On the free implicative semilattice extension of a Hilbert algebra

Hilbert algebras provide the equivalent algebraic semantics in the sense of Blok and Pigozzi to the implication fragment of intuitionistic logic. They are closely related to implicative semilattices. Porta proved that every Hilbert algebra has a free implicative semilattice extension. In this paper we introduce the notion of an optimal deductive filter of a Hilbert algebra and use it to provide a different proof of the existence of the free implicative semilattice extension of a Hilbert algebra as well as a simplified characterization of it. The optimal deductive filters turn out to be the traces in the Hilbert algebra of the prime filters of the distributive lattice free extension of the free implicative semilattice extension of the Hilbert algebra. To define the concept of optimal deductive filter we need to introduce the concept of a strong Frink ideal for Hilbert algebras which generalizes the concept of a Frink ideal for posets.Fil: Celani, Sergio Arturo. Universidad Nacional del Centro de la Provincia de Buenos Aires; Argentina. Universidad Central de Barcelona; España. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Tandil; ArgentinaFil: Jansana, Ramon. Universidad Nacional del Centro de la Provincia de Buenos Aires; Argentina. Universidad de Barcelona; Españ

Lattice of closure endomorphisms of a Hilbert algebra

A closure endomorphism of a Hilbert algebra A is a mapping that is simultaneously an endomorphism of and a closure operator on A. It is known that the set CE of all closure endomorphisms of A is a distributive lattice where the meet of two elements is defined pointwise and their join is given by their composition. This lattice is shown in the paper to be isomorphic to the lattice of certain filters of A, anti-isomorphic to the lattice of certain closure retracts of A, and compactly generated. The set of compact elements of CE coincides with the adjoint semilattice of A, conditions under which two Hilbert algebras have isomorphic adjoint semilattices (equivalently, minimal Brouwerian extensions) are discussed. Several consequences are drawn also for implication algebras.Comment: 16 pages, no figures, submitted to Algebra Universalis (under review since 24.11.2015

Variations of the free implicative semilattice extension of a Hilbert algebra

Celani and Jansana (Math Log Q 58(3):188–207, 2012) give an explicit description of the free implicative semilattice extension of a Hilbert algebra. In this paper, we give an alternative path conducing to this construction. Furthermore, following our procedure, we show that an adjunction can be obtained between the algebraic categories of Hilbert algebras with supremum and that of generalized Heyting algebras. Finally, in the last section, we describe a functor from the algebraic category of Hilbert algebras to that of generalized Heyting algebras, of possible independent interest.Fil: Castiglioni, José Luis. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Matemáticas; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: San Martín, Hernán Javier. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Matemáticas; Argentin

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

EXISTENTIALLY CLOSED BROUWERIAN SEMILATTICES

The variety of Brouwerian semilattices is amalgamable and locally finite, hence by well-known results, it has a model completion (whose models are the existen- tially closed structures). In this paper, we supply a finite and rather simple axiomatization of the model completio