Modal meet-implication logic

We extend the meet-implication fragment of propositional intuitionistic logic with a meet-preserving modality. We give semantics based on semilattices and a duality result with a suitable notion of descriptive frame. As a consequence we obtain completeness and identify a common (modal) fragment of a large class of modal intuitionistic logics. We recognise this logic as a dialgebraic logic, and as a consequence obtain expressivity-somewhere-else. Within the dialgebraic framework, we then investigate the extension of the meet-implication fragment of propositional intuitionistic logic with a monotone modality and prove completeness and expressivity-somewhere-else for it

Martindale quotients of Jordan algebras

AbstractIn this paper we introduce Martindale quotients of Jordan algebras over arbitrary rings of scalars with respect to denominator filters of ideals. For any denominatored algebra, we show the existence of maximal Martindale quotients naturally containing all Martindale quotients of the algebra with respect to the given denominator filter

Nuclearity of semigroup C*-algebras and the connection to amenability

We study C*-algebras associated with subsemigroups of groups. For a large class of such semigroups including positive cones in quasi-lattice ordered groups and left Ore semigroups, we describe the corresponding semigroup C*-algebras as C*-algebras of inverse semigroups, groupoid C*-algebras and full corners in associated group crossed products. These descriptions allow us to characterize nuclearity of semigroup C*-algebras in terms of faithfulness of left regular representations and amenability of group actions. Moreover, we also determine when boundary quotients of semigroup C*-algebras are UCT Kirchberg algebras. This leads to a unified approach to Cuntz algebras and ring C*-algebras.Comment: 42 pages; revised version, corrected typo

Modal meet-implication logic

We extend the meet-implication fragment of propositional intuitionistic logic with a meet-preserving modality. We give semantics based on semilattices and a duality result with a suitable notion of descriptive frame. As a consequence we obtain completeness and identify a common (modal) fragment of a large class of modal intuitionistic logics. We recognise this logic as a dialgebraic logic, and as a consequence obtain expressivity-somewhere-else. Within the dialgebraic framework, we then investigate the extension of the meet-implication fragment of propositional intuitionistic logic with a monotone modality and prove completeness and expressivity-somewhere-else for it

Axiomatizing complex algebras by games

Submitted versio

Deductive systems and finite axiomatization properties

The notions of a deductive system, equational logic and Gentzen system can be generalized into the notion of a K-deductive system. A universal Horn logic is also a K-deductive system. In Part I the relationship between the existence of equivalence K-terms in a K-deductive system and some semantical properties of these systems is studied. In particular, a K-deductive system S has a finite system of equivalence formulas with parameters if the Leibniz operator on the filter lattice of every S-matrix is monotone. An equivalent semantics theorem, characterizing K-deductive systems that are equivalent to some Birkhoff-like systems, is proved and used to characterize algebraizable K-deductive systems. The connection between the implication terms and semantical properties of one-deductive systems is investigated.;In Part II a finite basis theorem for finitely generated filter-distributive proto-quasivarieties is proved. It says that if the language has only finitely many symbols, and if a K-deductive, filter-distributive, protoalgebraic system S is determined by a finite set of finite matrices, then S has a basis consisting of finitely many axioms and rules of inference. This theorem extends Pigozzi\u27s finite basis theorem for relatively congruence-distributive quasivarieties and therefore also Baker\u27s finite basis theorem for congruence-distributive varieties. If all tautologies of a finite matrix can be derived using only finitely many axioms and rules, then the matrix is called finitely axiomatizable. In particular, a finite algebra A is called finitely axiomatizable if there is a finite set of quasi-identities of A from which every identity of A can be derived. In Part III we consider the problem of finite axiomatizability of finite matrices and finite algebras. Three-element nonfinitely axiomatizable matrices are given. This solves the problem of finding a smallest and simplest possible non-finitely axiomatizable matrix that was posed by Rautenberg, independently by Wojtylak and restated by Dziobiak. Examples, that show that the underlying algebra of a finite nonfinitely axiomatizable matrix can be finitely axiomatizable, are given. The notion of the second-order finite axiomatization is proposed and two sufficient conditions for a finite algebra to be second-order finitely axiomatizable are presented

Completeness Theorems via the Double Dual Functor

The aim of this paper is to apply properties of the double dual endofunctor on the category of bounded distributive lattices and some extensions thereof to obtain completeness of certain non-classical propositional logics in a unified way. In particular, we obtain completeness theorems for Moisil calculus, n-valued Łukasiewicz calculus and Nelson calculus. Furthermore we show some conservativeness results by these methods.Facultad de Ciencias Exacta