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We follow the ideas given by Chen and Grätzer to represent Stone algebras and adapt them for the case of Stonean residuated lattices. Given a Stonean residuated lattice, we consider the triple formed by its Boolean skeleton, its algebra of dense elements and a connecting map. We define a category whose objects are these triples and suitably defined morphisms, and prove that we have a categorical equivalence between this category and that of Stonean residuated lattices. We compare our results with other works and show some applications of the equivalence. Fil: Busaniche, Manuela. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Matemática Aplicada del Litoral. Universidad Nacional del Litoral. Instituto de Matemática Aplicada del Litoral; Argentina Fil: Cignoli, Roberto Leonardo Oscar. Universidad de Buenos Aires; Argentina Fil: Marcos, Miguel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Matemática Aplicada del Litoral. Universidad Nacional del Litoral. Instituto de Matemática Aplicada del Litoral; Argentina

Given an integral commutative residuated lattice L, the product L × L can be endowed with the structure of a commutative residuated lattice with involution that we call a twist-product. In the present paper, we study the subvariety KK of commutative residuated lattices that can be represented by twist-products. We give an equational characterization of KK , a categorical interpretation of the relation among the algebraic categories of commutative integral residuated lattices and the elements in KK , and we analyze the subvariety of representable algebras in KK . Finally, we consider some specific class of bounded integral commutative residuated lattices GG , and for each fixed element L∈GL∈G , we characterize the subalgebras of the twist-product whose negative cone is L in terms of some lattice filters of L, generalizing a result by Odintsov for generalized Heyting algebras. Fil: Busaniche, Manuela. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Santa Fe. Instituto de Matemática Aplicada "Litoral"; Argentina Fil: Cignoli, Roberto Leonardo Oscar. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina

Let $${\mathbb{BRL}}$$ denote the variety of commutative integral bounded residuated lattices (bounded residuated lattices for short). A Boolean retraction term for a subvariety $${\mathbb{V}}$$ of $${\mathbb{BRL}}$$ is a unary term t in the language of bounded residuated lattices such that for every $${{\bf A} \in \mathbb{V}, t^{A}}$$ , the interpretation of the term on A, defines a retraction from A onto its Boolean skeleton B(A). It is shown that Boolean retraction terms are equationally definable, in the sense that there is a variety $${\mathbb{V}_{t} \subsetneq \mathbb{BRL}}$$ such that a variety $${\mathbb{V} \subsetneq \mathbb{BRL}}$$ admits the unary term t as a Boolean retraction term if and only if $${\mathbb{V} \subseteq \mathbb{V}_{t}}$$ . Moreover, the equation s(x) = t(x) holds in $${\mathbb{V}_{s} \cap \mathbb{V}_{t}}$$ . The radical of $${{\bf A} \in \mathbb{BRL}}$$ , with the structure of an unbounded residuated lattice with the operations inherited from A expanded with a unary operation corresponding to double negation and a a binary operation defined in terms of the monoid product and the negation, is called the radical algebra of A. To each involutive variety $${\mathbb{V} \subseteq \mathbb{V}_{t}}$$ is associated a variety $${\mathbb{V}^{r}}$$ formed by the isomorphic copies of the radical algebras of the directly indecomposable algebras in $${\mathbb{V}}$$ . Each free algebra in such $${\mathbb{V}}$$ is representable as a weak Boolean product of directly indecomposable algebras over the Stone space of the free Boolean algebra with the same number of free generators, and the radical algebra of each directly indecomposable factor is a free algebra in the associated variety $${\mathbb{V}^{r}}$$ , also with the same number of free generators. A hierarchy of subvarieties of $${\mathbb{BRL}}$$ admitting Boolean retraction terms is exhibited.

Let Γ be Mundici's functor from the category $${\mathcal{LG}}$$ whose objects are the lattice-ordered abelian groups (l-groups for short) with a distinguished strong order unit and the morphisms are the unital homomorphisms, onto the category $${\mathcal{MV}}$$ of MV-algebras and homomorphisms. It is shown that for each strong order unit u of an l-group G, the Boolean skeleton of the MV-algebra Γ(G, u) is isomorphic to the Boolean algebra of factor congruences of G.

The class of NPc-lattices is introduced as a quasivariety of commutative residuated lattices, and it is shown that the class of pairs (A,A+) such that A is an NPc-lattice and A+ is its positive cone, is a matrix semantics for Nelson paraconsistent logic.

Spinks and Veroff have shown that constructive logic with strong negation (CLSN for short), can be considered as a substructural logic. We use algebraic tools developed to study substructural logics to investigate some axiomatic extensions of CLSN. For instance, we prove that Nilpotent minimum logic is the extension of CLSN by the prelinearity axiom. This generalizes the well-known result by Monteiro and Vakarelov that three-valued ukasiewicz logic is an extension of CLSN. A Glivenko-like theorem relating CLSN and three-valued ukasiewicz logic is proved. Fil: Busaniche, Manuela. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Matemática Aplicada del Litoral. Universidad Nacional del Litoral. Instituto de Matemática Aplicada del Litoral; Argentina Fil: Cignoli, Roberto Leonardo Oscar. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina

Given a variety $$\mathbb{V}$$ of bounded residuated lattices satisfying the Stone identity $$\neg x \lor \neg\neg x = \top$$ , the free algebras in $$\mathbb{V}$$ over a set X of cardinality |X| are represented as weak Boolean products over the Cantor space 2|X| of a family of free algebras in an associated variety of (not necessarily bounded) residuated lattices with a bottom added.

For each integer n ≥ 2, MVn denotes the variety of MV-algebras generated by the MV-chain with n elements. Algebras in MVn are represented as continuous functions from a Boolean space into a n-element chain equipped with the discrete topology. Using these representations, maximal subalgebras of algebras in MVn are characterized, and it is shown that proper subalgebras are intersection of maximal subalgebras. When A ∈ MV3, the mentioned characterization of maximal subalgebras of A can be given in terms of prime filters of the underlying lattice of A, in the form that was conjectured by A. Monteiro. Fil: Cignoli, Roberto Leonardo Oscar. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderon; Argentina Fil: Monteiro, Luis. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca. Instituto de Matemática Bahía Blanca. Universidad Nacional del Sur. Departamento de Matemática. Instituto de Matemática Bahía Blanca; Argentina

Building on the Goodearl–Handelman–Lawrence functional representation theorem, we provide a purely topological representation (specifically, a categorical duality) for a large class of Dedekind σ-complete ℓ-groups G with order-unit u, including all G where u has a finite index of nilpotence. Our duality is a far-reaching generalization of the well-known Stone duality between σ-complete boolean algebras and basically disconnected compact Hausdorff spaces. Fil: Cignoli, Roberto Leonardo Oscar. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderon; Argentina Fil: Mundici, Daniele. Universita Degli Studi Di Firenze; Italia

The aim of this paper is to give a description of the free algebras in some varieties of Glivenko MTL-algebras having the Boolean retraction property. This description is given in terms of weak Boolean products over Cantor spaces. We prove that in some cases the stalks can be obtained in a constructive way from free kernel DL-algebras, which are the maximal radical of directly indecomposable Glivenko MTL-algebras satisfying the equation in the title. We include examples to show how we can apply the results to describe free algebras in some well known varieties of involutive MTL-algebras and of pseudocomplemented MTL-algebras. Fil: Cignoli, Roberto Leonardo Oscar. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderon; Argentina Fil: Torrens Torrell, Antoni. Universidad de Barcelona; España

Stone duality between boolean algebras and inverse limits of finite sets is extended to a duality between locally finite MV-algebras and a category of multisets naturally arising as inverse limits of finite multisets.

The classical Glivenko theorem asserts that a propositional formula admits a classical proof if and only if its double negation admits an intuitionistic proof. By a natural expansion of the BCK-logic with negation we understand an algebraizable logic whose language is an expansion of the language of BCK-logic with negation by a family of connectives implicitly defined by equations and compatible with BCK-congruences. Many of the logics in the current literature are natural expansions of BCK-logic with negation. The validity of the analogous of Glivenko theorem in these logics is equivalent to the validity of a simple one-variable formula in the language of BCK-logic with negation. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Using the theory of BL-algebras, it is shown that a propositional formula ϕ is derivable in Łukasiewicz infinite valued Logic if and only if its double negation ˜˜ϕ is derivable in Hajek Basic Fuzzy logic. If SBL is the extension of Basic Logic by the axiom (φ & (φ→˜φ)) → ψ, then ϕ is derivable in in classical logic if and only if ˜˜ ϕ is derivable in SBL. Axiomatic extensions of Basic Logic are in correspondence with subvarieties of the variety of BL-algebras. It is shown that the MV-algebra of regular elements of a free algebra in a subvariety of BL-algebras is free in the corresponding subvariety of MV-algebras, with the same number of free generators. Similar results are obtained for the generalized BL-algebras of dense elements of free BL-algebras.

In this paper we study the subclass of left-continuous t-norms *n which are definable by an arbitrary continuous t-norm * and a weak (i.e. non necessarily involutive) negation n by putting x *n y = 0 if x ≤ n(y), x *n y = x * y otherwise, thus generalizing the construction of the so-called nilpotent minimum t-norms. We provide the characterization of weak negations compatible with a given continuous t-norm and conversely which are the continuous t-norms compatible with a given weak negation function. © 2002 Elsevier Science B.V. All rights reserved., Part of this paper was written when Roberto Cignoli was in 2000 a visiting professor at the University of Siena with the support of GNSAGA (Gruppo Nazionale Struuture Algebriche e Geometriche e loro Applicazioni). Francesc Esteva and Lluís Godo acknowledge partial support of the bilateral cooperation project IIIA — Universidad de Buenos Aires, ACI99-25, funded by the Generalitat de Catalunya, and the Spanish project e-Institutor, TIC-2000-1414.

The aim of this paper is to give a description of the free algebras in varieties of BL-algebras having the Boolean retraction property. This description is given in terms of weak Boolean products over Cantor spaces. The stalks are obtained in a constructive way from free radical algebras. Radical algebras are obtained endowing the maximal radicals of BL-algebras with a unary operation corresponding to double negation. The radical algebras obtained from a variety of BL-algebras form themselves a variety, that in the cases of PL-algebras and bipartite MV-algebras can be identified with the class of cancellative hoops.

It is shown that axiomatic extensions of intuitionistic propositional calculus defining univocally new connectives, including those proposed by Gabbay, are strongly complete with respect to valuations in Heyting algebras with additional operations. In all cases, the double negation of such a connective is equivalent to a formula of intuitionistic calculus. Thus, under the excluded third law it collapses to a classical formula, showing that this condition in Gabbay's definition is redundant. Moreover, such connectives can not be interpreted in all Heyting algebras, unless they are already equivalent to a formula of intuitionistic calculus. These facts relativize to connectives over intermediate logics. In particular, the intermediate logic with values in the chain of length n may be “completed” conservatively by adding a single unary connective, so that the expanded system does not allow further axiomatic extensions by new connectives.

The aim of this short note is to give a description of the free cancellative hoops in terms of piecewise linear functions.

In this paper we prove that Basic Logic (BL) is complete w.r.t. the continuous t-norms on [0, 1], solving the open problem posed by Hajek in [4]. In fact, Hajek proved that such completeness theorem can be obtained provided two new axioms, B1 and B2, were added to the original axioms of BL. The main result of the paper is to show that B1 and B2 axioms are indeed redundant. We also obtain an improvement of the decomposition theorem for saturated BL-chains as ordinal sums whose components are either MV, product or Godel chains, in an analogous way as for continuous t-norms. Finally we provide equational characterizations of the variety of BL-algebras generated by the three basic BL subvarieties, as well as of the varieties generated by each pair of them, together with completeness results of the calculi corresponding to all these subvarieties.

We prove that for each ℓ-group G, the topological space Spec(G) satisfies a condition Idω. Generalising a previous construction of Delzell and Madden we show that for each nondenumerable cardinal there is a completely normal spectral space that is not homeomorphic to Spec(G) for any ℓ-group G. We show also that a stronger form of property Idω, called Id, suffices to ensure that a completely normal spectral space is homeomorphic to Spec(G) for some ℓ-group G. © 1999 Elsevier Science B.V. All rights reserved. Fil:Gluschankof, D. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina.

Aim of this paper is to provide a self-contained presentation of the natural equivalence Γ between MV-algebras and lattice-ordered abelian groups with strong unit.

The interpretation of propositions in Lukasiewicz's infinite-valued calculus as answers in Ulam's game with lies--the Boolean case corresponding to the traditional Twenty Questions game--gives added interest to the completeness theorem. The literature contains several different proofs, but they invariably require technical prerequisites from such areas as model-theory, algebraic geometry, or the theory of ordered groups. The aim of this paper is to provide a self-contained proof, only requiring the rudiments of algebra and convexity in finite-dimensional vector spaces.

The dual spaces of the free distributive lattices with a quantifier are constructed, generalizing Halmos' construction of the dual spaces of free monadic Boolean algebras.

A symmetric residuated lattice is an algebra A = (A, ∨, ∧, *, →, ∼, 1, 0) such that (A, ∨, ∧, *, →, 1, 0) is a commutative integral bounded residuated lattice and the equations ∼ ∼ x = x and ∼ (x ∨ y) = ∼ x ∧ ∼ y are satisfied. The aim of the paper is to investigate the properties of the unary operation ε defined by the prescription ε x = ∼ x → 0. We give necessary and sufficient conditions for ε being an interior operator. Since these conditions are rather restrictive (for instance, on a symmetric Heyting algebra ε is an interior operator if and only the equation (x → 0) ∨ ((x → 0) → 0) = 1 is satisfied) we consider when an iteration of ε is an interior operator. In particular we consider the chain of varieties of symmetric residuated lattices such that the n iteration of ε is a boolean interior operator. For instance, we show that these varieties are semisimple. When n = 1, we obtain the variety of symmetric stonean residuated lattices. We also characterize the subvarieties admitting representations as subdirect products of chains. These results generalize and in many cases also simplify, results existing in the literature. © 2009 Elsevier B.V. All rights reserved., The research communicated in this paper was partially supported by a bilateral Argentinean - Spanish project CONICET - CSIC

The notion of a Priestley relation between Priestley spaces is introduced, and it is shown that there is a duality between the category of bounded distributive lattices and 0-preserving join-homomorphisms and the category of Priestley spaces and Priestley relations. When restricted to the category of bounded distributive lattices and 0-1-preserving homomorphisms, this duality yields essentially Priestley duality, and when restricted to the subcategory of Boolean algebras and 0-preserving join-homomorphisms, it coincides with the Halmos-Wright duality. It is also established a duality between 0-1-sublattices of a bounded distributive lattice and certain preorder relations on its Priestley space, which are called lattice preorders. This duality is a natural generalization of the Boolean case, and is strongly related to one considered by M. E. Adams. Connections between both kinds of dualities are studied, obtaining dualities for closure operators and quantifiers. Some results on the existence of homomorphisms lying between meet and join homomorphisms are given in the Appendix.

The infinite-valued logic of Łukasiewicz was originally defined by means of an infinite-valued matrix. Łukasiewicz took special forms of negation and implication as basic connectives and proposed an axiom system that he conjectured would be sufficient to derive the valid formulas of the logic; this was eventually verified by M. Wajsberg. The algebraic counterparts of this logic have become know as Wajsberg algebras. In this paper we show that a Wajsberg algebra is complete and atomic (as a lattice) if and only if it is a direct product of finite Wajsberg chains. The classical characterization of complete and atomic Boolean algebras as fields of sets is a particular case of this result.

In this note we characterize free algebras in varieties of MV-algebras generated by a finite chain L n as algebras of continuous functions from the spectrum of the Boolean skeleton of the free algebra into L n . Fil: Busaniche, Manuela. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Matemática Aplicada del Litoral. Universidad Nacional del Litoral. Instituto de Matemática Aplicada del Litoral; Argentina Fil: Cignoli, Roberto Leonardo Oscar. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina

Canonical completeness results for Ł\((\mathcal{C})\), the expansion of Łukasiewicz logic Ł with a countable set of truth-constants \(\mathcal{C}\), have been recently proved in [5] for the case when the algebra of truth constants \(\mathcal{C}\) is a subalgebra of the rational interval [0, 1] ∩ ℚ. The case when \(C \not \subseteq [0, 1] \cap \mathbb{Q}\) was left as an open problem. In this paper we solve positively this open problem by showing that Ł\((\mathcal{C})\) is strongly canonical complete for finite theories for any countable subalgebra \(\mathcal{C}\) of the standard Łukasiewicz chain [0,1]Ł.

An outline of the history of the algebras corresponding to Łukasiewicz many-valued logic from the pioneering work by G. Moisil in 1940 until D. Mundici's work in 1986.

Free algebras with an arbitrary number of free generators in varieties of BL-algebras generated by one BL-chain that is an ordinal sum of a finite MV-chain Ln and a generalized BL-chain B are described in terms of weak Boolean products of BL-algebras that are ordinal sums of subalgebras of L and free algebras in the variety of basic hoops generated by B. The Boolean products are taken over the Stone spaces of the Boolean subalgebras of idempotents of free algebras in the variety of MV-algebras generated by L N. Fil: Busaniche, Manuela. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina Fil: Cignoli, Roberto Leonardo Oscar. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina

Product Logic Π is an axiomatic extension of Hájek's Basic Fuzzy Logic BL coping with the 1-tautologies when the strong conjunction & and implication → are interpreted by the product of reals in [0, 1] and its residuum respectively. In this paper we investigate expansions of Product Logic by adding into the language a countable set of truth-constants (one truth-constant for each r in a countable Π-subalgebra of [0, 1]) and by adding the corresponding book-keeping axioms for the truthconstants. We first show that the corresponding logics Π() are algebraizable, and hence complete with respect to the variety of Π()-algebras. The main result of the paper is the canonical standard completeness of these logics, that is, theorems of Π() are exactly the 1-tautologies of the algebra defined over the real unit interval where the truth-constants are interpreted as their own values. It is also shown that they do not enjoy the canonical strong standard completeness, but they enjoy it for finite theories when restricted to evaluated Π-formulas of the kind → φ, where is a truth-constant and φ a formula not containing truth-constants. Finally we consider the logics Π(Δ)(), the expansion of Π() with the well-known Baaz's projection connective Δ, and we show canonical finite strong standard completeness for them., The authors thanks the anonymous reviewers for their helpful comments and suggestions. They are also indebted to Rostislav Horcˇík for his helpful comments and for pointing out the reference [14]. P. Savicky´ acknowledges the support from the Czech Ministry of Education, Youth and Sports by the project 1M0021620808, Institute for Theoretical Computer Science - ITI. R. Cignoli, F. Esteva and L. Godo also acknowledge a bilateral Argentinian-Spanish project CONICET-CSIC, and F. Esteva and L. Godo a bilateral Czech-Spanish project CSAV-CSIC. Finally, F. Esteva, L. Godo and C. Noguera also acknowledge partial support from the Spanish project TIN2004-07933-C0301

The crucial problem of interpreting n truth values when n > 2 was investigated, among others, by Łukasiewicz himself. As shown in this chapter, a simple interpretation is given by Ulam game, the variant of the game of Twenty Questions where n - 2 lies, or errors, are allowed in the answers. The case n = 2 corresponds to the traditional game without lies. The game is originally described by Ulam on page 281 of his book [235] as follows: Someone thinks of a number between one and one million (which is just less than 220). Another person is allowed to ask up to twenty questions, to each of which the first person is supposed to answer only yes or no. Obviously the number can be guessed by asking first: Is the number in the first half million? then again reduce the reservoir of numbers in the next question by one-half, and so on. Finally the number is obtained in less than log2(1000000). Now suppose one were allowed to lie once or twice, then how many questions would one need to get the right answer?

As proved at the beginning of Chapter 2, Γ is a functor from the category A of l-groups with a distinguished strong unit, to the category MV of MV-algebras. In this chapter we shall prove that Γ is a natural equivalence (i.e., a full, faithful and dense functor) between A and MV. As a consequence, a genuine addition can be uniquely recovered from the MV-algebraic structure. Several applications will be discussed.

In this chapter we study properties that are strongly related to the lattice structure of MV-algebras. We start by considering relations between the ideals of an MV-algebra A and the ideals of the lattice L(A). A stonean ideal of a bounded distributive lattice L is an ideal generated by complemented elements of L. We shall show that the minimal prime lattice ideals of L(A), as well as the stonean ideals of L(A), are always ideals of A.

In this chapter we shall prove Chang’s completeness theorem stating that if an equation holds in the unit real interval [0,1], then the equation holds in every MV-algebra. Thus, intuitively, the two element structure {0,1} stands to boolean algebras as the interval [0,1] stands to MV-algebras. Our proof is elementary, and makes use of tools (such as “good sequences”) that shall also find applications in a subsequent chapter to show the equivalence between MV-algebras and lattice-ordered abelian groups with strong unit.

Free algebras are universal objects: every n-generated MV-algebra A is a homomorphic image of the free MV-algebra Free n over n generators; if an equation is satisfied by Free n then the equation is automatically satisfied by all MV-algebras. As a consequence of the completeness theorem, Free n is easily described as an MV-algebra of piecewise linear continuous [0,1]-valued functions defined over the cube [0, 1] n . Known as McNaughton functions, they stand to MV-algebras as {0,1}-valued functions stand to boolean algebras. Many interesting classes of MV-algebras can be described as algebras of [0, l]-valued continuous functions over some compact Hausdorff space. The various representation theorems presented in this chapter all depend on our concrete representation of free MV-algebras.

A class C of MV-algebras is said to be a variety (or, an equational class), iff there is a set e of MV-equations such that for every MV-algebra A, A ∈ C iff A satisfies all equations in e. For instance, when e = o, we obtain the variety MV of MV-algebras. When e = { x = y}, we obtain the variety of trivial MV-algebras. The main aim of this chapter is to describe all varieties of MV-algebras.

Proper n-valued Łukasiewicz algebras are obtained by adding some binary operators, fulfilling some simple equations, to the fundamental operations of n-valued Łukasiewicz algebras. They are the s-algebras corresponding to an axiomatization of Łukasiewicz n-valued propositional calculus that is an extention of the intuitionistic calculus.

The aim of this paper is to characterize the injective objects in both the category of De Morgan algebras and the subcategory of Kleene algebras.