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We give sufficient conditions for a BL-algebra to be isomorphic to a poset product of BL-chains which are poset product indecomposable. 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: Gomez, Conrado Javier. 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

Motivated by Kalman residuated lattices, Nelson residuated lattices and Nelson paraconsistent residuated lattices, we provide a natural common generalization of them. Nelson conucleus algebras unify these examples and further extend them to the non-commutative setting. We study their structure, establish a representation theorem for them in terms of twist structures and conuclei that results in a categorical adjunction, and explore situations where the representation is actually an isomorphism. In the latter case, the adjunction is elevated to a categorical equivalence. By applying this representation to the original motivating special cases we bring to the surface their underlying similarities.

Different constructions of BL-chains are compared. We establish when the ordinal sum and the poset product of the same family of BL-chains coincide. We also compare the poset product of MV-chains and product chains with saturated BL-chains. 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: Gomez, Conrado Javier. 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

A representation theorem for a class of BL-algebras with finite spectrum is presented. Although the class comprised by our result is not the whole class of BL-algebras with finite spectrum, it applies to some important classes such as finite BL-algebras and BL-chains with finite spectrum among others. Our representation constitutes a generalization of the ordinal sum construction, since we decompose each algebra in terms of totally ordered Wajsberg hoops. Fil: Aguzzoli, Stefano. Università degli Studi di Milano; Italia 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: Castiglioni, José Luis. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Matemáticas; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina Fil: Lubomirsky, Noemí. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Matemáticas; Argentina

Consider any subvariety of BL-algebras generated by a single BL-chain which is the ordinal sum of the standard MV-algebra on [0,1] and a basic hoop H. We present a geometrical characterization of elements in the finitely generated free algebra of each of these subvarieties. In this characterization there is a clear insight of the role of the regular and dense elements of the generating chain. As an application, we analyze maximal and prime filters in the free algebra. 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: Castiglioni, José Luis. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Matemáticas; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata; Argentina Fil: Lubomirsky, Noemí. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Matemáticas; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata; Argentina

We introduce the notion of generalized rotation of a residuated lattice and characterize the varieties of bounded residuated lattices they generate, which we name MVR n . These algebras have a retraction onto a hyperarchimedean MV-algebra. Then we characterize algebras in MVR n as triples made of an MV-algebra, a residuated lattice with a nucleus, and an operator joining them. 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: Marcos, Miguel Andrés. Universidad Nacional del Litoral; Argentina Fil: Ugolini, Sara. Università degli Studi di Pisa; Italia

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

Fuzzy possibilistic logic is an important formalism for approximate reasoning. It extends the well-known basic propositional logic BL, introduced by Hájek, by offering the ability to reason about possibility and necessity of fuzzy propositions. We consider an algebraic approach to study this logic, introducing Pseudomonadic BL-algebras. These algebras turn to be a generalization of both Pseudomonadic algebras introduced by Bezhanishvili (Math Log Q 48:624?636, 2002) and serial, Euclidean and transitive Bimodal Gödel algebras proposed by Caicedo and Rodriguez (J Log Comput 25:37?55, 2015). We present the connection between this class of algebras and possibilistic BL-frames, as a first step to solve an open problem proposed by Hájek (Metamathematics of fuzzy logic. Trends in logic, Kluwer, Dordrecht, 1998, Chap. 8, Sect. 3). 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: Cordero González, Penélope. 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: Rodriguez, Ricardo Oscar. 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

In their recent seminal paper published in the Annals of Pure and Applied Logic, Dubuc and Poveda call an MV-algebra A strongly semisimple if all principal quotients of A are semisimple. All boolean algebras are strongly semisimple, and so are all finitely presented MV-algebras. We show that for any 1-generator MV-algebra semisimplicity is equivalent to strong semisimplicity. Further, a semisimple 2-generator MV-algebra A is strongly semisimple if and only if its maximal spectral space m(A) does not have any rational Bouligand-Severi tangents at its rational points. In general, when A is finitely generated and m(A) has a Bouligand-Severi tangent then A is not strongly semisimple.

A polyhedron in R^n is a finite union of simplexes in R^n. An MV-algebra is polyhedral if it is isomorphic to the MV-algebra of all continuous I-valued piecewise linear functions with integer coefficients, defined on some polyhedron P in R^n. We characterize polyhedral MV-algebras as finitely generated subalgebras of semisimple tensor products of a simple MV-algebra and a finitely presented MV-algebra. We establish a duality between the category of polyhedral MV-algebras and the category of polyhedra with Z-maps. We prove that polyhedral MV-algebras are preserved under various kinds of operations, and have the amalgamation property. Strengthening the Hay-Wojcicki theorem, we prove that every polyhedral MV-algebra is strongly semisimple, in the sense of Dubuc-Poveda. 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: Cabrer, Leonardo. Universita Degli Studi Di Firenze; Italia Fil: Mundici, Daniele. Universita Degli Studi Di Firenze; Italia

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

We establish the Robinson joint consistency theorem for the infinite-valued propositional logic of Łukasiewicz. As a corollary we easily obtain the amalgamation property for MV-algebras-the algebras of Łukasiewicz logic: all pre-existing proofs of this latter result make essential use of the Pierce amalgamation theorem for abelian lattice-ordered groups (with strong unit) together with the categorical equivalence Γ between these groups and MV-algebras. Our main tools are elementary and geometric. 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: Mundici, Daniele. Università degli Studi di Firenze; Italia

In the present paper we give a description of the free algebra over an arbitrary set of generators in the variety of nilpotent minimum algebras. Such description is given in terms of a weak Boolean product of directly indecomposable algebras over the Boolean space corresponding to the Boolean subalgebra of the free NM-algebra. 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

A simple and self contained proof of decomposition of BL-chains into ordinal sums of Wajsberg hoops is given.

In this paper we describe finitely generated free algebras in varieties of BL-algebras generated by one BL-chain which is an ordinal sum of a finite MV-chain and a generalized BL-chain. We also give some particular examples of these free algebras.

A unital ℓ-group (G; u) is an abelian group G equipped with a translationinvariant lattice-order and a distinguished element u, called order-unit, whose positive integer multiples eventually dominate each element of G. It is shown that, for direct systems S and T of finitely presented unital ℓ-groups, confluence is a necessary condition for lim S ≅ lim T . (Sufficiency is an easy byproduct of a general result). When (G; u) is finitely generated we equip it with a sequence W (G;u) = (W 0;W 1; : : ) of weighted abstract simplicial complexes, where W t+1 is obtained from W t either by the classical Alexander binary stellar operation, or by deleting a maximal simplex of W t. We show that the map (G; u) → W (G;u) has an inverse. A confluence criterion is given to recognize when two sequences arise from isomorphic unital ℓ-groups. 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: Cabrer, Leonardo Manuel. Universidad Nacional del Centro de la Provincia de Buenos Aires; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina Fil: Mundici, Daniele. Università degli Studi di Firenze; Italia

In the present paper we extend previous results by completely characterizing dual canonical subvarieties of BL-algebras. These are subvarieties of algebras that satisfy the equation x^k=x^{k+1} for some integer k>0. As a corollary we get a full description of subvarieties of BL-algebras that admit completions. 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: Cabrer, Leonardo Manuel. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional del Centro de la Provincia de Buenos Aires; Argentina

We prove that every subvariety of BL-algebras which is not finitely generated is not σ-canonical. We also prove π-canonicity for an infinite family of subvarieties of BL-algebras that are not finitely generated. To do so we study the behavior of canonical extensions of ordered sums of posets. 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: Cabrer, Leonardo Manuel. Universidad Nacional del Centro de la Provincia de Buenos Aires; Argentina

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

We establish a categorical duality for the finitely generated Lindenbaum-Tarski algebras of prepositional nilpotent minimum logic. The latter's conjunction is semantically interpreted by a left-continuous (but not continuous) triangular norm; implication is obtained through residuation. Our duality allows one to transfer to nilpotent minimum, logic several known results about inutitionistic logic with the prelinearity axiom (also called Gödel-Dummett logic), mutatis mutandis. We give several such applications. Fil: Aguzzoli, Stefano. Università degli Studi di Milano; Italia 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: Marra, Vincenzo. Università degli Studi di Milano; Italia

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