Joan Gispert

Adding truth-constants to logics of continuous t-norms: Axiomatization and completeness results

In this paper we study generic expansions of logics of continuous t-norms with truth-constants, taking advantage of previous results for Lukasiewicz logic and more recent results for Godel and Product logics. Indeed, we consider algebraic semantics for expansions of logics of continuous t-norms with a set of truth-constants {r@?|r@?C}, for a suitable countable C@?[0,1], and provide a full description of completeness results when (i) the t-norm is a finite ordinal sum of Lukasiewicz, Godel and Product components, (ii) the set of truth-constants covers all the unit interval in the sense that each component of the t-norm contains at least one value of C different from the bounds of the component, and (iii) the truth-constants in Lukasiewicz components behave as rational numbers.

On Some Varieties of MTL-algebras

The study of perfect, local and bipartite IMTL-algebras presented in [29] is generalized in this paper to the general non-involutive case, i. e. to MTL-algebras. To this end we describe the radical of MTL-algebras and characterize perfect MTL-algebras as those for which the quotient by the radical is isomorphic to the two-element Boolean algebra, and a special class of bipartite MTL-algebras, BP0, as those for which the quotient by the radical is a Boolean algebra. We prove that BP0 is the variety generated by all perfect MTL-algebras and give some equational bases for it. We also introduce a new way to build MTL-algebras by adding a negation fixpoint to a perfect algebra and also by adding some set of points whose negation is the fixpoint. Finally, we consider the varieties generated by those algebras, giving equational bases for them, and we study which of them define a fuzzy logic with standard completeness theorem.

Perfect and bipartite IMTL-algebras and disconnected rotations of prelinear semihoops

IMTL logic was introduced in [12] as a generalization of the infinitely-valued logic of Lukasiewicz, and in [11] it was proved to be the logic of left-continuous t-norms with an involutive negation and their residua. The structure of such t-norms is still not known. Nevertheless, Jenei introduced in [20] a new way to obtain rotation-invariant semigroups and, in particular, IMTL-algebras and left-continuous t-norm with an involutive negation, by means of the disconnected rotation method. In order to give an algebraic interpretation to this construction, we generalize the concepts of perfect, bipartite and local algebra used in the classification of MV-algebras to the wider variety of IMTL-algebras and we prove that perfect algebras are exactly those algebras obtained from a prelinear semihoop by Jeneis disconnected rotation. We also prove that the variety generated by all perfect IMTL-algebras is the variety of the IMTL-algebras that are bipartite by every maximal filter and we give equational axiomatizations for it.

On triangular norm based axiomatic extensions of the weak nilpotent minimum logic

In this paper we carry out an algebraic investigation of the weak nilpotent minimum logic (WNM) and its t-norm based axiomatic extensions. We consider the algebraic counterpart of WNM, the variety of WNM-algebras ([MATHEMATICAL DOUBLE-STRUCK CAPITAL W]ℕ[MATHEMATICAL DOUBLE-STRUCK CAPITAL M]) and prove that it is locally finite, so all its subvarieties are generated by finite chains. We give criteria to compare varieties generated by finite families of WNM-chains, in particular varieties generated by standard WNM-chains, or equivalently t-norm based axiomatic extensions of WNM, and we study their standard completeness properties. We also characterize the generic WNM-chains, i. e. those that generate the variety [MATHEMATICAL DOUBLE-STRUCK CAPITAL W]ℕ[MATHEMATICAL DOUBLE-STRUCK CAPITAL M], and we give finite axiomatizations for some t-norm based extensions of WNM.

Bounded BCK‐algebras and their generated variety

In this paper we prove that the equational class generated by bounded BCK-algebras is the variety generated by the class of finite simple bounded BCK-algebras. To obtain these results we prove that every simple algebra in the equational class generated by bounded BCK-algebras is also a relatively simple bounded BCK-algebra. Moreover, we show that every simple bounded BCK-algebra can be embedded into a simple integral commutative bounded residuated lattice. We extend our main results to some richer subreducts of the class of integral commutative bounded residuated lattices and to the involutive case. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Universal Classes of MV-chains with Applications to Many-valued Logics

In this paper we characterize, classify and axiomatize all universal classes of MV-chains. Moreover, we accomplish analogous characterization, classification and axiom- atization for congruence distributive quasivarieties of MV-algebras. Finally, we apply those results to study some finitary extensions of theinfinite valued propositional cal- culus. Mathematics Subject Classification: 03G35, 03B50, 06D35, 08C15, 08C99.

Boolean representation of bounded BCK-algebras

We define the Boolean center and the Boolean skeleton of a bounded BCK-algebra, and we use the Boolean skeleton to obtain a representation of bounded BCK-algebras, called (weak) Pierce

Axiomatic Extensions of IMT3 Logic

b\mathbb{BCK}