From Lambda Calculus to Universal Algebra and Back

(1)

From Lambda Calculus to Universal Algebra and Back

^∗

Giulio Manzonetto

^†◦

and Antonino Salibra

^◦

† Laboratoire PPS (Universit´e Paris 7) ◦ Dip. Informatica (Universit`a di Venezia) [email protected] [email protected]

Abstract

We generalize to universal algebra concepts originat- ing from lambda calculus and programming in order first to prove a new result on the lattice ofλ-theories, and sec- ond a general theorem of pure universal algebra which can be seen as a meta version of the Stone Representation The- orem. The interest of a systematic study of the latticeλT of λ-theories grows out of several open problems on lambda calculus. For example, the failure of certain lattice identities in λT would imply that the problem of the order- incompleteness of lambda calculus raised by Selinger has a negative answer. In this paper we introduce the class of Church algebras(which includes all Boolean algebras, combinatory algebras, rings with unit and the term algebras of all λ-theories) to model the if-then-else instruction of programming and to extend some properties of Boolean algebras to general universal algebras. The interest of Church algebras is that each has a Boolean algebra of central elements, which play the role of the idempotent elements in rings. Central elements are the key tool to represent any Church algebra as a weak Boolean product of directly indecomposable Church algebras and to prove the meta representation theorem mentioned above. We generalize the notion of easyλ-term and prove that any Church algebra with an “easy set” of cardinalitynadmits (at the top) a lattice interval of congruences isomorphic to the free Boolean algebra with n generators. This theorem has the following consequence forλT: for every recursively enumerableλ- theoryφand eachn, there is aλ-theoryφn ≥φsuch that {ψ:ψ≥φn}“is” the Boolean lattice with2ⁿelements.

1. Introduction

The lambda calculus is not a genuine equational theory since the variable-binding properties of lambda abstraction prevent variables in lambda calculus from operating as real

∗ Work partially supported by Equipe PPS of the University Paris 7- Denis Diderot, and by LIX Laboratoire d’Informatique de l’Ecole Polytechnique (Palaiseau, France).

algebraic variables. Consequently the general methods that have been developed in universal algebra, for defining the semantics of an arbitrary algebraic theory for instance, are not directly applicable. There have been several attempts to reformulate the lambda calculus as a purely algebraic theory. The earliest, and best known, class of algebraic models is that of Curry’s combinatory algebras [7] (CAs, for short).

This class has a simple equational characterization and was used to provide an intrinsic first-order, but not equational, characterization of the models of lambda calculus [1, Def.

5.2.7].

Lambda theories are equational extensions of the untyped lambda calculus closed under derivation. They arise by syntactical or semantic considerations. Indeed, a λ- theory may correspond to a possible operational semantics of lambda calculus, as well as it may be induced by a model of lambda calculus through the kernel congruence relation of the interpretation function. The set ofλ-theories is nat- urally equipped with a structure of complete lattice (see [1, Ch. 4]). The bottom element of this lattice is the least λ-theory λβ, while the top element is the inconsistentλ- theory. Although researchers have mainly focused their interest on a limited number of them, the lattice ofλ-theories has a very rich and complex structure (see e.g. [1, 15]).

The remark that the lattice of λ-theories is isomorphic to the congruence lattice of the term algebra of the least λ-theoryλβ is the starting point for studying lambda calculus by universal algebraic methods. In [22] Salibra has shown that the variety (i.e., equational class) generated by the term algebra ofλβ is axiomatized by the finite schema of identities characterizing the class ofλ-abstraction algebras (LAAs). The equational theory ofλ-abstraction algebras has been introduced by Pigozzi and Salibra in [20] and constitutes a purely algebraic theory of the untyped lambda calculus which keeps the lambda notation and hence all the functional intuitions. In [22] Salibra has shown that, for every variety ofLAAs, there exists exactly oneλ-theory whose term algebra generates the variety. Thus, the properties of a λ-theory can be studied by means of the variety of LAAs generated by its term algebra.

The axioms characterizingCAs andLAAs are suggested by an analysis of recursive processes, not by logic (as for

(2)

Boolean algebras) or by algebra (as for rings). The results developed in universal algebra in the last thirty years are not directly applicable to CAs and LAAs because these algebras are never commutative, associative, finite or recursive [1, Prop. 5.1.15]. In addition, Lusin and Salibra [15] have shown that the only lattice identities satisfied by all congruence lattices ofCAs (respectivelyLAAs) are the trivial ones.

This seemed to confirm the common belief that lambda calculus and combinatory logic are algebraically pathological.

In fact lambda calculus and combinatory logicdo satisfy interesting algebraic properties. One of the milestones of modern algebra is the Stone representation theorem for Boolean algebras, which was generalized by Pierce to commutative rings with unit and next by Comer to the class of algebras with “Boolean factor congruences” (see [19, 6]).

In [17] we have shown that Comer’s generalization of Stone representation theorem holds also for combinatory algebras:

any combinatory algebra is isomorphic to a weak Boolean product of directly indecomposable combinatory algebras (i.e., algebras which cannot be decomposed as the Carte- sian product of two other non-trivial algebras). The proof of this representation theorem was based on the fact that every combinatory algebra hascentral elements, i.e., elements from which one can define a direct decomposition of the algebra as the Cartesian product of two other combinatory algebras, just like idempotent elements do for rings. The central elements of a combinatory algebra constitute a Boolean algebra, and have been used in [17] to prove that the semantics of lambda calculus given in terms of directly indecomposable models, although it includes the Scott con- tinuous semantics and its refinements (see [1, 25, 21]), is equationally incomplete (i.e., it does not induce all consis- tentλ-theories).

The interest of a systematic study of the lattice λT of λ-theories grows out of several open problems on lambda calculus. For example, the problem of the order-incompleteness of lambda calculus, raised by Selinger, asks for the existence of a λ-theory not aris- ing as the equational theory of a non-trivially partially or- dered model of lambda calculus. This problem can be proved equivalent to the existence of a recursively enumerable (r.e., for short) λ-theory φ whose term algebra generates ann-permutable variety of LAAs for some natural number n ≥ 2 (see [24] and the remark after Thm. 3.4 in [26]). Lipparini [14] has found out interesting non-trivial lattice identitites that hold in the congruence lattices of all algebras living in an n-permutable variety. The failure of Lipparini’s lattice identities in λT would imply that Selinger’s problem has a negative answer.

Techniques of universal algebra were applied in [22, 15, 2] to study the structure of the lattice ofλ-theories. It was shown that the lattice of λ-theories does not satisfy the

modular law, but that it satisfies non-trivial lattice quasi- identities and admits (at the top) a distributive lattice interval with a continuum of elements. All these results sup- port the idea that universal algebra can be fruitfully applied to lambda calculus.

In this paper we would like to validate the inverse slo- gan: lambda calculus can be fruitfully applied to universal algebra. By generalizing to universal algebra concepts orig- inating from lambda calculus and programming, we create a zigzag path from lambda calculus to universal algebra and back. All the algebraic properties we have shown in [17]

for combinatory algebras, hold for a wider class of algebras, calledChurch algebras, that includes, beside combinatory algebras, all Boolean algebras, λ-abstraction algebras and rings with unit. Church algebras model the if-then- else instruction of programming by two constants0,1and a ternary termq(x, y, z)satisfying the following identities:

q(1, x, y) =x; q(0, x, y) =y.

Bloom and Tindell [4] and Manes [16] have introduced many different axiomatizations for the “if-then-else” instruction, but they lack the algebraic properties studied in this paper.

The interest of Church algebras is that each has a Boolean algebra of central elements, which can be used to represent a Church algebra as a weak Boolean product of directly indecomposable algebras. We generalize the notion of easy λ-term from lambda calculus and prove that: (i) any Church algebra with an “easy set” of car- dinality κ admits a congruence φ such that the lattice reduct of the free Boolean algebra with κgenerators em- beds into the lattice interval [φ); (ii) If κ is finite, this embedding is an isomorphism. This theorem applies directly to all Boolean algebras and rings with units. ForλT it has the following consequence: for every r.e. λ-theory φand each natural numbern, there is aλ-theoryφn ≥ φ such that the lattice interval [φn)is the finite Boolean lattice with 2ⁿ elements. It is the first time that it is found an interval of the lattice of λ-theories whose cardinal- ity is not1,2or2^ℵ⁰. We leave open the problem of whether the existence of such Boolean lattice intervals is inconsistent with the existence of an n-permutable variety of LAAs. If yes, this would negatively solve Selinger’s problem.

We also prove a meta version of the Stone Representa- tion Theorem that applies to all varieties of algebras and not only to the classic ones. Indeed, we show that any variety of algebras can be decomposed as a weak Boolean product of directly indecomposable subvarieties. This means that, given a variety V, there exists a family of “directly indecomposable” subvarietiesV_i(i ∈ I) ofV for which every algebra ofVis isomorphic to a weak Boolean product of algebras ofVi(i∈I).

(3)

Applications of the theory of Church algebras to the lattices of equational theories are outlined in Section 6.

2. Preliminaries

We will generally use the notation of Barendregt’s classic work [1] for lambda calculus and combinatory logic, and the notation of McKenzie et al. [18] for universal algebra.

A latticeLisboundedif it has a top element1and a bottom element0.a ∈ Lis anatom(coatom) if it is a mini- mal element inL− {0}(maximal element inL− {1}). For a∈L, we setL_a={b∈L− {0}:a∧b= 0}.Lis called:

lower semicomplementedifL_a 6=∅for alla6= 1;pseudo- complementedif eachL_ahas a greatest element (called the pseudocomplementofa).

We write[a)for{b:a≤b≤1}.

2.1. Algebras

Analgebraic similarity typeΣis constituted by a non- empty set of operator symbols together with a function as- signing to each operatorf ∈Σa finitearity. Operator symbols of arity 0 are callednullary operatorsorconstants.

AΣ-algebraAis determined by a non-empty setAto- gether with an operationf^A :Aⁿ →Afor everyf ∈Σof arityn.Aistrivialif its underlying set is a singleton.

A compatible equivalence relationφon aΣ-algebraA is called acongruence. We often writeaφbora =φ bfor (a, b)∈φ. The set{b:aφb}is denoted by[a]φ.

If φ ≤ ψ are congruences on A, then ψ/φ = {([a]φ,[b]φ) :aψb}is a congruence on the quotientA/φ.

IfX⊆A×A, then we writeθ(X)for the least congruence includingX. We writeθ(a, b)forθ({(a, b)}). Ifa∈A andY ⊆A, then we writeθ(a, Y)forθ({(a, b) :b∈Y}).

We denote byCon(A)the coatomic algebraic complete lattice of all congruences ofA, and by∇and∆the top and the bottom element of Con(A). A congruenceφonA is called:trivialif it is equal to∇or∆;consistentifφ6=∇;

compactifφ=θ(X)for some finite setX ⊆A×A.

An algebraAisdirectly decomposableif there exist two non-trivial algebrasB,Csuch thatA∼=B×C, otherwise it is calleddirectly indecomposable.

An algebra A is a subdirect product of the algebras (Bi)_i∈I, writtenA ≤ Π_i∈IBi, if there exists an embeddingf of Ainto the direct productΠi∈IBi such that the projectionπi◦f :A→Biis onto for everyi∈I.

A non-empty classVof algebras is avarietyif it is closed under subalgebras, homomorphic images and direct products or, equivalently, if it is axiomatizable by a set of equations. A varietyV⁰is asubvarietyof the varietyVifV⁰ ⊆ V.

We will denote byV(A)the variety generated by an al- gebraA, i.e.,B∈ V(A)if every equation satisfied byAis also satisfied byB.

LetV be a variety. We say thatAisthe freeV-algebra over the setX of generatorsiffA ∈ V,Ais generated by Xand for every mappingg:X→B∈ V, there is a unique homomorphismf : A → B that extendsg(i.e.,f(x) = g(x)for every x ∈ X). A free algebra in the class of all Σ-algebras is calledabsolutely free.

Given two congruencesσandτonA, we can form their relative product:τ◦σ={(a, c) :∃b∈A aσbτ c}.

Definition 1 A congruenceφon an algebraAis afactor congruenceif there exists another congruenceψsuch that φ∧ψ= ∆andφ◦ψ=∇. In this case we call(φ, ψ)apair of complementary factor congruencesorcfc-pair, for short.

Under the hypotheses of the above definition the homomorphism f : A → A/φ×A/ψ defined by f(x) = ([x]φ,[x]ψ)is an isomorphism. So, the existence of factor congruences is just another way of saying “this algebra is a direct product of simpler algebras”.

The set of factor congruences ofAis not, in general, a sublattice ofCon(A).∆and∇ are thetrivialfactor congruences, corresponding toA∼=A×B, whereBis a trivial algebra. An algebraAis directly indecomposable if, and only if,Ahas no non-trivial factor congruences.

Factor congruences can be characterized in terms of certain algebra homomorphisms calleddecomposition operators(see [18, Def. 4.32] for more details).

Definition 2 Adecomposition operationfor an algebraA is an algebra homomorphismf :A×A→Asuch that

• f(x, x) =x;

• f(f(x, y), z) =f(x, z) =f(x, f(y, z)).

There exists a bijective correspondence between cfc- pairs and decomposition operations, and thus, between decomposition operations and factorizations likeA∼=B×C.

Proposition 3 [18, Thm. 4.33] Given a decomposition operatorf, the relationsφ, ψdefined by:

x φ yifff(x, y) =y; x ψ yifff(x, y) =x, form a cfc-pair. Conversely, given a cfc-pair(φ, ψ), the map f defined by:

f(x, y) =u if, and only if, x φ u ψ y, (1) is a decomposition operation (for allxand ythere is just one elementusuch thatx φ u ψ y).

The Boolean product construction allows us to transfer numerous fascinating properties of Boolean algebras into other varieties of algebras (see [5, Ch. IV]). We recall that a Boolean space is a compact, Hausdorff and totally discon- nected topological space, and that clopen means open and closed.

(4)

Definition 4 Aweak Boolean product of a family(A)i∈I

of algebras is a subdirect productA ≤ Πi∈IAi, whereI can be endowed with a Boolean space topology such that:

(i) the set{i∈I:ai=bi}is open for alla, b∈A, and (ii) ifa, b∈AandNis a clopen subset ofI, then the ele-

mentc, defined byc_i=a_ifor everyi∈Nandc_i=b_i for everyi∈I−N, belongs toA.

ABoolean productis a weak Boolean product such that the set{i∈I:a_i =b_i}is clopen for alla, b∈A.

2.2. λ-calculus and combinatory logic We consider two fixed countable non-empty sets;

namely, the set Na of names and the set Va of algebraic variables. The elements of Na will be denoted by a, b, c, . . ., while the elements of Va byx, y, z, . . ..

The algebraic similarity typeΣ_λ is constituted by a binary operator symbol “·” of “application” and, for every a ∈Na, a nullary operator symbol “a” and a unary operator symbol “λa” of “lambda abstraction”.

Theλ-termsof lambda calculus are theΣ_λ-terms without occurrences of algebraic variables. We recall that aλ- term isclosedif every occurrence of a nameain it is under the scope of aλ-abstractionλa.

We will writeλabc.M forλa(λb(λc(M))). The dot “·”

of the application operator is usually omitted and associa- tion is made on the left, so that, for example,(((a·b)·c)·d)·e is writtenabcde.

We remark thatΣλ-terms and algebraic variables correspond respectively to the contexts and holes of Barendregt’s book [1, Def. 14.4.1].

Let Λ = (Λ,·, λa, a)_a∈Na be the absolutely free Σλ- algebra over an empty set of generators, whereΛis the set of λ-terms. Aλ-theoryis any congruence onΛincluding (α)- and(β)-conversion (see [1, Ch. 2]). It can also be seen as a (specific) set of equations between λ-terms. We use for λ-theories the same notational convention as for congruences (see Section 2.1). The set of allλ-theories is nat- urally equipped with a structure of complete lattice, hereafter denoted byλT, with meet defined as set theoretical in- tersection. The least element ofλT is denoted byλβ, while the top element ofλT is the inconsistentλ-theory∇.

The quotient of the absolutely free algebra Λ by a λ- theoryφis calledthe term algebra ofφand is denoted by Λ_φ. The latticeλT ofλ-theories is isomorphic to the congruence lattice of the term algebraΛ_λβofλβ, while the lattice interval[φ)is isomorphic to the congruence lattice of the term algebraΛ_φ.

The varietyV(Λ_λβ)generated by the term algebraΛ_λβ of λβ is the starting point for studying the lambda calculus by universal algebraic methods. It can be axiomatized by seven equations between Σλ-terms. The first six ones

express precisely the meta-mathematical content of (β)- conversion. The last one is an algebraic translation of(α)- conversion.

Theorem 5 (Salibra [22]) The varietyV(Λλβ) is axiomatized by the following identities, where a, b, c are names withb6=a, c6=bandx, y, zare variables:

(β₁) (λa.a)x=x;

(β₂) (λa.b)x=b;

(β₃) (λa.x)a=x;

(β4) (λaa.x)y=λa.x;

(β5) (λa.xy)z= ((λa.x)z)((λa.y)z);

(β6) (λab.x)((λb.y)c) =λb.(λa.x)((λb.y)c));

(α) λa.(λb.x)c=λb.(λa.(λb.x)c)b.

These seven identities were isolated by Pigozzi and Salibra in [20]. They define the varietyLAAofλ-abstraction algebras, which areΣλ-algebras satisfying(β1)-(β6)and(α).

The varietyCAofcombinatory algebras(see [7]) con- sists of the algebras C = (C,·,k,s), where·is a binary operation and k,sare constants, satisfyingkxy = xand sxyz=xz(yz)(as usual, the symbol “·” is omitted and as- sociation is made on the left).

3. Church algebras

Many algebraic structures, such as combinatory algebras,λ-abstraction algebras, Boolean algebras etc., have in common the fact that all are Church algebras. In this section we study the algebraic properties of this class of algebras. Applications are given in Section 5 and in Section 6.

Definition 6 An algebra Ais called a Church algebra if there are two constants 0,1 ∈ A and a ternary term q(e, x, y)such thatAsatisfies the following identities:

q(1, x, y) =x; q(0, x, y) =y.

A varietyVis called aChurch varietyif every algebra inV is a Church algebra with respect to the same termq(e, x, y) and constants0,1.

Example 7 The following are easily checked to be Church algebras:

1. λ-abstraction algebras:

q(e, x, y)≡(e·x)·y; 1≡λab.a; 0≡λab.b 2. Combinatory algebras:

q(e, x, y)≡(e·x)·y; 1≡k; 0≡sk

3. Boolean algebras:q(e, x, y)≡(e∨y)∧(e⁻∨x) 4. Heyting algebras:q(e, x, y)≡(e∨y)∧((e→0)∨x) 5. Rings with unit:q(e, x, y)≡(y+e−ey)(1−e+ex)

(5)

3.1. Central elements

LetA= (A,+,·,0,1)be a commutative ring with unit.

Every idempotent elementa∈A(i.e., satisfyinga·a=a) induces a cfc-pairθ(1, a), θ(a,0). In other words, the ring Acan be decomposed asA∼=A/θ(1, a)×A/θ(a,0).Ais directly indecomposable if0and1are the unique idempotent elements. Vaggione [28, 29] generalized idempotent elements to any universal algebra whose top congruence∇ is compact, and called them central elements. Central elements were used, among the other things, to investigate the closure of varieties of algebras under Boolean products.

Here we give a new characterization based on decomposition operators (see Def. 2).

As a matter of notation, we define θe≡θ(1, e); θe≡θ(e,0).

Definition 8 We say that an elementeof a Church algebra Aiscentral, and we writee ∈ Ce(A), ifθeandθeare a cfc-pair. A central elementeis callednon-trivialife6= 0,1.

We now show how to internally represent in a Church algebra factor congruences as central elements. We start with a lemma.

Lemma 9 LetAbe a Church algebra ande∈A. Then we have, for allx, y∈A:

(a) x θ_eq(e, x, y)θ_ey.

(b) xθeyiffq(e, x, y) (θe∧θe)y.

(c) xθeyiffq(e, x, y) (θe∧θe)x.

(d) θ_e◦θ_e=θ_e◦θ_e=∇.

Proof.(a) From1θee θe0.

(b) By (a) we have x θe q(e, x, y). Then xθey iff q(e, x, y)θey.

(c) Analogous to (b).

(d) By (a).

Proposition 10 LetAbe a ChurchΣ-algebra ande∈ A.

The following conditions are equivalent:

(i) eis central;

(ii) θe∧θe= ∆;

(iii) For allxand y,q(e, x, y)is the unique element such thatxθ_eq(e, x, y)θ_ey;

(iv) esatisfies the following identities:

1.q(e, x, x) =x.

2.q(e, q(e, x, y), z) =q(e, x, z) =q(e, x, q(e, y, z)).

3.q(e, f(x), f(y)) =f(q(e, x1, y1), . . . , q(e, xn, yn)), for everyn-ary function symbolf ∈Σ.

4.e=q(e,1,0).

(v) The functionf_e defined byf_e(x, y) = q(e, x, y)is a decomposition operator such thatf_e(1,0) =e.

Proof.(i)⇔(ii)From Lemma 9(d).

(ii) ⇒ (iii)By Lemma 9(d)θe andθe are a cfc-pair.

Then the conclusion follows from Lemma 9(a).

(iii)⇒(ii)First note thatq(e, x, x) =x. Ifx(θe∧θe)y thenxθeyθex, that isy=q(e, x, x) =x.

(iv)⇔(v)By Prop. 3.

(i) ⇒ (v) First we recall that (i) is equivalent to (iii). fe is a decomposition operator because (θe, θe) is a cfc-pair and q(e, x, y) is the unique element satisfying x θe q(e, x, y)θey. Moreover,fe(1,0) = q(e,1,0) = e follows from1θee θe0.

(v)⇒(i)Let(φ, φ)be the cfc-pair associated withfe. From Prop. 3 and fromfe(1,0) =q(e,1,0) =eit follows that1φeφ0, so thatθe, θe ⊆φ. For the opposite direction, letxφy, which is equivalent toq(e, x, y) = y by Prop. 3.

Then by 1θeewe derivex = q(1, x, y)θe q(e, x, y) = y.

Similarly, forφ.

Corollary 11 LetAbe a Church algebra ande∈Asuch thatθe 6= ∇,∆. Then the equivalence class ofeis a non- trivial central element in the algebraA/θe∧θe.

Thus a Church algebraAis directly indecomposable iff Ce(A) ={0,1}iffθ_e∧θ_e6= ∆for alle6= 0,1.

Example 12 • All elements of a Boolean algebra are central by Prop. 10(iv) and by Example 7.

• An element is central in a commutative ring with unit if, and only if, it is idempotent. This characterization does not hold for non-commutative rings with unit.

• LetΩ≡(λa.aa)(λa.aa)be the usual looping term of lambda calculus. It is well-known that theλ-theories θΩ = θ(Ω, λab.a) and θΩ = θ(Ω, λab.b) are consistent (see [1]). Then by Corollary 11 the termΩis a non-trivial central element in the term algebra of θ_Ω∧θ_Ω.

We now show that the partial ordering on the central elements, defined by:

e≤dif, and only if,θe⊆θd (2) is a Boolean ordering and that the meet, join and comple- mentation operations are internally representable. 0and1 are respectively the bottom and top element of this ordering.

Theorem 13 Let A be a Church algebra. The algebra (Ce(A),∧,∨,⁻,0,1)of central elements ofA, defined by

e∧d=q(e, d,0); e∨d=q(e,1, d); e⁻=q(e,0,1), is a Boolean algebra isomorphic to the Boolean algebra of factor congruences ofA.

(6)

Proof. (Outline) If A ∼= B ×C, then it is easy to check, by using the definition of a central element, that Ce(A) = Ce(B)×Ce(C). In the terminology of universal algebra one says that Ahas no “skew factor congruences”. From this and [3, Prop. 1.3] the factor congruences of A form a Boolean sublattice of the congruence lattice Con(A). It follows that the partial ordering on central elements, defined in (2), is a Boolean ordering. Then it is possible to show that, for all central elementseandd, the ele- mentse⁻,e∧dande∨dare central and are respectively associated with the cfc-pairs(θe, θe),(θe∨θd, θe∧θd)and (θe∧θd, θe∨θd).

We check the details only for e⁻. Similar reasonings work fore∧dande∨d. Sinceeis central then(θe, θe)is a cfc-pair. The complement of(θe, θe)is the pair(θe, θe). We have thate⁻ is the unique element such that0θee⁻ θe1.

Then1θee⁻θe0for the pair(θe, θe). This means thate⁻ is the central element associated with(θe, θe).

The Stone representation theorem for Church algebras is an easy corollary of Thm. 13 and of theorems by Comer [6]

and by Vaggione [29].

LetA be a Church algebra. IfI is a maximal ideal of the Boolean algebra Ce(A), then φ_I denotes the congruence onAdefined by:φ_I =∪e∈Iθ_e. Moreover,X denotes the Boolean space of maximal ideals ofCe(A).

Theorem 14 (The Stone Representation Theorem) Let A be a Church algebra. Then, for allI∈Xthe quotient alge- braA/φ_Iis directly indecomposable and the map

f :A→Π_I∈X(A/φI), defined by

f(x) = ([x]φ_I :I∈X),

gives a weak Boolean product representation ofA.

Proof.By the proof of Thm. 13 the factor congruences of Aconstitute a Boolean sublattice ofCon(A). Then by Comer’s generalization [6] of Stone representation theo- remf gives a weak Boolean product representation ofA.

The quotient algebras A/φ_I are directly indecomposable by [29, Thm. 8].

Note that, in general, Thm. 14 does not give a (non- weak) Boolean product representation. This was shown in [17] for combinatory algebras.

4. The main theorem

It is well known that lambda calculus haseasyλ-terms, i.e., terms that can be consistently equated with any other closed λ-term. In this section we generalize the notion of easiness to Church algebras to show that any Church algebra with an easy set of cardinalitynadmits a congruenceφ such that the lattice interval of all congruences greater than

φis isomorphic to the free Boolean algebra withngenera- tors.

We recall thatθ(1, Y)denotes the congruence generated by the set{(1, a) :a∈Y}.

Definition 15 LetA be a Church algebra. We say that a subsetX ofAis aneasy setif, for everyY ⊆X,

θ(1, Y)∨θ(0, X−Y)6=∇ (by definitionθ(1,∅) =θ(0,∅) = ∆).

We say that an elementaiseasyif{a}is an easy set. Thus, a is easy if the congruencesθ_a andθ_a are both different from∇.

Example 16 • A finite subsetXof a Boolean algebra is an easy set if it holds: (i)WX 6= 1; (ii)VX6= 0; (iii) for allY ⊂X,WY 6≥ V(X −Y). Thus, for example,{{1,2},{2,3}}is an easy set in the powerset of {1,2,3,4}.

• The term algebra of every r.e.λ-theory has a countable infinite easy set. This will be shown in Section 5.

The following three lemmas are used in the proof of the main theorem. We recall that asemicongruenceon an alge- braAis a reflexive compatible binary relation onA.

Lemma 17 The semicongruences of a Church algebra per- mute with its factor congruences, i.e., φ◦ψ = ψ◦φfor every semicongruenceφand factor congruenceψ.

Proof. Let ψ = θe for a central element e and let a φ b θe c for some b. We get the conclusion of the lemma if we show that a θe q(e, a, c) φ c. Notice that a θe q(e, a, c) is a consequence of Lemma 9(a). We now prove that q(e, a, c) φ c. First we remark that by b θe c and by Prop. 10(iii) we have q(e, b, c) = c. From this last equality and from a φ b it follows the conclusion q(e, a, c)φ q(e, b, c) =c.

Lemma 18 LetAbe a Church algebra. Then the congruence lattice ofAsatisfies the Zipper condition, i.e., for all Iand for allδ_i, ψ, φ∈Con(A)(i∈I):

If _

i∈I

δi=∇andδi∧ψ=φ(i∈I), thenψ=φ.

Proof. By [11], where it is shown that the congruence lattice of every0,1-algebra (i.e., an algebra having a binary term with a right unit and a right zero) satisfies the Zipper condition.

Lemma 19 LetBbe a Church algebra andφ∈ Con(B).

Then, B/φ is also a Church algebra and the map cφ : Ce(B)→Ce(B/φ), defined by

cφ(x) = [x]φ

is a homomorphism of Boolean algebras.

(7)

Proof.It is not difficult to show thatcφis a homomorphism with respect to the Boolean operations defined in Thm. 13.

We denote byP(X)the powerset of a setX.

Theorem 20 LetAbe a Church algebra andXbe an easy subset ofA. Then there exists a congruenceφX satisfying the following conditions:

1. The lattice reduct of the free Boolean algebra with a setX of generators can be embedded into the lattice interval[φ_X);

2. IfXhas finite cardinalityn, then the above embedding is an isomorphism and[φX)has2²ⁿelements.

Proof.We start by definingφX. If we let δY ≡θ(1, Y)∨θ(0, X−Y), (Y ⊆X) then by the hypothesis of easiness we have thatδ_Y 6= ∇.

We consider an enumeration (e_γ)_γ<κ of A×A, where κ is the cardinal ofA×A. Define by transfinite induction an increasing sequenceψ_γ(γ≤κ) of congruences onA:

• ψ0=∩Y⊆XδY.

• ψγ+1=ψγifψγ∨θ(eγ)∨δY =∇for someY ⊆X.

• ψ_γ+1=ψ_γ∨θ(e_γ)otherwise.

• ψγ =∪β<γψβfor every limit ordinalγ≤κ.

We defineφX≡ψκ.

We now prove that the free Boolean algebra with a setX of generators can be embedded into the interval[ψκ).

Claim 21 ψ_γ∨δ_Y 6=∇for allγ≤κandY ⊆X. Proof.The result can be shown by transfinite induction on γ. It is true for γ = 0 by definition of an easy set. If ψγ = ∪β<γψβ, then it follows from inductive hypothesis, because(∪β<γψβ)∨δY = ∪β<γ(ψβ ∨δY). Finally, ifψγ =ψγ−1∨θ(eγ−1)then by definition ofψγ we have ψγ−1∨θ(eγ−1)∨δY =ψγ∨δY 6=∇for allY ⊆X.

LetAγ ≡A/ψγand[x]γ ≡[x]ψ_γ (x∈A).

Claim 22 [x]_γ ∈Ce(A_γ)for everyx∈X andγ≤κ.

Proof.If we prove that[x]0is central inA0, then byψ0 ≤ ψγ and by Lemma 19 we get the same conclusion forγ.

Since the elementx∈X is equivalent either to1or to0in each congruenceδ_Y, then[x]_δ_Y is a trivial central element in the algebraA/δ_Y. Thenh[x]δY : Y ⊆ Xiis central in the Cartesian productΠ_Y_⊆XA/δ_Y. Sinceψ₀=∩Y⊆X δ_Y then by [5, Lemma II.8.2]A₀≡A/ψ₀is a subdirect product of the algebrasA/δ_Y, so thatA₀can be embedded into Π_Y_⊆XA/δ_Y. It follows that[x]₀is central inA₀.

Let B(X)be the free Boolean algebra over the set X of generators and f_γ : B(X) → Ce(A_γ)be the unique Boolean homomorphism satisfyingfγ(x) = [x]γ.

Claim 23 fγ :B(X)→Ce(Aγ)is an embedding.

Proof.LetY ⊆X. By Claim 21 the algebraA/ψγ∨δY is non-trivial, while by Lemma 19 there exists a Boolean homomorphism (denoted byhY in this proof) fromCe(Aγ) into Ce(A/ψγ∨δY). Since (x,1) ∈ ψγ ∨δY for every x∈Y and(y,0)∈ψγ∨δY for everyy∈X−Y, then the kernel ofhY ◦fγ is an ultrafilter ofB(X). By the arbitrari- ness ofY ⊆X, every ultrafilter ofB(X)can be the kernel of a suitableh_Y ◦f_γ. This is possible only iff_γ is an embedding.

This concludes the proof of (1).

Recall that the lattice interval[ψ_κ)ofCon(A)is isomorphic to the congruence latticeCon(A_κ).

If Y ⊆ X, we denote by δ_Y^κ the congruence (ψκ ∨ δY)/ψκ∈Con(Aκ).

Claim 24 Letσ ∈Con(Aκ). Ifσ∨δ_Y^κ 6=∇for allY ⊆ X, thenσ= ∆.

Proof.Letx, y∈Asuch that(x, y)∈/ ψκand([x]κ,[y]κ)∈ σ. From the hypothesis it follows thatψκ∨δY ∨θ(x, y)∈ Con(A)is non-trivial for allY ⊆X. By definition ofψκ

this last condition implies(x, y)∈ψκ. Contradiction.

Hereafter, we assume thatXis a finite easy set of cardinalityn.

We show that the setAt_κ of atoms ofCon(A_κ)is not empty and has∇as join.

Claim 25 W{β ∈Atκ:βis a factor congruence}=∇.

Proof.SinceX has cardinalityn, then the free Boolean al- gebraB(X)is finite, atomic and hasngenerators. Letabe an atom ofB(X)and letfκ(a)∈Aκbe the central element determined by the embeddingfκof Claim 23. Consider the factor congruenceτ =θ(fκ(a),0)∈Con(Aκ)associated withfκ(a). We claim that τ is an atom inCon(Aκ). By the way of contradiction, assume that σ ∈ Con(Aκ)is a non-trivial congruence which is strictly underτ. By Claim 23 and Lemma 19 we have a chain of Boolean homomorphisms:

B(X)−→^f^κ Ce(A_κ)−→^c^σ Ce(A_κ/σ)−−−→^c^{τ /σ} Ce(A_κ/τ) such thatcτ =c_{τ /σ}◦cσ. Sinceais an atom ofB(X)and τ = θ(fκ(a),0), then the set{0, a} is the Boolean ideal associated with the kernel of cτ ◦ fκ. If cσ(fκ(a)) = 0, then σ contains the pair (fκ(a),0), i.e., σ = τ. Then cσ(fκ(a))6= 0and the mapcσ◦fκ :B(X)→Ce(Aκ/σ) is an embedding. Then the images of the elements ofXinto Ce(A_κ/σ)are distinct central elements, so thatσ∨δ_Y^κ 6=∇ for allY. By Claim 24 we getσ= ∆, that contradicts the non-triviality ofσ. Thenτis an atom. Finally,W{β∈At_κ: β is a factor congruence} =∇follows because the join of all atoms ofB(X)is the top element.

(8)

Claim 26 The congruence latticeCon(Aκ)is pseudocomplemented, complemented, atomic, and the coatoms form a finite irredundant decomposition of the least element∆.

Proof.The coatomic and complete latticeCon(Aκ)satisfies the Zipper condition (by Lemma 18) andW

Atκ =∇ (by Claim 25). Then by [8, Prop. 2] Con(A_κ)is complemented, atomic and every coatom has a complement which is an atom. It is also pseudocomplemented by [8, Prop. 1].

Since the top element∇is compact, by [8, Prop. 3] we get that the coatoms form a finite irredundant decomposition of the least element.

Claim 27 Let ξ ∈ Con(A_κ)be a non-trivial congruence andγ=W{δ∈At_κ:δ≤ξ}. Ifβ ∈At_κis a factor congruence which is not underξ, thenξ∧(β∨γ) =γ.

Proof.We always haveγ ≤ ξ∧(β∨γ). We show the opposite direction. Let(x, y)∈ξ∧(β∨γ), i.e.,x ξ yand x(β∨γ)y. We have to show thatxγy. Sinceβ is a factor congruence, by Lemma 17 we haveβ∨γ = β◦γ. Then x β z γ yfor somez. Sinceγ≤ξthenz ξ y, that together withx ξ yimpliesx ξ z. Thenx(ξ∧β)z. Sinceβis an atom andβ 6≤ξ, we getx=z. This last equality andzγyimply xγy. In other words,ξ∧(β∨γ) =γ.

Claim 28 The congruence lattice Con(Aκ) is a finite Boolean algebra.

Proof.By Claim 26Con(A_κ)is complemented, atomic and pseudocomplemented. If we can show that each ele- mentξ 6= ∇is a join of atoms, thenCon(Aκ)is isomorphic to the power set of Atκ. LetAtξ be the set of atoms underξandγ=WAtξ. We will show thatγ=ξby apply- ing the Zipper condition of Lemma 18. By Claim 27 and by the definition of γ we have: W

{ν : ξ∧ν = γ} ≥ W{β∨γ : β ∈ Atκ, β 6≤ ξ, βis a factor congruence} ≥ W{β : β ∈Atκis a factor congruence}. By Claim 25 this last element is equal to∇, so thatW

{ν :ξ∧ν =γ}=∇.

By the Zipper condition this entailsξ=γ.

Since[ψκ)∼= Con(Aκ), then[ψκ)is Boolean.

Claim 29 The Boolean lattice [ψκ)has exactly 2ⁿ atoms and2ⁿcoatoms.

Proof.Sinceψκ∨δY 6=∇for everyY ⊆X,[ψκ)has at least 2ⁿ coatoms. For everyY ⊆ X, letcY be a coatom including ψκ∨δY. Assume now that there is a coatomξ distinct fromcY for everyY ⊆X. Consider the intersec- tion∩(Coκ− {ξ}), whereCoκdenotes the set of coatoms of [ψκ). By Claim 26 and by [ψκ) ∼= Con(Aκ)we have that ∩(Coκ− {ξ}) 6= ψ_κ, so that there is a pair(a, b) ∈

∩(Coκ− {ξ})−ψ_κ. Sinceψ_κ∨δ_Y ∨θ(a, b)≤c_Y 6=∇ for allY ⊆X, then(a, b)∈ψ_κby the inductive definition ofψ_κ. Contradiction. In conclusion, we have2ⁿcoatoms. A Boolean lattice has the same number of atoms and coatoms.

This concludes the proof of the main theorem.

The following proposition explains why the main theorem cannot be improved.

Proposition 30 LetAbe a Church algebra. Then there exists no congruenceφsuch that the interval sublattice[φ)is isomorphic to an infinite Boolean lattice.

Proof.From [8, Prop. 4], where it is shown that a complete coatomic Boolean lattice satisfying the Zipper condition, and whose top element is compact, is finite.

5. The lattice of λ-theories

The fact that the term algebra of everyλ-theoryφis a Church algebra has the interesting consequence that the lat- ticeλT admits (at the top) Boolean lattice intervals of car- dinality2ⁿfor everyn.

Berline and Salibra have noticed in [2] that there exists a countable infinite sequence of λ-terms that can be consistently equated to any other arbitrary infinite sequence of closedλ-terms. In the following lemma we generalize this result to any r.e.λ-theory.

Letωbe the set of natural numbers andΛôbe the set of closed λ-terms. As a matter of notation, if M = hMk ∈ Λô : k ∈ ωiandN = hNk ∈ Λô : k ∈ ωiare infinite sequences, we write(M,N)for{(Mk, Nk) :k∈ω}.

Lemma 31 For every r.e. λ-theoryφ, there exists an infinite sequence M = hM_k ∈ Λ^o : k ∈ ωi, called φ-easy sequence, satisfying the following conditions:

• M_n6=φM_kfor everyn6=k;

• For all sequences N = hNk ∈ Λ^o : k ∈ ωitheλ- theory generated byφ∪(M,N)is consistent.

Proof.By [1, Prop. 17.1.9] there is aφ-easyλ-term, i.e., a termPsuch that theλ-theory generated byφ∪ {P =Q}

is consistent for every closed termQ. We defineMn ≡P n, wherenis the Church numeral ofn(see [1, Def. 6.4.4]).

LetN =hNk ∈Λ^o:k∈ωibe an arbitrary sequence. By compactness we get the conclusion of the lemma if theλ- theory generated byφ∪{(Mi, Ni) :i≤n}is consistent for every natural numbern. Fixn. It is routine to find aλ-term Rsuch thatRi=λβ Nifor alli≤n. SinceP is aφ-easy λ-term, then theλ-theoryψgenerated byφ∪ {P =R}is consistent andP i=_ψN_i(i.e.,M_i=_ψN_i) for alli≤n.

Theorem 32 For every r.e. λ-theory φ and each natural numbern, there is a λ-theory φn ≥ φ such that the lattice interval[φn)is isomorphic to the finite Boolean lattice with2ⁿelements.

Proof.The term algebra ofφis a Church algebra by Ex- ample 7. LetM =hM_k ∈Λ^o : k ∈ωibe theφ-easy sequence of Lemma 31. Then the set{[Mk]φ : k ∈ ω}is a

(9)

countable infinite easy subset of the term algebra ofφ. From Thm. 20 there exists a congruenceψnsuch thatψn≥φand [ψ_n)is isomorphic to the free Boolean algebra with2²ⁿele- ments. The congruenceφ_nof the theorem can be defined by usingψ_nand the following facts: (a) Every filter of a finite Boolean algebra is a Boolean lattice; (b) The free Boolean algebra with2²ⁿ elements has filters of arbitrary cardinal- ity2^k(k≤2ⁿ).

6. Lattices of equational theories

An equational theory is a set of identities closed under the rules of the equational calculus. It is an open problem raised by Birkhoff to characterize abstractly the lattices L(T)of all equational theories containing a fixed equational theoryT, or dually, the lattices of all subvarieties of a variety. We say thatLis alattice of equational theoriesiffL is isomorphic to someL(T). Such lattices are algebraic and coatomic, possessing a compact top element; but stronger properties were not known before Lampe’s discovery [11]

that any lattice of equational theories obeys the Zipper condition (see Lemma 18). Other conditions were successively discovered by Ern`e, Lampe and Tardos (see [9, 12]).

In this section we show the existence of Boolean lattice intervals in the lattices of equational theories, and a meta version of the Stone representation theorem that holds for all varieties of algebras.

It is well known that a lattice of equational theories is isomorphic to a congruence lattice (see [5, 18]). In- deed, the lattice L(T) of all equational theories contain- ingT is isomorphic to the congruence lattice of the algebra(F_T, f)_f∈End(F_T₎, whereF_T is the free algebra over a countable set of generators in the variety axiomatized byT, and End(F_T)is the set of its endomorphisms (i.e., homomorphisms fromF_T intoF_T).

We expand (without changing the congruence lattice) the algebra(FT, f)_f∈End(F_T₎by the operationqdefined as follows (x1, x2are two fixed variables):

q(t, s₁, s₂) =t[s₁/x₁, s₂/x₂], (3) wheret[s1/x1, s2/x2]is the term obtained by substituting term si for variable xi (i = 0,1) within t. The algebra (FT, f, q)_f∈End(F_T)was defined, but not directly used, by Lampe in the proof of McKenzie Lemma in [11]. If we de- fine1≡x₁and0≡x₂, from the identitiesq(x₁, s₁, s₂) = s₁andq(x₂, s₁, s₂) =s₂we get that(F_T, f, q)_f∈End(F_T₎is a Church algebra. It will be denoted byC_T and called hereafter theChurch algebra ofT.

In the following lemma, whose proof is omitted, we characterize the central elements of the Church algebra of an equational theory.

Lemma 33 LetTbe an equational theory andVbe the variety of Σ-algebras axiomatized byT. Then the following conditions are equivalent, for every element e ∈ C_T and termt(x₁, x₂)∈e:

(i) eis a central element of the Church algebra ofT. (ii) T contains the identities:

t(x, x) =x; t(x, t(y, z)) =t(x, z) =t(t(x, y), z)

t(f(x), f(y)) =f(t(x1, y1), . . . , t(xn, yn)), f ∈Σ.

(iii) For everyA∈ V, the functiont^A :A×A→Ais a decomposition operator.

(iv) T =T1∩T2, whereTiis the theory axiomatized (over T) byt(x1, x2) =xi(i= 1,2).

If the equivalence class oft(x1, x2)is a central element ofCT, then by Lemma 33(iii)-(iv) every algebraA∈ Vcan be decomposed asA ∼= A/φ×A/φ, where(φ, φ)is the cfc-pair associated with the decomposition operatort^A, and the algebrasA/φ andA/φ satisfy respectively the equational theoriesT₁andT₂. In such a case, we say thatV is decomposableas a product of the two subvarieties axiomatized respectively byT₁andT₂(see [27]).

We say that a variety isdirectly indecomposable if the Church algebra of its equational theory is a directly indecomposable algebra.

Theorem 34 LetT be an equational theory. Assume there existnbinary termst0, . . . , tn−1such that, for every function k : n → {1,2}, the theory axiomatized (over T) by ti(x1, x2) = xk(i) (i = 0, . . . , n−1) is consistent. Then there exists a theory T⁰ ≥ T such that the latticeL(T⁰) of all equational theories extendingT⁰is isomorphic to the free Boolean lattice with2²ⁿelements.

Proof.The equivalence classes of the termst0, . . . , t_n−1 constitute a finite easy set in the Church algebra ofT. The conclusion follows from Thm. 20.

The set of all factor congruences of an algebra does not constitute in general a sublattice of the congruence lattice.

We now show that in every algebra there is a subset of factor congruences which always constitutes a Boolean sublattice of the congruence lattice.

We denote byt^A_e the decomposition operator associated with the central elementeby Lemma 33(iii).

Lemma 35 LetTbe an equational theory andVbe the variety axiomatized byT. Then, for every algebraA∈ V, the functionh: Ce(CT)→Con(A), defined by:

h(e) ={(x, y) :t^A_e(x, y) =x},

is a lattice homomorphism from the Boolean algebra of central elements ofC_T into the set of factor congruences ofA such that(h(e), h(e⁻))is a cfc-pair for alle ∈ Ce(C_T).

The range ofhconstitutes a Boolean sublattice ofCon(A).

(10)

We say that a variety V is decomposable as a weak Boolean product of directly indecomposable subvarietiesif there exists a familyhVi :i ∈Xiof directly indecomposable subvarietiesVi of V such that every algebra A ∈ V is isomorphic to a weak Boolean productΠ_i∈XB_iof alge- brasB_i∈ V_i.

Theorem 36 (The Meta-Representation Theorem) Every varietyV of algebras is decomposable as a weak Boolean product of directly indecomposable subvarieties.

Proof. (Outline) Let T be the equational theory of V andCT be the Church algebra ofT. By Thm. 14 we can represent CT as a weak Boolean product f : CT → ΠI∈X(CT/φI), whereXis the Stone space of the Boolean algebraCe(CT)of central elements ofCT,I∈ X ranges over the maximal ideals ofCe(CT),φI =∪e∈Iθe, andθe

is the factor congruence associated with the central element e∈I. Since the latticeL(T)of the equational theories ex- tendingTis isomorphic to the congruence lattice ofC_T, the congruenceφ_Icorresponds to an equational theory, sayT_I. The Church algebra ofT_I is isomorphic toC_T/φ_I, so that it is directly indecomposable. Then by Lemma 33 the vari- etyVI axiomatized byTI is directly indecomposable.

Let A ∈ V andh : Ce(CT) → Con(A)be the lattice homomorphism defined in Lemma 35. For every maximal ideal I of Ce(CT), consider the congruence φ^A_I =

∪_e∈Ih(e). The map f : A → Π_I∈X(A/φ^A_I )defined by f(x) = ([x]_φA

I :I ∈X), determines a weak Boolean representation ofA, whereA/φ^A_I ∈ VI. The algebraA/φ^A_I may be directly decomposable also if it belongs to the directly indecomposable varietyVI.

We remark that, if an algebraA∈ Vhas Boolean factor congruences, thenAcan be represented as a weak Boolean product in two different way, by using either the Comer representation theorem or the meta-representation theorem.

The meta representation is in general weaker than Comer’s representation.

References

[1] H.P. Barendregt.The lambda calculus: Its syntax and semantics. North-Holland Publishing Co., Amsterdam, 1984.

[2] C. Berline, A. Salibra. Easiness in graph models.Theo. Comp.

Sci.354:4–23, 2006.

[3] D. Bigelow, S. Burris. Boolean algebras of factor congruences.Acta Sci. Math.54:11–20, 1990.

[4] S. L. Bloom, R. Tindell. Varieties of “if-then-else”.Siam J.

Comput.12: 677–707, 1983.

[5] S. Burris, H.P. Sankappanavar.A course in universal algebra.

Springer-Verlag, Berlin, 1981.

[6] S. Comer. Representations by algebras of sections over boolean spaces.Pacific J. Math.38:29–38, 1971.

[7] H.B. Curry and R. Feys. Combinatory logic, Vol. I. North- Holland, Amsterdam, 1958.

[8] V. Diercks, M. Ern`e and J. Reinhold. Complements in lattices of varieties and equational theories.Algebra Universalis 31(4):506–515, 1994

[9] M. Ern`e. Weak distributive laws and their role in lattices of congruences and equational theories.Algebra Universalis 25:290–321, 1988

[10] R. Freese and R. McKenzie,Commutator theory for congruence modular varieties. Cambridge University Press, 1987.

[11] W. A. Lampe. A property of the lattice of equational theories.Algebra Universalis23: 61–69, 1986.

[12] W. A. Lampe. Further properties of lattices of equational theories.Algebra Universalis28: 459–486, 1991.

[13] P. Lipparini. Commutator theory without join-distributivity.

Trans. American Math. Society346: 177–202, 1994.

[14] P. Lipparini. n-permutable varieties satisfy nontrivial congruence identities.Algebra Universalis33(2):, 159–168, 1995.

[15] S. Lusin, A. Salibra. The lattice of lambda theories.Journal of Logic and Computation14:373–394, 2004.

[16] E. Manes. Adas and the equational theory of if-then-else.Al- gebra Universalis30: 373–394, 1993.

[17] G. Manzonetto, A. Salibra. Boolean algebras for lambda calculus.21th Annual IEEE Symposium on Logic in Computer Science (LICS 2006), IEEE Computer Society Press, 2006.

[18] R.N. McKenzie, G.F. McNulty and W.F. Taylor.Algebras, Lattices, Varieties, Volume I. Wadsworth Brooks, 1987.

[19] R.S. Pierce. Modules over commutative regular rings.Mem- oirs Amer. Math. Soc., 1967.

[20] D. Pigozzi and A. Salibra. Lambda abstraction algebras: representation theorems.Theo. Comp. Sci.140: 5–52, 1995.

[21] G.D. Plotkin. Set-theoretical and other elementary models of theλ-calculus.Theo. Comp. Sci.121:351–409, 1993.

[22] A. Salibra. On the algebraic models of lambda calculus.

Theo. Comp. Sci.249:197–240, 2000.

[23] A. Salibra. Nonmodularity results for lambda calculus.Fun- damenta Informaticae45:379–392, 2001.

[24] A. Salibra. Topological incompleteness and order incompleteness of the lambda calculus. (LICS’01 Special Issue).

ACM Transactions on Computational Logic4:379–401, 2003.

[25] D.S. Scott. Lambda calculus: some models, some philoso- phy.The Kleene Symposium, North-Holland, 1980.

[26] P. Selinger. Order-incompleteness and finite lambda reduc- tion models.Theo. Comp. Sci.309:43–63, 2003.

[27] W. Taylor. The fine spectrum of a variety.Algebra Univer- salis5: 263–303, 1975.

[28] D. Vaggione. V with factorable congruences and V = IΓ^a(VDI)implyVis a discriminator variety.Acta Sci. Math., 62:359-368, 1996.

[29] D. Vaggione. Varieties in which the Pierce stalks are directly indecomposable.Journal of Algebra184:424–434, 1996.