Natural dualities for varieties generated by a set of subalgebras of a semi-primal algebra

The main contribution of this paper is the construction of a strong duality for the varieties generated by a set of subalgebras of a semi-primal algebra. We also obtain an axiomatization of the objects of the dual category and develop some algebraic consequences (description of the dual of the fini...

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Datum:	2007
Hauptverfasser:	Mathonet, P., Niederkorn, P., Teheux, B.
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Sprache:	English
Veröffentlicht:	Інститут прикладної математики і механіки НАН України 2007
Schriftenreihe:	Algebra and Discrete Mathematics
Online Zugang:	http://dspace.nbuv.gov.ua/handle/123456789/157346
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spelling	irk-123456789-1573462019-06-21T01:30:10Z Natural dualities for varieties generated by a set of subalgebras of a semi-primal algebra Mathonet, P. Niederkorn, P. Teheux, B. The main contribution of this paper is the construction of a strong duality for the varieties generated by a set of subalgebras of a semi-primal algebra. We also obtain an axiomatization of the objects of the dual category and develop some algebraic consequences (description of the dual of the finite structures and algebras, construction of finitely generated free algebras,. . . ). Eventually, we illustrate this work for the finitely generated varieties of MV-algebras. 2007 Article Natural dualities for varieties generated by a set of subalgebras of a semi-primal algebra / P. Mathonet, P. Niederkorn, B. Teheux // Algebra and Discrete Mathematics. — 2007. — Vol. 6, № 1. — С. 67–85. — Бібліогр.: 18 назв. — англ. 1726-3255 2000 Mathematics Subject Classification: 06D35; 08B. http://dspace.nbuv.gov.ua/handle/123456789/157346 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	The main contribution of this paper is the construction of a strong duality for the varieties generated by a set of subalgebras of a semi-primal algebra. We also obtain an axiomatization of the objects of the dual category and develop some algebraic consequences (description of the dual of the finite structures and algebras, construction of finitely generated free algebras,. . . ). Eventually, we illustrate this work for the finitely generated varieties of MV-algebras.
format	Article
author	Mathonet, P. Niederkorn, P. Teheux, B.
spellingShingle	Mathonet, P. Niederkorn, P. Teheux, B. Natural dualities for varieties generated by a set of subalgebras of a semi-primal algebra Algebra and Discrete Mathematics
author_facet	Mathonet, P. Niederkorn, P. Teheux, B.
author_sort	Mathonet, P.
title	Natural dualities for varieties generated by a set of subalgebras of a semi-primal algebra
title_short	Natural dualities for varieties generated by a set of subalgebras of a semi-primal algebra
title_full	Natural dualities for varieties generated by a set of subalgebras of a semi-primal algebra
title_fullStr	Natural dualities for varieties generated by a set of subalgebras of a semi-primal algebra
title_full_unstemmed	Natural dualities for varieties generated by a set of subalgebras of a semi-primal algebra
title_sort	natural dualities for varieties generated by a set of subalgebras of a semi-primal algebra
publisher	Інститут прикладної математики і механіки НАН України
publishDate	2007
url	http://dspace.nbuv.gov.ua/handle/123456789/157346
citation_txt	Natural dualities for varieties generated by a set of subalgebras of a semi-primal algebra / P. Mathonet, P. Niederkorn, B. Teheux // Algebra and Discrete Mathematics. — 2007. — Vol. 6, № 1. — С. 67–85. — Бібліогр.: 18 назв. — англ.
series	Algebra and Discrete Mathematics
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first_indexed	2025-07-14T09:47:28Z
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fulltext	Algebra and Discrete Mathematics RESEARCH ARTICLE Number 1. (2007). pp. 67 – 85 c© Journal “Algebra and Discrete Mathematics” Natural dualities for varieties generated by a set of subalgebras of a semi-primal algebra P. Mathonet, P. Niederkorn, B. Teheux Communicated by L. Marki Abstract. The main contribution of this paper is the con- struction of a strong duality for the varieties generated by a set of subalgebras of a semi-primal algebra. We also obtain an axiomati- zation of the objects of the dual category and develop some alge- braic consequences (description of the dual of the finite structures and algebras, construction of finitely generated free algebras,. . . ). Eventually, we illustrate this work for the finitely generated vari- eties of MV-algebras. 1. Introduction The most famous example of equivalence between algebras and topologi- cal spaces appeared in 1936 in a pioneering paper [18] of Stone in which he developed a duality between the category of Boolean algebras and the category of Boolean spaces. This duality gives a way to translate the algebraic properties of Boolean algebras in the language of topology and vice versa. His ideas were later adapted to a wide range of algebras, especially to classes of algebras arising from logic. Let us quote, for example, Pon- tryagin’s duality for Abelian groups (see [14] and [15]), Priestley’s dualities for distributive bounded lattices (see [16] and [17]) or Heyt- ing algebras etc. In all these situations, the results were obtained for a quasi-variety of algebras on the one hand and a category of topological structures on the other hand. 2000 Mathematics Subject Classification: 06D35; 08B. Key words and phrases: MV-algebras, Natural duality, Semi-primal algebras. 68 Natural dualities for varieties generated by... In the early eighties Davey and Werner initiated a systematic ap- proach of the problem of construction of dualities for classes of algebras (see [9] for one of the first papers and [4] for a monograph on the subject). This was the starting point of the theory of natural dualities. One of the main features of this theory is to provide sufficient conditions for the ex- istence of a duality for a finitely generated quasi-variety of algebras, and a canonical construction of the duality. The objects of the dual category are topological spaces with a structure, and the theory suggests which structure one has to add to these spaces to obtain a duality. When a class of algebras is dualisable, one can translate algebraic problems into equivalent topological problems that are sometimes more easily solved in this language. The most interesting cases arise when the duality is strong. Indeed, it means that the representation theorem can be extended to a categorical equivalence between the categories of algebras and topologi- cal structures. Universal problems can then be solved by duality. This theory has been widely developed and applied since its birth (see [7], [10], [5], [8], [13] for example or [4] and the bibliography therein). One of the most beautiful applications of the theory of natural duali- ties is the case of a variety generated by a semi-primal algebra. Indeed, in this particular situation, the dual category admits a very nice axiomati- zation (see theorem 3.14 in [4]). Moreover, some interesting varieties fall in this scope. For instance, any MV-chain Ln (n ∈ N) is a semi-primal al- gebra and generates a dualisable variety (which is the variety of algebras of the n + 1-valued Lukasiewicz logic). As noticed by R. Cignoli in [3], this result can be seen as a consequence of Theorem 6.5 in [11]. The paper [13] deals with some of the applications of this duality such as the description of the finitely (or countably) generated free algebras, or the finite projective members in HSP( Ln). The first motivation of the present paper is to generalize the strong dualities for the varieties HSP( Ln) to strong dualities for varieties HSP( Ln1 , . . . , Lnr) generated by a finite number of finite MV-chains. These latter varieties deserve a special interest for they are exactly, ac- cording to Komori’s classification work [12], the finitely generated sub- varieties of the variety of MV-algebras. Now, remark that if n is the least common multiple of {n1, . . . , nr}, then for 1 ≤ i ≤ r, Lni is a subalgebra of Ln and the variety HSP( Ln1 , . . . , Lnr) = ISP( Ln1 , . . . , Lnr) is a subvariety of HSP( Ln) = ISP( Ln). Therefore, with a duality for HSP( Ln), any member of HSP( Ln1 , . . . , Lnr) can be represented as a concrete algebra of morphisms and a duality for HSP( Ln1 , . . . , Lnr) might seem to be useless. But a strong duality is much more than a representation theorem : it is the P. Mathonet, P. Niederkorn, B. Teheux 69 expression of a (dual) equivalence between two categories. So, in order to translate properties that involve the category HSP( Ln1 , . . . , Lnr) and no longer a single object (such as the universal problems), it is the duality for HSP( Ln) that would be useless. The method we develop to construct a strong duality for HSP( Ln1 , . . . , Lnr) relies on the following observation: for any algebra A of HSP( Ln1 , . . . , Lnr), if n is the least common multiple of {n1, . . . , nr}, then for any homomorphism u from A to Ln, there exists a i in {1, . . . , r} such that u is a homomorphism form A to Lni . It means that, roughly speaking, the information contained in the union for 1 ≤ i ≤ r of the sets of the homomorphisms from A to Lni is the same as the informa- tion contained in the set of the homomorphisms from A to Ln. Thus, by transporting the structure of the dual of any algebra A under the duality for HSP( Ln) to the disjoint union for 1 ≤ i ≤ r of the sets of the homo- morphisms from A to Lni (which is the base set for the construction of the dual of A in the multi-sorted approach), there is a hope to obtain a duality. The interesting point is that, by proceeding carefully (by iden- tifying what should be identified; this is obviously made precise in the paper), it is possible to obtain a strong duality. Now, it appears clearly that this line of argument does not depend on the nature of the algebras Ln. The semi-primality of the algebras is not even necessary. So, one can restate the problem in these more gen- eral settings. Suppose that Π is a set of subalgebras of a finite algebra D and that a duality for ISP(D) already exists. Can one hope to con- struct a duality for ISP(Π) by transporting the structure of the duality for ISP(D)? In the very first part of the paper, we study some conditions under which one can construct a duality for ISP(Π) by this argument. Then, the results are particularized to varieties generated by a set of subalgebras of a semi-primal algebra. In this case, the obtained duality reveals to be a strong one. In Theorem 2.5, we even obtain a nice axiomatization of the objects of the dual category (this axiomatization is the counterpart of Theorem 3.14 in [4]). Then, we study some algebraic consequences of the duality. For in- stance, when we deal with the strong dualities for the varieties generated by subalgebras of a semi-primal algebra, we are able to describe the dual of finite structures and finite algebras, to construct the finitely generated free algebras, and to characterize finite projective algebras in HSP(Π). The final part of the paper is dedicated to a concrete illustration of our developments. We indeed use our results to construct a strong duality for any of the finitely generated subvarieties of the variety of MV-algebras and give some of its applications such as the description of 70 Natural dualities for varieties generated by... finitely generated free algebras in these varieties. 2. Construction of a duality 2.1. Notations Let us first set the general notations and assumptions that are in use throughout the paper. We use the standard notations of the theory of natural dualities and category theory. Hence, we denote the algebras by underlined Roman capital letters and the topological structures by “undertilded” Roman capital letters. We start with a finite dualisable algebra D on the language L. We denote by D the category with algebras of ISP(D) for objects and homo- morphisms for arrows. We set a structure D ˜ = 〈D;G0, H0, R0, τ〉 that generates a duality for D where τ is the discrete topology and G0 (resp. H0, R0) is a set of operations (resp. partial operations, relations). We denote by E the dual category: objects of E are the topological structures of IScP(D ˜ ) and arrows of E are the continuous maps that preserve the structure defined by G0, H0 and R0. The canonical functors of this duality are denoted by H : D → E and K : E → D. For the first part of the paper, we do not assume that D is a semi- primal algebra. Our goal is to construct a (strong) duality for the quasi-variety A = ISP(Π) where Π = {P 1, . . . , P r} is a set of subalgebras of D. To this aim, our approach in natural duality theory is the multi-sorted one. It means that the objects of the dual category are the members of IScP(Π ˜ ) for a suitable Π-indexed structure Π ˜ and that the dual of an algebra A of A is a Π-indexed topological structure defined on ∪· i∈{1,...,r}A(A,P i) (where ∪· stands for the disjoint union). We are going to restrict ourselves to some particular sets of subalge- bras Π. Indeed, in this paper, we only consider the sets Π such that ∀ A ∈ A, D(A,D) = ⋃ i∈{1,...,r} A(A,P i), (2.1) (note that D(A,P i) = A(A,P i) for every A in A and P i in Π). This is a rather strong restriction. It means that, disregarding the structure that could be defined on ∪· i∈{1,...,r}A(A,P i), this set contains the same information as H(A). But, as we shall see, condition (2.1) is satisfied for any set Π of proper subalgebras of a semi-primal algebra D. Note that the converse property always holds: if A is an algebra of D such that ∪i∈{1,...,r}A(A,P i) = D(A,D), then A is an algebra of A. P. Mathonet, P. Niederkorn, B. Teheux 71 Now, in order to obtain a duality for A, it suffices to define a structure Π ˜ on Π in such a way that the algebra of morphisms from ∪· i∈{1,...,r}A(A,P i) (seen as a structure of IScP(Π ˜ )) to Π ˜ is isomorphic to the algebra of morphisms from H(A) to D ˜ (which is the bidual of A under the duality between D and E). The idea is to define Π ˜ by transporting in Π the structure of D ˜ and to identify the elements of ∪· i∈{1,...,r}Pi which are equal in D. This should ensure that any morphism from the Π- indexed structure ∪· i∈{1,...,r}A(A,P i) to Π ˜ can be seen as an E-morphism and conversely. More precisely, we define Π ˜ as the Π-indexed structure Π ˜ = 〈 ⋃· i∈{1,...,r} Pi; {fij \| i, j ∈ {1, . . . , r}}, ⋃ s∈R+ 0 {s−1 i1,...,iks \| 1 ≤ i1 ≤ · · · ≤ iks ≤ r}; τ〉, (2.2) where • τ is the discrete topology; • for every i and j in {1, . . . , r}, the map fij is a partial operation de- fined on the subset Pi ∩ Pj of the jth summand Pj of ∪· k∈{1,...,r}Pk, which is valued in the subset Pi ∩ Pj of the ith summand Pi of ∪· k∈{1,...,r}Pk and which identifies the element that are equal in ∪k∈{1,...,r}Pk (the latter union is not taken disjoint): fij : Pi ∩ Pj ⊂ Pj → Pi : q 7→ q; • R+ 0 denotes the set whose elements are the relations of R0 and the graphs of the (partial) operations of G0 and H0; • for every relation s in R+ 0 with arity ks and every i1, . . . , iks ∈ {1, . . . , r}, the relation s−1 i1,...,iks is defined by s−1 i1,...,iks = {(q1, . . . , qks) ∈ Pi1 × · · · × Piks \| (q1, . . . , qks) ∈ s}. We denote by R+,−1 0 the set of the relations s−1 i1,...,iks with s ∈ R+ 0 and 1 ≤ i1 ≤ · · · ≤ iks ≤ r. We denote by X the category of the topological structures of IScP(Π ˜ ) with the obvious morphisms (the continuous maps that preserve the struc- ture). If A is an algebra of A, we denote by D(A) the Π-indexed structure induced by Π ˜ A on ∪· i∈{1,...,r}A(A,P i). If X ˜ is an element of IScP(Π ˜ ), we denote by E(X ˜ ) the A-algebra of the X -morphisms from X ˜ to Π ˜ . 72 Natural dualities for varieties generated by... Note that, strictly speaking, since we compute the structure Π ˜ on ∪· i∈{1,...,r}Pi, we should have replaced Pi by Pi×{i} for every i in {1, . . . , r} in order to ensure that the Pi are disjoint. By doing so, the definition of fij would have become fij : {(q, j) \| q ∈ Pi ∩ Pj} → Pi × {i} : (q, j) 7→ (q, i). Since the idea behind the construction of Π ˜ is clear, we have decided not to bother with such heavier notations. 2.2. The duality theorem With the notations and assumptions introduced in the previous section in mind, we can prove the following theorem. Theorem 2.1. If Π = {P 1, . . . , P r} is a set of subalgebras of a finite algebra D that satisfies condition (2.1) and if Π ˜ is the Π-indexed structure defined by (2.2), then the structure Π ˜ generates a duality between A = ISP(Π) and X = IScP(Π ˜ ). Proof. Assume that A is an algebra of A. Since Π ˜ is a Π-indexed structure that is algebraic over Π, it suffices to prove that the evaluation map eAA : A →֒ ED(A) is onto. In fact, one can prove that ψ : ED(A) → KH(A) defined by ψ(α)(u) = α(u) for every α ∈ ED(A) and every u ∈ H(A) is an isomorphism such that ψ ◦ eAA = eDA (where eDA is the evaluation map eDA : A→ KH(A)). The details are left the reader. Note that the structure Π ˜ could also have been defined with the help of the Piggyback duality theorem (see [6]). Naturally, we denote by D : A → X and E : X → A the functors defined by the duality of Theorem 2.1. Theorem 2.1 provides a tool to obtain a representation of any algebra of A as a concrete algebra of morphisms. But we could already obtain such a representation with the duality between D and E . So, the advan- tage of this result must be found elsewhere. In fact, the best way to derive benefit from Theorem 2.1 is to consider it as a step toward the construction of a strong duality for A (and so a dual equivalence for the category A). Indeed, the category X that we have defined can be seen as the category of structures of E that have nothing to share with algebras of D \ A. It means exactly that the dual E(X ˜ ) of any structure X ˜ of X is an algebra of A while the dual K(Y ˜ ) of a structure Y ˜ of E is not necessarily an algebra of A. Of course, we would be very lucky if the duality of Theorem 2.1 were a strong duality (even when the duality between D and E is strong). But in the next section, we prove that it can happen. P. Mathonet, P. Niederkorn, B. Teheux 73 2.3. When D is a semi-primal algebra Expression of the duality When D is a semi-primal algebra, it is possible to simplify the expression of the dual D(A) of an algebra A of A (which proves in this case to be a variety), similarly as in the uni-sorted case (see [4] for example). Moreover, this duality turns out to be a strong duality. Finally, we can obtain an axiomatization of the dual class of A. For the sake of convenience, we suppose that the language of the algebras contains at least two constants (as a consequence, there is no one-element subalgebra of D). The duality that we consider for D is the strong duality of Theorem 3.14 of Chapter 3 in [4]. Recall that we denote by Pn and Bn the set of the n-ary (partial) operations and relations respectively which are algebraic on Π. Let us also recall that if F is a finite algebra, we define irr(F ) as the least n such that the zero congruence on F is a meet of n meet-irreducible non zero congruences. The irreducibility index Irr(P ) of a finite algebra P is the maximum of the irr(F ) where F is a subalgebra of P . The irreducibility index Irr(Π) of the set Π = {P 1, . . . , P r} is Irr(Π) = max1≤i≤r Irr(P i). Proposition 2.2. If Π = {P 1, . . . , P r} is a set of subalgebras of a semi- primal algebra D and if Π ˜ denotes the structure Π ˜ =< ⋃· i∈{1,...,r} Pi; {fij \| i, j ∈ {1, . . . , r}}, r⋃ i=1 S(P i); τ >, where • for every i in {1, . . . , r}, S(P i) denotes the set of subalgebras of P i (viewed as unary relations); • for every i and j in {1, . . . , r}, the partial operation fij : Pi ∩ Pj ⊆ Pj → Pi is defined by fij(q) = q (see (2.2) for details); • τ is the discrete topology; then Π ˜ generates a strong natural duality on A. Proof. It is an application of the Multi Sorted N.U. Strong Duality The- orem (see Theorem 1.2 of Chapter 7 in [4]). Indeed, the algebras P i (i ∈ {1, . . . , r}) have a common Pixley term, and so a common ternary near-unanimity term. Furthermore, since Π ˜ generates a natural duality on A (it is the content of Theorem 2.1 in this more restricted context of semi-primal algebras), this structure entails all the finitary algebraic 74 Natural dualities for varieties generated by... relations on Π and in particular the relations of B2. Finally, since the algebras of Π are hereditarily simple, we have Irr(Π) = 1. It then follows that P1 = ⋃ 1≤i,j≤r{f ∈ A(B,P j) \| B ∈ S(P i)}. So, if i and j are in {1, . . . , r}, if B is an element of S(P i) and if f is a homomorphism of A(B,P j) then we can deduce that B is an algebra of S(P i∩P j) and that f = fji\|B. We can thus conclude that B2 ∪ P1 is strongly entailed by {fij \| 1 ≤ i, j ≤ r} ∪ ⋃r i=1 S(P i). As in the previous section, we denote by D : A → X and E : X → A the canonical functors defined by this strong duality (where X = IScP(Π ˜ )). Axiomatization of the dual class Our next task is to use this strong duality to obtain an axiomatization of the class D(A) = X . First, we study the properties of the members of X . Proposition 2.3. For any algebra A of A, any element i of {1, . . . , r} and any subalgebras F and F ′ of P i, 1. the interpretation FD(A)i of F on the ith summand of D(A) is a closed subspace of A(A,P i); 2. P D(A)i i = A(A,P i) ; 3. (F ∩ F ′)D(A)i = FD(A)i ∩ F ′D(A)i ; Proof. The proof is straightforward. The next proposition gives the properties of the partial operations fij . Once more, these are direct consequences of the definitions. Proposition 2.4. If A is a member of A, if i, j and k are elements of {1, . . . , r} and if F is a subalgebra of A, then 1. the composition f D(A) kj ◦f D(A) ji is equal to f D(A) ki on (P i∩P j∩P k) D(A)i and f D(A) ii = id\|A(A,P i) ; 2. the partial operation f D(A) ji is a homeomorphism from (P i∩P j) D(A)i to (P i ∩ P j) D(A)j ; 3. if x is an element of FD(A)i, then f D(A) ji (x) is an element of FD(A)j . P. Mathonet, P. Niederkorn, B. Teheux 75 In fact, these properties are exactly the ones required on a Π-indexed structure of the same type as Π ˜ to be a member of X . Thus, the following proposition is the counterpart to the well-known axiomatization of the dual class of a variety generated by a single semi-primal algebra (see Proposition 3.14 in Chapter 3 of [4]). Theorem 2.5. If Π ˜ denotes the structure defined in Proposition 2.2, then a topological structure X ˜ is a member of X = IScP(Π ˜ ) if and only if X ˜ =< ⋃· 1≤i≤r Xi; {f X ĩj \| 1 ≤ i, j ≤ r}, r⋃ i=1 {r X ˜i F \| F ∈ S(P i)}; τ >, where for any i and j in {1, . . . , r} and any subalgebras F and F ′ of P i, 1. τ is the disjoint union of Boolean topologies τi on each Xi; 2. the set r X ˜i F is a closed subspace of Xi and • r X ˜i P i = Xi • r X ˜i F ∩ r X ˜i F ′ = r X ˜i F∩F ′ ; 3. the partial operation f X j̃i is a homeomorphism from r X ˜i P i∩P j to r X ˜j P i∩P j such that • f X k̃j ◦ f X j̃i = f X k̃i for every k ∈ {1, . . . , r} • f X ĩi = id\|Xi • if x ∈ r X ˜i F∩P j then f X j̃i (x) ∈ r X ˜j F . Proof. The necessity of the condition is the content of Propositions 2.3 and 2.4. Now, let us consider the equivalence Θ whose elements are the (xi, xj) ∈ Xi ×Xj (where i, j ∈ {1, . . . , r}) such that xi ∈ r X ˜i P i∩P j and xj ∈ r X ˜j P j∩P i with f X j̃i (xi) = xj . We are about to define a subset r X/Θ F of X/Θ for every subalgebra F of D in such a way that 〈X/Θ, {r X/Θ F \| F ∈ S(D)}; τ〉, (where τ is the quotient topology) is a member H(A) of E (recall that the duality between D and E is strong since D is a semi-primal algebra) with D(A) ∼= X ˜ . 76 Natural dualities for varieties generated by... For every F in S(D) we define r X/Θ F as the set that collects the classes xΘ such that x is in ∪i∈{1,...r}r X ˜i F . With this definition, the only non trivial part consists in proving that X/Θ is a Boolean space. Let us consider two points x, y ∈ X and suppose that they are not equivalent. We will show that there exists a saturated clopen set Ω of X such that x ∈ Ω and y 6∈ Ω. We may suppose that x ∈ X1, and say y ∈ Xj . For any k ∈ {1, . . . , r} such that y ∈ r X ˜j P j∩Pk , we set yk = f X k̃j(y). Now, we will build by induction clopen sets Ωk of Xk (k ∈ {1, . . . , r}) such that 1. x ∈ Ω1 2. ∀k ∈ {1, . . . , r}, yk 6∈ Ωk 3. ∀k, l ∈ {1, . . . , r}, f X k̃l(Ωl ∩ r X ˜l Pk∩P l ) = Ωk ∩ r X ˜k Pk∩P l . Then the set Ω = ∪1≤k≤rΩk will allow us to conclude the proof. The existence of Ω1 is obvious since X1 is a Boolean space and x 6= y1. Now, let us suppose that we have constructed clopen sets Ω1, . . . ,Ωi−1 (i ≤ r) fulfilling conditions (1), (2) and (3) and let us show how to define Ωi. For any k < i we set ωi,k = f X ĩk(Ωk ∩ r X ˜k Pk∩P i ) and ω = ∪1≤k<i ωi,k. In view of condition (2), the set ω does not contain yi. It is also a clopen subset of ∪1≤k<i r X ˜i Pk∩P i . Indeed, on the one hand, for k < i, ωi,k is a closed subset of r X ˜i Pk∩P i and therefore of Xi. On the other hand, since f X k̃i(ωi,l ∩ r X ˜i Pk∩P i ) ⊂ f X k̃i ◦ f X ĩl (Ωl ∩ r X ˜l P i∩Pk∩P l ) ⊂ Ωk ∩ r X ˜k P i∩Pk holds for any k, l < i, we have ωi,l ∩ r X ˜i Pk∩P i ⊂ ωi,k, and then (∪k<ir X ˜i Pk∩P i ) \ (∪k<iωi,k) = ∪k<i(r X ˜i Pk∩P i \ ωi,k). This set is open since ωi,k is closed in r X ˜i Pk∩P i . Then, there exists a clopen set Ωi of Xi such that Ωi ∩ ( ⋃ k<i r X ˜i Pk∩P i ) = ω, P. Mathonet, P. Niederkorn, B. Teheux 77 so that for any k < i, Ωi ∩ r X ˜i Pk∩P i = ω ∩ r X ˜i Pk∩P i = ωi,k = fik(Ωk ∩ r X ˜k Pk∩P i ). Eventually, we can require in addition that yi 6∈ Ωi since yi 6∈ ω. 3. Algebraic aspects of the duality In this section, we gather algebraic consequences of the duality between A and X . 3.1. General Results We do not assume initially the semi-primality of D. Our first result shows that if the duality between D and E is strong, then there is a natural bijection between onto A-morphisms and X -embeddings. Proposition 3.1. Suppose that the duality between D and E is strong and consider two algebras A and B of A. If ψ : D(B) → D(A) is an embedding, then E(ψ) : ED(A) → ED(B) is an onto A-morphism. Con- versely, if u : A → B is an onto A-morphism then D(u) : D(B) → D(A) is an X -embedding. Proof. For the first part of the assertion, it suffices to prove that if ψ : D(B) → D(A) is an embedding, then the map ψ′ : H(B) → H(A) : x 7→ ψ(x) is a well defined E-embedding and satisfies E(ψ) = K(ψ′) (up to the canonical isomorphism f : ED(A) → KH(A) defined by f(α) : x 7→ α(x)). Indeed, the map K(ψ′) is an onto homomorphism according to Lemma 2.6 of Chapter 3 in [4]. Conversely, its suffices to note that the map D(u) : D(B) → D(A) is in fact defined by (D(u))(x) = (H(u))(x). The result follows from the fact that H(u) is an E-embedding following the previously cited Lemma. The connection between A-embeddings and onto X -morphisms is not so perfect, since the dual of an X -morphism is an A-embedding but not conversely. Moreover, this result requires more restrictive hypotheses. Proposition 3.2. Suppose that the duality between D and E is strong and that D is injective in D. Consider two algebras A and B of A. If ψ : D(B) → D(A) is an onto X -morphism, then E(ψ) : ED(A) → ED(B) is an A-embedding. Proof. Follow the idea of the proof of Proposition 3.1 and apply Lemma 2.8 of Chapter 3 in [4]. Example 4.2 shows that the converse of the previous proposition does not hold. 78 Natural dualities for varieties generated by... 3.2. Results involving the semi-primality of D Much more consequences can be derived from the duality between A and X when D is a semi-primal algebra. Our first result is a description of finite dual spaces. To obtain it, we need to introduce some new unary relations. Definition 3.3. For each member B of D and each subalgebra F of D, the unary relation s H(B) F is defined by s H(B) F = FH(B) \ ⋃ F ′∈S(F )\{F} F ′H(B). Similarly, for each algebra A of A, each i in {1, . . . , r} and each subalgebra F of P i, we define the unary relation s D(A)i F on the ith summand of D(A) by s D(A)i F = FD(A)i \ ⋃ F ′∈S(F )\{F} F ′D(A)i . If F is a subalgebra of D and if B is a member of D, then r H(B) F = ⋃· F ′∈S(F ) s H(B) F ′ , and a similar result holds for the relations r D(A)i F (where A ∈ A, i ∈ {1, . . . , r} and F ∈ S(P i)). With these new relations, we can give a new expression of Proposition 2.2 in [13]. Proposition 3.4. If B ∼= ∏ F∈S(D) F fB(F ) (where the fB(F ) are non- negative integers) then H(B) is a discrete topological space such that \|s H(B) F \| = fB(F ) for every F in S(D). Conversely, if Y ˜ is a finite (dis- crete) member of E, then E(Y ˜ ) ∼= ∏ F∈S(D) F \|s Y F̃ \|. With the help of this proposition we can describe the dual of the finite members of A. Proposition 3.5. If A = ∏ F∈ ⋃ 1≤i≤r S(P i) F fA(F ) (where the fA(F ) are non negative integers) is a finite member of A, then D(A) is a finite member of X such that for every i ∈ {1, . . . , r} and F ∈ S(P i) \|D(A)i\| = ∑ F∈S(P i) fA(F ) and \|s D(A)i F \| = fA(F ). P. Mathonet, P. Niederkorn, B. Teheux 79 Conversely, if X ˜ is a finite (thus discrete) member of X , then E(X ˜ ) = ∏ F∈ ⋃ i∈{1,...,r} S(P i) F \|s X ˜i F \|. Proof. By the previous proposition, we know that H(A) is a discrete space such that \|s H(A) F \| = { fA(F ) if F ∈ ⋃ 1≤i≤r S(P i) 0 if F ∈ S(D) \ ⋃ 1≤i≤r S(P i). It follows that \|D(A)i\| = \|P H(A) i \| = ∑ F∈S(P i) \|s H(A) F \| = ∑ F∈S(P i) fA(F ). We get the first part of the proposition if we note that \|s D(A)i F \| = \|s H(A) F \| for every i ∈ {1, . . . , r} and F ∈ S(P i). Conversely, if X ˜ is a finite member of X and if X/Θ is the E-structure defined in the proof of Theorem 2.5, we see that K(X/Θ) ∼= ∏ F∈ ⋃ 1≤i≤r S(P i) F \|s X/Θ F \|, since \|s X/Θ F \| = 0 if F ∈ S(D) \ ⋃ 1≤i≤r S(P i). Then, E(X ˜ ) ∼= K(X/Θ) ∼= ∏ F∈ ⋃ 1≤i≤r S(P i) F \|s X/Θ F \| ∼= ∏ F∈ ⋃ 1≤i≤r S(P i) F \|s Xi F \|, and the conclusion follows. Our next purpose is to construct the finitely generated free algebras over A. We first recall the definition of the classical Möbius’ function of lattice theory. Definition 3.6. If O is a finite partially ordered set, the Möbius’ func- tion associated to O is the function µO defined on O ×O by µO(x, y) =    0 if x � y 1 if x = y − ∑ x≤z<y µO(x, z) if x < y. We denote by Sub(D) the lattice of the subalgebras of D. Let us recall that we denote by FA(k) the free algebra with k generators over the class A. 80 Natural dualities for varieties generated by... Proposition 3.7. For every positive integer k, we have FA(k) ∼= ∏ F∈ ⋃ 1≤i≤r S(P i) F fA(k,F ), where fA(k, F ) = ∑ F ′∈S(F ) µSub(D)(F ′, F ).\|F ′\|k. Proof. Since, by Proposition 3.5, FA(k) ∼= E(Π ˜ k) ∼= ∏ F∈ ⋃ 1≤i≤r S(P i) F \|s Pk i F \|, we just have to prove that \|s Pk i F \| is equal to the proposed fA(k, F ) for every subalgebra F of P i. An application of the Möbius’ inversion formula to \|F \|k = \|FP k i \| = ∑ F ′∈S(F ) \|s Pk i F ′ \| allows us to give the desired value for each of the fA(k, F ). Our last application is a characterization of finite projective alge- bras in A. In the following proposition, we denote by F 0 the algebra⋂ i∈{1,...,r} P i (this intersection is not empty since we have assumed that the language L of the algebras contains at least two constants). Proposition 3.8. If A is a member of A, the following conditions are equivalent: 1. A is a finite projective member of A; 2. there is an element i of {1, . . . , r} such that r D(A)i F 0 is not empty; 3. there is a finite member B of A such that A is isomorphic to F 0×B. Proof. Since the duality between A and X is strong, a finite algebra A of A is projective in A if and only if D(A) is a finite injective member of X . Suppose that t1 belongs to r D(A)1 F 0 and denote by ti the element fi1(t1) of D(A)i for every i in {2, . . . , r}. If ψ : Y ˜ → Z ˜ is an X -embedding be- tween two members Y ˜ and Z ˜ of X and if φ : Y ˜ → D(A) is an X -morphism, we know that there is a continuous application θ : Z → D(A) respecting the Π-indexed structure such that θ ◦ψ = φ (this is a consequence of the injectivity of finite Boolean spaces in the class of Boolean spaces). As φ is an X -morphism, we thus know that θ\|ψ(Y ) is also an X -morphism. P. Mathonet, P. Niederkorn, B. Teheux 81 Now, for each i in {1, . . . , r} and each x in D(A)i, denote by F x the unique subalgebra of P i such that x belongs to s D(A)i Fx . Let us also define the closed subspace rx of Z ˜ i by rx = θ−1(x)∩ ∩ {( ⋃ F∈S(P i) F�Fx r Z ˜ i F ) ∪ ( ⋃ j∈{1,...,r} i6=j ⋃ z∈r D(A)j Pi∩Pj z 6=fji(x) {y ∈ r Z ˜ i P i∩P j \| θ(fji(y)) = z})}. Denote by ri the subspace ⋃ x∈D(A)i rx and by r the subspace ⋃ i∈{1,...,r} ri. These are closed in Zi and Z respectively. Since θ \|ψ(Y ) is an X -morphism, the intersection ψ(Y ) ∩ r is empty. So, for every i in {1, . . . , r}, we can find an open subset ωi of Zi which contains ri but no element of ψ(Y )i. Thus, the map λ : Z → D(A) : x ∈ Zi 7→ { ti if x ∈ ωi θ(x) if x /∈ ωi is an X -morphism such that ψ ◦ θ = φ. Conversely, let us consider Y ˜ = D(P 1 × · · · × P r) and Z ˜ = D(F 0 × P 1 × · · · × P r). According to Proposition 3.5, the sets Yi and Zi are discrete spaces with Yi = sP i = {yi} and Zi = sF 0 ∪ sP i = {fi} ∪ {zi} for all i in {1, . . . , r}. If D(A) is a finite injective member of X , if ψ denotes the unique X -morphism from Y to Z and if φ : Y → D(A) is an X -morphism, there exists an X -morphism θ : Z → D(A) such that θ ◦ ψ = φ. It follows that θ(fi) belongs to F D(A)i 0 . The equivalence between (2) and (3) is a consequence of Proposi- tion 3.5 4. Natural dualities on finitely generated varieties of MV- algebras In this section we illustrate the previous developments by giving an ap- plication to the finitely generated varieties of MV-algebras. In this way, we finish the work begun in [13]. MV-algebras were introduced in 1958 by Chang (see [1] and [2]) as a many-valued counterpart of Boolean algebras. Their study in a logical and algebraic aspect led to numerous interesting results, as for instance an algebraic proof of the completeness theorem of Lukasiewicz’s infinite- valued propositional calculus (see [2]). An MV-algebra can be viewed as an algebra A = 〈A;⊕,⊙,¬, 0, 1〉 of type (2,2,1,0,0) such that 〈A;⊕, 0〉 is an Abelian monoid and that satisfies 82 Natural dualities for varieties generated by... the following identities: ¬¬x = x, x⊕1 = 1, ¬0 = 1, x⊙y = ¬(¬x⊕¬y), (x⊙ ¬y) ⊕ y = (y ⊙ ¬x) ⊕ x. One of the most simple (and most important) example of MV-algebra is the real interval [0, 1] endowed with the operations x⊕y = min(x+y, 1), x⊙ y = max(x+ y − 1, 0) and ¬x = 1 − x. Komori’s classification of the subvarieties of the variety M of MV- algebras (see [12]) underlines the importance of the subalgebras Ln = {0, 1 n , . . . , n−1 n , 1} (where n is a positive integer) of [0, 1]. Indeed, the finitely generated subvarieties of M are exactly the ones generated by a finite number of Ln. For example, the variety of MV-algebras that satisfies the additional equation (m+ 1).x = m.x for an integer m ≥ 1 is exactly the variety generated by L1, . . . , Lm. We thus now apply our results to the finitely generated varieties of MV-algebras. From now on, we denote by r an element of N \ {1} and by n1, . . . , nr some positive integers such that ni does not divide nj for every i 6= j in {1, . . . , r}. The set Π is the set of the algebras Ln1 , . . . , Lnr and m is the lowest common multiple to n1, . . . , nr. The class A is the variety ISP(Π) and D is the variety ISP( Lm) (on which a natural duality was developed in [13]). We also make use of the previously defined notations concerning functors and duality. The duality is obtained with the help of the results of section 2.3. In the sequel of the paper, we denote by div(n) the set of the positive divisors of the integer n. Proposition 4.1. Let us define the structure Π ˜ =< ⋃· 1≤i≤r Lni ; {fji \| 1 ≤ i, j ≤ r}, r⋃ i=1 { Lk \| k ∈ div(ni)}; τ >, where τ is the discrete topology and where for every i, j in {1, . . . , r}, fji : Lgcd(ni,nj) ⊂ Lni → Lnj : q 7→ q. Then Π ˜ generates a strong duality on A. The results of Proposition 3.1 and 3.2 apply here. We have announced a counterexample for the converse of Proposition 3.2. We are now able to produce it. Example 4.2. Let us set Π = { L6, L10} and D = L30. Consider the two algebras A = L2 × L2 and B = L6 × L10 of A = ISP(Π). With the help of Proposition 3.5, we obtain on the one hand that the dual spaces H(A) and H(B) are two discrete spaces with two elements. On the other P. Mathonet, P. Niederkorn, B. Teheux 83 hand, each of the two summands of D(A) is made of two elements, but each summand of D(B) contains a single element. If we consider the dual H(p) of the canonical A-embedding of A into B, we obtain an onto E-morphism, since L30 is injective in D = ISP( L30). On the other hand, it is clear that D(p) : D(B) → D(A) can not be onto. Using Proposition 3.7 and the Möbius function µ defined on N by µ(n) = { 0 if n is not square free (−1)\|P (n)\| if n is square free, where P (n) is the set of the prime divisors of n, we can compute the free algebras with a finite number of generators. We denote by ∏ X the product of the elements of the finite set of integers X. Proposition 4.3. For every positive integer k, we have FA(k) ∼= ∏ q∈ ⋃ 1≤i≤r div(ni) L f(k,q) q , where f(k, q) = ∑ X⊆P (q) (−1)\|X\| · ( q∏ X + 1)k. Proof. By Proposition 3.5, we know that f(k, q) = fA(k, Lq) = ∑ Ls∈S( Lq) µ Sub( Lm)( Ls, Lq) · \| Ls\| k. Since Sub( Lq) is isomorphic to the lattice of divisors of q, we can write successively µ Sub( Lm)( Ls, Lq) = µdiv(q)(s, q) = µdiv( q s )(1, q s ) = µ( q s ), and since \| Ls\| = s+ 1, it follows that f(k, q) = ∑ s∈div(q) µ( q s )(s+ 1)k. The integer q s is not square free if and only if there is a subset X of P (q) such that q s = ∏ X. In this case, we have µ( qs ) = (−1)\|X\| and s = q∏ X , which allows us to draw the desired conclusion. 84 Natural dualities for varieties generated by... Acknowledgments We want to thank the anonymous referee for his fruitful suggestions. References [1] C. C. Chang. Algebraic analysis of many valued logics. Trans. Amer. Math. Soc., 88:467–490, 1958. [2] C. C. Chang. A new proof of the completeness of the lukasiewicz axioms. Trans. Amer. Math. Soc., 93:74–80, 1959. [3] R. Cignoli. Natural dualities for the algebras of lukasiewicz finite-valued logics. Bull. Symbolic Logic, 2(2):218, 1996. [4] D. M. Clark and B. A. Davey. Natural dualities for the working algebraist, vol- ume 57 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1998. [5] D. M. Clark, P. M. Idziak, L. R. Sabourin, C. Szabó, and R. Willard. Natu- ral dualities for quasivarieties generated by a finite commutative ring. Algebra Universalis, 46(1-2):285–320, 2001. [6] B. A. Davey and H. A. Priestley. Generalised piggyback dualities and applications to Ockham algebras. Houston J. Math., 13(2):151–198, 1987. [7] B. A. Davey and H. A. Priestley. Optimal natural dualities for varieties of Heyting algebras. Studia Logica, 56(1-2):67–96, 1996. [8] B. A. Davey and R. W. Quackenbush. Natural dualities for dihedral varieties. J. Austral. Math. Soc. Ser. A, 61(2):216–228, 1996. [9] B. A. Davey and H. Werner. Dualities and equivalences for varieties of algebras. In Contributions to lattice theory (Szeged, 1980), volume 3 3 of Colloq. Math. Soc. János Bolyai, pages 101–275. North-Holland, Amsterdam, 1983. [10] A. Di Nola and P. Niederkorn. Natural dualities for varieties of BL-algebras. Arch. Math. Logic, 44(8):995–1007, 2005. [11] K. Keimel and H. Werner. Stone duality for varieties generated by quasi-primal algebras. In Recent advances in the representation theory of rings and C ∗-algebras by continuous sections (Sem., Tulane Univ., New Orleans, La., 1973), pages 59– 85. Mem. Amer. Math. Soc., No. 148. Amer. Math. Soc., Providence, R.I., 1974. [12] Y. Komori. Super- lukasiewicz propositional logics. Nagoya Math. J., 84:119–133, 1981. [13] P. Niederkorn. Natural dualities for varieties of MV-algebras. I. J. Math. Anal. Appl., 255(1):58–73, 2001. [14] L. S. Pontryagin. Sur les groupes abГ c©liens continus. C. R. Acad. Sci. Paris, 198:238–240, 1934. [15] L. S. Pontryagin. The theory of topological commutative groups. Ann. Math., 35:361–388, 1934. [16] H. A. Priestley. Representation of distributive lattices by means of ordered stone spaces. Bull. London Math. Soc., 2:186–190, 1970. [17] H. A. Priestley. Ordered topological spaces and the representation of distributive lattices. Proc. London Math. Soc. (3), 24:507–530, 1972. [18] M. H. Stone. The theory of representations for Boolean algebras. Trans. Amer. Math. Soc., 40(1):37–111, 1936. P. Mathonet, P. Niederkorn, B. Teheux 85 Contact information P. Mathonet, P. Niederkorn, B. Teheux University of Liège, Department of Mathe- matics Grande Traverse, 12, Liège 4000, Belgium E-Mail: p.mathonet@ulg.ac.be, p.niederkorn@gmail.com, b.teheux@ulg.ac.be Received by the editors: 23.04.2007 and in final form 25.05.2007.

Natural dualities for varieties generated by a set of subalgebras of a semi-primal algebra

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