On Heyting algebras and dual BCK-algebras

Pacific Journal of Mathematics, 1985

Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 1987

Journal of Universal Computer Science, 2007

Since all the algebras connected to logic have, more or less explicitely, an associated order relation, it follows that they have two presentations, dual to each other. We classify these dual presentations in "left" and "right" ones and we consider that, when dealing with several algebras in the same research, it is useful to present them unitarily, either as "left" algebras or as "right" algebras. In some circumstances, this choice is essential, for instance if we want to build the ordinal sum (product) between a BL algebra and an MV algebra. We have chosen the "left" presentation and several algebras of logic have been redefined as particular cases of BCK algebras.

Zenodo (CERN European Organization for Nuclear Research), 2008

Since all the algebras connected to logic have, more or less explicitely, an associated order relation, it follows that they have two presentations, dual to each other. We classify these dual presentations in "left" and "right" ones and we consider that, when dealing with several algebras in the same research, it is useful to present them unitarily, either as "left" algebras or as "right" algebras. In some circumstances, this choice is essential, for instance if we want to build the ordinal sum (product) between a BL algebra and an MV algebra. We have chosen the "left" presentation and several algebras of logic have been redefined as particular cases of BCK algebras. We introduce several new properties of algebras of logic, besides those usually existing in the literature, which generate a more refined classification, depending on the properties satisfied. In this work (Parts I-V) we make an exhaustive study of these algebras-with two bounds and with one bound-and we present classes of finite examples, in bounded case. In this Part I, divided in two because of its length, after surveying chronologically several algebras related to logic, as residuated lattices, Hilbert algebras, MV algebras, divisible residuated lattices, BCK algebras, Wajsberg algebras, BL algebras, MTL algebras, WNM algebras, IMTL algebras, NM algebras, we propose a methodology in two steps for the simultaneous work with them (the first part of Part I). We then apply the methodology, redefining those algebras as particular cases of reversed left-BCK algebras. We analyse among others the properties Weak Nilpotent Minimum and Double Negation of a bounded BCK(P) lattice, we introduce new corresponding algebras and we establish hierarchies (the subsequent part of Part I).

For every consequence (or closure) operator Cn on a set S, the family C of all Cn-closed sets, partially ordered by set inclusion, forms a complete lattice, called the lattice of logics. If the lattice C is distributive, then, it forms a Heyting algebra, since it has the zero element Cn(0) and is complete. Logics determined by this Heyting algebra is studied in the second part. In Part I it is shown (for Cn finitary or compact) that the lattice C is distributive iff its dual space is topological. Moreover a representation theorem for lattices of logics is given.

Studia Logica, Volume 56, 263-276, 1996

Mathematica Slovaca, 2013

We introduce the notion of gi-algebra as a generalization of dual BCK-algebra, and define the notions of strong, commutative and transitive gi-algebra, and then we show that an interval ↑l = {a ∈ P | l ≤ a} in a strong and commutative gi-algebra P is a lattice. Also, we define a congruence relation ∼D on a transitive gi-algebra P and show that the quotient set P/∼D is a gi-algebra and a dual BCK-algebra.

Mathematical Logic Quarterly, 2006

The present paper introduces and studies the variety ℋn of n-linear weakly Heyting algebras. It corresponds to the algebraic semantic of the strict implication fragment of the normal modal logic K with a generalization of the axiom that defines the linear intuitionistic logic or Dummett logic. Special attention is given to the variety ℋ2 that generalizes the linear Heyting algebras studied in [10] and [12], and the linear Basic algebras introduced in [2]. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

St. Petersburg Mathematical Journal, 2008

Let Θ be an arbitrary variety of algebras and H an algebra in Θ. Along with algebraic geometry in Θ over the distinguished algebra H, a logical geometry in Θ over H is considered. This insight leads to a system of notions and stimulates a number of new problems. Some logical invariants of algebras H ∈ Θ are introduced and logical relations between different H 1 and H 2 in Θ are analyzed. The paper contains a brief review of ideas of logical geometry (§1), the necessary material from algebraic logic (§2), and a deeper introduction to the subject (§3). Also, a list of problems is given. 0.1. Introduction. The paper consists of three sections. A reader wishing to get a feeling of the subject and to understand the logic of the main ideas can confine himself to §1. A more advanced look at the topic of the paper is presented in § §2 and 3. In §1 we give a list of the main notions, formulate some results, and specify problems. Not all the notions used in §1 are well formalized and commonly known. In particular, we operate with algebraic logic, referring to §2 for precise definitions. However, §1 is self-contained from the viewpoint of ideas of universal algebraic geometry and logical geometry. Old and new notions from algebraic logic are collected in §2. Here we define the Halmos categories and multisorted Halmos algebras related to a variety Θ of algebras. §3 is a continuation of §1. Here we give necessary proofs and discuss problems. The main problem we are interested in is what are the algebras with the same geometrical logic. The theory described in the paper has deep ties with model theory, and some problems are of a model-theoretic nature. We emphasize once again that §1 gives a complete insight on the subject, while §2 and §3 describe and decode the material of §1. §1. Preliminaries. General view 1.1. Main idea. We fix an arbitrary variety Θ of algebras. Throughout the paper we consider algebras H in Θ. To each algebra H ∈ Θ one can attach an algebraic geometry (AG) in Θ over H and a logical geometry (LG) in Θ over H. In algebraic geometry we consider algebraic sets over H, while in logical geometry we consider logical (elementary) sets over H. These latter sets are related to the elementary logic, i.e., to the first order logic (FOL). Consideration of these sets gives grounds to geometries in an arbitrary variety of algebras. We distinguish algebraic and logical geometries in Θ. However, there is very 2000 Mathematics Subject Classification. Primary 03G25.

Logica Universalis, 2013

The goal of this paper is to generalize a notion of characteristic (or Jankov) formula by using finite partial Heyting algebras instead of the finite subdirectly irreducible algebras: with every finite partial Heyting algebra we associate a characteristic formula, and we study the properties of these formulas. We prove that any intermediate logic can be axiomatized by such formulas. We further discuss the correlations between characteristic formulas of finite partial algebras and canonical formulas. Then with every well-connected Heyting algebra we associate a set of characteristic formulas that correspond to each finite relative subalgebra of this algebra. Finally, we demonstrate that in many respects these sets enjoy the same properties as regular characteristic formulas. In the last section we outline an approach how to generalize these obtained results to the broad classes of algebras.