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{0, a}, {1, a′}, {u} is principal and generated by the open pair (0, a), but it fails tobe open for (u, 1) = (u,�u) /∈ θ.

Lemma 7.3. If V is a quasi-discriminator variety and A ∈ V , for a, b, c, d ∈ A,

upon settingc:

φa,b = {(c, d) : [a = b] ⊆ [c = d]},

φ′a,b = {(c, d) : [�a = �b] ⊆ [c = d]}

we get

θ(�a,�b) = φ′a,b ≤ θ(a, b) ≤ φa,b.

Proof. Evidently, φa,b, φ′a,b ∈ Con(A), and θ(�a,�b) ≤ φ′a,b, θ(a, b) ≤ φa,b,

θ(�a,�b) ≤ θ(a, b). As regards the missing inclusion, if [�a = �b] ⊆ [c = d],then

d = n(�a,�b, c, d)θ(�a,�b)n(�a,�a, c, d) = c.

The following theorem is obtained.

Theorem 7.4. If V has EDPOC and A ∈ V , then the join semilattice of compactopen congruences on A is dually relatively pseudocomplemented.

Proof. It is proved in [38] that the join semilattice of compact congruencesin an algebra with EDPC is dually relatively pseudocomplemented. However,Lemma 3.1(3) implies that the join semilattice of compact open congruences ofA is isomorphic to the join semilattice of all compact congruences of A/ker�,which is in a discriminator variety by Lemma 5.2 and therefore has EDPC, whenceour conclusion.

8. A Purely Algebraic Characterization

One of the deepest results in the theory of discriminator varieties gives the followingpurely algebraic characterization thereof: a variety is discriminator if and only if it iscongruence permutable, semisimple, and has EDPC [29; 6, Corollary 3.4]. One maywonder if a suitable analog of this theorem holds in the case under investigation.The aim of the present section is answering the question in the affirmative (at leastunder some modest assumptions).

Let A be an algebra in a variety with EDPOC. From Theorem 7.4 it followsthat the dual relative pseudocomplement θ ∗ φ exists for compact members θ, φ

cTaking a subdirect representation of A with index set I, for a, b ∈ A the notation [a = b] refersas usual to the set {i ∈ I : ai = bi}.

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of Conσ(A). If Cpσ(A) denotes the lattice of all such members (or, by abuse ofnotation, its universe), we define a new operation in Cpσ(A) ∪ {∆} as follows:

θ � φ =

θ ∗ φ if φ �≤ θ and θ, φ ∈ Cpσ(A);

∆ if φ ≤ θ;

φ if θ = ∆.

It can be checked that this operation is well-defined. We write Cp∆σ (A) in place

of Cpσ(A) ∪ {∆} for the sake of brevity.

Lemma 8.1. Let V be a variety with EDPOC, and A ∈ V. Then Cp∆σ (A) is a

dually relatively pseudocomplemented semilattice.

Proof. Let θ1, θ2 ∈ Cp∆σ (A). We prove that � is a dual relative pseudocomplement.

To this purpose, we verify that θ1 � θ2 ≤ θ3 if and only if θ2 ≤ θ1 ∨ θ3, forθ3 ∈ Cp∆

σ (A).

• If θ1, θ2, θ3 ∈ Cpσ(A) and θ2 �≤ θ1, and θ3 �= ∆, then, θ1 � θ2 = θ1 ∗ θ2 ≤ θ3 ifand only if θ2 ≤ θ1 ∨ θ3. The case θ3 = ∆ yields an equivalence between falsesentences.

• If θ2 ≤ θ1, then immediately ∆ = θ1 � θ2 ≤ θ3 if and only if θ2 ≤ θ1 ∨ θ3.• The case θ1 = ∆ is straightforward.

We now introduce a useful variant of the concept of quaternary deduction term(QD-term), introduced by Blok and Pigozzi [6, Sec. 3] as a generalization of thequaternary discriminator to non-semisimple varieties.

Definition 8.2. Let V be a variety whose type ν includes a unary term �. Aquaternary term q of the same type is called a �-quaternary deduction (�-QD)term for A ∈ V if and only if for all a, b, c, d ∈ A:

qA(a, b, c, d) =

c if �a = �b;d if c ≡θσ(a,b)d and �a �= �b;arbitrary otherwise.

q is called a �-QD term for V if and only if it is a �-QD term for any A ∈ V .

In the next theorem, θσ(x, y)2 abbreviates θσ(x, y) � θσ(x, y).

Theorem 8.3. Every �-permutable variety V with EDPOC has a �-QD term q.

Proof. Let F be the free V (K)-algebra over the four free generators x, y, z, w.By Lemma 8.1, both θσ(x, y) � θσ(z, w) and θσ(x, y) � θσ(z, w)2 = (θσ(x, y) �θσ(z, w))� θσ(z, w) exist in Cp∆

σ . By the theory of dually relatively pseudocomple-mented semilattices, (z, w) ∈ (θσ(x, y) � θσ(z, w)) ∨ (θσ(x, y) � θσ(z, w)2). So, by

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�-permutability, (z, w) ∈ (θσ(x, y)�θσ(z, w))◦(θσ(x, y)�θσ(z, w)2). Consequently,there is quaternary term q such that

wθσ(x, y) � θσ(z, w)q(x, y, z, w)θσ(x, y) � θσ(z, w)2z.

If we evaluate this condition on an algebra A we obtain that, in case �a = �b:θσ(a, b) � θσ(c, d)2 = ker� � θσ(c, d)2 = θσ(c, d) � θσ(c, d) = ∆. Therefore in A,q(a, b, c, d) = c. Otherwise, if (c, d) ∈ θσ(a, b), then θσ(a, b) � θσ(c, d) = ∆, whenceq(a, b, c, d) = d.

Definition 8.4. A variety V of the appropriate type is σ-semisimple if and only ifall s.i. members of V are σ-simple.

Theorem 8.5. For a variety V of the appropriate type, the following are equivalent:

(1) V is quasi-discriminator;(2) V is �-permutable, σ-semisimple and has EDPOC.

Proof. (1)⇒ (2) If V is quasi-discriminator, then V is �-permutable (byLemma 5.3), is σ-semisimple (by Theorem 5.7(3)) and has EDPOC (byTheorem 7.2).

(2)⇒ (1) In the light of Theorem 8.3, V has a �-QD term q. Let a, b ∈ A, whereA is a s.i. member of V . Then, whenever �a = �b, q(a, b, c, d) = c. Otherwise if�a �= �b, then θσ(a, b) = ∇ because A is σ-simple, and thus q(a, b, c, d) = d. So, qacts as a quasi-discriminator term on A.

This much is implicit in the proof of the above results.

Theorem 8.6. Let V (K) be a quasi-discriminator variety and A ∈ V (K). Thefollowing are equivalent:

(1) A is a quasi-discriminator algebra;(2) A is σ-simple.

What we just pointed out affords an alternative proof of Theorem 4.4.

Theorem 8.7. Let K be a class of algebras with a common ternary quasi-discriminator t. Then V (K) has EDPOC, with principal open congruencesdefined by

t(x, y,�z) ≈ t(x, y,�w).

Proof. For A in V (K) and a, b, c, d ∈ A set

φ = {(c, d) : t(a, b,�c) = t(a, b,�d)}

Apparently, φ is a congruence, since V (K) satisfies

t(x, y, f(�z)) ≈ t(x, y, f(t(x, y, z1), . . . , t(x, y, zn)))

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for any function f in the type. Further, mimicking the proof of Theorem 7.2, φ =θσ(a, b).

If K is as above, then every A ∈ K is σ-simple. In fact, if θ > ker� and a, b aredistinct open elements such that aθb, we have that: a = t(a, a, c)θt(a, b, c) = c. Soθ = ∇. Also, V (K) is �-permutable. Therefore, it has a �-QD term q, by Theorem8.3. Combining all those ingredients together, we have the desired result because theQD-term q realizes the quaternary quasi-discriminator function on every σ-simplealgebra.

9. The Pure Pointed Quasi-Discriminator Variety

In this section, we investigate the pure pointed quasi-discriminator variety PQ1,i.e. the variety of algebras of type (4, 1, 0) generated by all pure pointed quasi-discriminator algebras A = (A, n,�, 1), where n is the quasi-discriminator on A,and � is the quasi-subtractive box.

Using the terminology of Sec. 3, PQ1 is contractive in virtue of Lemma 3.2.Moreover, borrowing some more terminology from [40].

Lemma 9.1. PQ1 is invertible: for every algebra A ∈ PQ1 and every congruenceφ on A�, there is a congruence θ on A such that φ = θ|A� .

Proof. Let A ∈ PQ1. By contractivity, A� = �A. Therefore, we have to showthat, for every congruence φ on �A, there is a congruence θ on A such that φ =θ|�A. Defining

aθb if and only if (a, b ∈ A� and aφb) or a = b

does the job.

Lemma 9.2. Let A = (A, n,�, 1) be an algebra in PQ1. Then A is subdirectlyirreducible if and only if either A is a discriminator algebra (hence simple), or|A| = |A�| + 1, with �A a discriminator algebra.

Proof. From left to right, if |A| > |A�| + 1, then there are distinct a, b ∈ A suchthat a, b �∈ A�. Thus, set

ξ = ∆ ∪ {(a,�a)},

ξ′ = ∆ ∪ {(b,�b)},

where ξ and ξ′ are congruences: it can be verified that n is preserved appealingonly to equational properties of n. Moreover, ξ ∩ ξ′ = ∆, which contradicts thefact that A is subdirectly irreducible. On the other hand, if neither A nor �A is adiscriminator algebra, then �A fails to be s.i. and thus there are atomic congruencesξ, ξ′ therein such that ξ∩ξ′ = ∆. By Lemma 9.1, such congruences extend to atomiccongruences θ, θ′ on the whole of A such that θ ∩ θ′ = ∆.

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Conversely, if |A| = |A�|+1, and a ∈ A but a �∈ A�, then ker� = ∆∪{(a,�a)}is the monolith. In fact, it is clearly an atom, and a coatom as well because �A isa discriminator algebra and thus simple. Moreover, if θ collapses a with the openelement b, then (b,�a) ∈ θ and thus ker� < θ = ∇.

Theorem 9.3. Let F2 be the quasi-discriminator algebra of Example 4.9. The fol-lowing varieties are mutually coincident:

(1) Mod(�x ≈ 1);(2) Mod(n(x, y, z, w) ≈ z);(3) V (F2).

Proof. Let e, b, c, d ∈ F2.(1 ⊆ 2) n(e, b, c, d) =(QD6) n(1, 1, c, d) = c.(1 ⊇ 2) �e =(QD5) n(e, 1, 1,�e) = 1.(1 ⊆ 3) Let A be a subdirectly irreducible algebra satisfying �x ≈ 1. Then

|A�| = 1, and, by Lemma 9.2 |A| = 2. That is A � F2.(1 ⊇ 3) Evidently, F2 |= �x ≈ 1.

Following customary usage, we denote by PD1 the pure pointed discriminatorvariety, i.e. the variety of algebras of type (4, 0) generated by the class of all purepointed discriminator algebras A = (A, n, 1), where n is the quaternary discrimi-nator on A and 1 is a constant. With a slight abuse, we will identify it with thevariety of type (4, 1, 0) axiomatized relative to PQ1 by the equation �x ≈ x.

Lemma 9.4. PD1 and V (F2) are independent.

Proof. Evidently, n(x, y, y, x) witnesses independence of the two varieties.

The already mentioned results in [33] imply the following corollary.

Corollary 9.5. The join of PD1 and V (F2), taken in the lattice of subvarieties ofPQ1, is their direct product.

Of course, this corollary can also be obtained as a special case of Theorem 6.5,in light of Theorem 9.3.

Definition 9.6. Let A = 〈A, n, 1〉 be a pure pointed discriminator algebra, a ∈ A

and a′ /∈ A. Define Aa′= 〈A∪{a′}, n′,�,1〉 to be the algebra whose operations are

defined as follows, for any b, c, d, e ∈ A ∪ {a′}:

�b =

{b if b ∈ A;

a otherwise;

n′(b, c, d, e) =

{d if �b = �c;e if �b �= �c.

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It follows directly that Aa′is a s.i. (by Lemma 9.2) quasi-discriminator algebra,

which is not discriminator, and A ≤ Aa′, since n′ �A= n.

Definition 9.7. Let A be an algebra in a quasi-discriminator variety. We set

P�k (A) = {(a1, . . . , ak) ∈ Ak : ∀ i, j ≤ k(�ai = �aj)}.

Observe that if A is an algebra in a discriminator variety, then for every k,P�

k (A) only contains constant sequences and is thus isomorphic to A, while ifA ∈ V (F2) it follows that P�

k (A) = Ak. In general, P�k (A) is a subalgebra of the

k-th power of A.

Lemma 9.8. If A is a quasi-discriminator algebra, then for every k ≥ 1, P�k (A)

is a quasi-discriminator algebra.

Proof. Let −→a ,−→b ,−→c ,−→d ∈ P�k (A), and consider n(−→a ,−→b ,−→c ,−→d ). If �−→a = �−→

b ,then clearly n(−→a ,−→b ,−→c ,−→d ) = −→c , for it is an equational property that is at stake.If �−→a �= �−→

b , then for some i ≤ k �ai �= �bi, and by definition �aj �= �bj for anyother j, then n(−→a ,−→b ,−→c ,−→d ) =

−→d .

Taking up suggestions from [50], we now introduce two useful equations:

t(x, 1, 1) ≈ x; (9.1)

t(x, 1,�x) ≈ �x. (9.2)

Intuitively, Eq. (9.1) says that if �a = 1 then a = 1. On the other hand, Eq. (9.2)asserts that if �a �= 1 then �a = a.

One may verify that if Aa′,Ab′ are quasi-discriminator algebras as in Defini-

tion 9.6, and a �= b in A − {1}, then Aa′ � Ab′ .d Therefore, V (Aa′) = V (Ab′).

Furthermore, if A,B are discriminator algebras, with |A| = |B|, then A � B,because, for discriminator algebras, the discriminator is univocally determined bydefinition. It is known (see e.g. [2, 3]) that the lattice of subvarieties of PD1 = Dω

is a chain of order type ω + 1, as depicted in Fig. 4, where, for each n, An is thepure pointed discriminator algebra of cardinality n+ 1.

In light of [2, Theorem 4.12], each Di is axiomatized relative to Dω by theequation

εi :n∨

i=0

n∨

j=0,j �=i

xj ⇒ xi

≈ 1,

dA moment’s reflection shows that any permutation p : A ∪ {a′} → A ∪ {b′} such that p(1) = 1,p(a) = b and p(a′) = b′ preserves both n and the quasi-subtractive box �.

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406 F. Paoli et al.

Fig. 4. The pure pointed discriminator subvarieties chain.

where, given a0, . . . , an, a, b ∈ A ∈ Di:

0∨i=0

ai = a0,

n+1∨i=0

ai = an ∨(

n∨i=0

ai

),

a ∨ b = t(a, t(a, 1, b), b),

a⇒ b = t(b, a, 1).

In accordance with our previous diagram, upon setting for any i ∈ N+, D1′i =

V (A1′i ), Db′

i = V (Ab′i ) and DFb′

i = V (Ab′i ,F2) the following is obtained.

Lemma 9.9. Let b �= 1 be an element of Ai. Then:

(1) Di is axiomatized relative to D1′i by

�x ≈ x;

(2) D1′i � Db′

i , and Db′i � D1′

i ;(3) D1′

i ∧ Db′i = Di;

(4) D1′i ∨ Db′

i = DFb′i ;

(5) For all i, F2 belongs to D1′i but not to Db′

i .

Proof. (1) is immediate.(2) Observe that A1′

i �|= t(x, 1, 1) ≈ x, but Ab′i |= t(x, 1, 1) ≈ x. Conversely,

A1′i |= t(x, 1,�x) ≈ �x, but Ab′

i �|= t(x, 1,�x) ≈ �x.(3) In every s.i. algebra B satisfying Eqs. (9.1) and (9.2), if b ∈ B and �b = 1,

then �b = t(b, 1,�b) = t(b, 1, 1) = b. If �b �= 1, then b = t(b, 1,�b) = �b.

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(4) D1′i ∨Db′

i ≤ DFb′i because A1′

i is a subalgebra of Ai×F2 via the map

f(c) =

{(c, 1) if �c = c,

(c, a) otherwise.

On the other hand, D1′i ∨ Db′

i ≥ DFb′i for F2 is a subalgebra of A1′

i .(5) That F2 belongs to D1′

i follows from the fact that F2 ∈ S(A1′i ). Also F2 /∈

Db′i because Ab′

i satisfies Eq. (9.1).

Lemma 9.10. For any i, j ∈ N+:

(1) Di ∨ V (F2) = D1′i ;

(2) If i < j, then D1′i ⊂ D1′

j ;(3) If i < j, then Db′

i ⊂ Db′j .

Proof. (1) Evidently, for any i ∈ N+, Ai belongs to D1′i and F2 is a subalgebra

of any A1′i . On the other hand, by the proof of Lemma 9.9, any A1′

i is isomorphicto a subalgebra of Ai × F2.

(2) and (3) are straightforward.

Lemmas 9.9 and 9.10 yield a complete description of the subvariety lattice ofPQ1, depicted in Fig. 5.

Now, let

�εi : �

n∨

i=0

n∨

j=0,j �=i

xj ⇒ xi

≈ 1.

Theorem 9.11. For each i < ω,

(1) DF b′i is axiomatized relative to PQ1 by �εi;

(2) Db′i is axiomatized relative to PQ1 by �εi and Eq. (9.1);

(3) D1′i is axiomatized relative to PQ1 by �εi and Eq. (9.2);

(4) Di is axiomatized relative to PQ1 by �εi and �x ≈ x.

Proof. (1) Since the generators of DF b′i satisfy �εi, DFb′

i ≤ Mod(�εi). Conversely,let A be a non-open s.i. algebra satisfying �εi. Then �A satisfies εi. Let c′ be theunique non-open element of A. If �c′ = 1, then A ∈ D1′

i ≤ DF b′i . If �c′ = b, then

A ∈ Db′i ≤ DF b′

i . The remaining items are proved similarly.

Observe that an alternative basis for Db′i (once again, relative to PQ1) is given

by the single equation εi. In fact, if A satisfies �εi and Eq. (9.1), it satisfies εi,since �εi has the form �q ≈ 1 for some term q. Conversely, it is clear that εiimplies �εi. Now, suppose ex absurdo that A satisfies εi and that there is 1′ �= 1such that �1′ = 1. Then εAi (

−→1′ ) = 1′ �= 1, a contradiction.

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408 F. Paoli et al.

Fig. 5. The subvarieties lattice of the pure pointed quasi-discriminator variety.

Lemma 9.12. If A is a nontrivial quasi-discriminator algebra, and e ∈ A, wheree �= 1, then A\({e} ∪ �−1(e)) is the universe of a quasi-discriminator subalgebraof A.

Proof. Since A is nontrivial quasi-discriminator algebra, then for elementsa, b, c, d ∈ A not in B = {e} ∪ �−1(e), either n(a, b, c, d) = c /∈ B or n(a, b, c, d) =d /∈ B. Moreover, if for a /∈ B, �a = b ∈ B, then either �a = e (a contradiction),or �a = c, with �c = e; then �a = �c = e, a contradiction again.

Since s.i. algebras in a quasi-discriminator variety are quasi-discriminator(because of Theorem 5.7), Lemma 9.12 holds true in case A is a s.i. algebra.Moreover, if A is a s.i. algebra in PQ1, �−1(e) is always a singleton or a pairby Lemma 9.2.

The next theorem generalizes an unpublished result of McKenzie from 1976asserting that the pure pointed discriminator variety is locally finite [45].

Theorem 9.13. The variety PQ1 is locally finite.

Proof. By [48, Theorem 1], it suffices to show that there exists a fixed integer-valued function f such that, for each nonnegative integer n, it is the case that every

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n-generated subdirectly irreducible algebra in PQ1 has at most f(n) elements. Inlight of Lemma 9.12 and the remark immediately following it, if A is a s.i. algebragenerated by n elements in PQ1, its cardinality is bounded above by n+ 2.

Acknowledgments

All authors gratefully acknowledge the useful comments of the anonymous referees.The second author acknowledges the support of the Italian Ministry of ScientificResearch (MIUR) within the FIRB project “Structures and Dynamics of Knowledgeand Cognition”, Cagliari: F21J12000140001. The third author gratefully acknowl-edges the support of ARC Future Fellowship grant FT100100952.

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