Semantic normal form

The idea of semantic normal form originally developed by Jankov [17] for Brouwerian semilattices is made applicable to the variety of equivalential algebras and thereby, to a broader family of locally finite and permutable varieties obeying the conditions of Fregeanity i.e. point regularity and congruence orderability. It is proved that every term in the language of such a variety can be equivalently expressed with the help of a relatively small set of building blocks manufactured from so-called monolith assignments

Closure properties for the class of behavioral models

Hidden k-logics can be considered as the underlying logics of program specification. They constitute natural generalizations of k-deductive systems and encompass deductive systems as well as hidden equational logics and inequational logics. In our abstract algebraic approach, the data structures are sorted algebras endowed with a designated subset of their visible parts, called filter, which represents a set of truth values. We present a hierarchy of classes of hidden k-logics. The hidden k-logics in each class are characterized by three different kinds of conditions, namely, properties of their Leibniz operators, closure properties of the class of their behavioral models, and properties of their equivalence systems. Using equivalence systems, we obtain a new and more complete analysis of the axiomatization of the behavioral models. This is achieved by means of the Leibniz operator and its combinatorial properties. © 2007 Elsevier Ltd. All rights reserved.FCT via UIM

Logics and operators

Two connectives are of special interest in metalogical investigations — the connective of implication which is important due to its connections to the notion of inference, and the connective of equivalence. The latter connective expresses, in the material sense, the fact that two sentences have the same logical value while in the strict sense it expresses the fact that two sentences are interderivable on the basis of a given logic. The process of identification of equivalent sentences relative to theories of a logic C defines a class of abstract algebras. The members of the class are called Lindenbaum-Tarski algebras of the logic C. One may abstract from the origin of these algebras and examine them by means of purely algebraic methods

Equivalential Algebras with Conjunction on Dense Elements

We study the variety generated by the three-element equivalential algebra with conjunction on the dense elements. We prove the representation theorem which let us construct the free algebras in this variety

Admissible equivalence systems

Whenever a logic is the set of theorems of some deductive system, where the latter has an equivalence system, the behavioral theorems of the logic can be determined by means of that equivalence system. In general, this original equivalence system may be too restrictive, because it su ces to check behavioral theorems by means of any admissible equivalence system (that is an equivalence system of the small- est deductive system associated with the given logic). In this paper, we present a range of examples, which show that: 1) there is an admissible equivalence system which is not an equivalence system for the initial deductive system, 2) there is a non- nitely equivalential deductive system with a nite admissible equivalence system, and 3) there is a deductive system with an admissible equivalence sys- tems, such that this deductive system is not even protoalgebraic itself. We use methods and results from algebraic and modal logic.FCT via UIMAFCT via KLog projec

Some remarks on axiomatizing logical consequence operations

In this paper we investigate the relation between the axiomatization of a given logical consequence operation and axiom systems defining the class of algebras related to that consequence operation. We show examples which prove that, in general there are no natural relation between both ways of axiomatization