Cylindric Algebras for Partial Relational Systems. Quasicylindric Algebras

Abstract

A fundamental inspiration of this paper is the problem of the construction of the class of algebras which is an algebraic counterpart for partial relational systems. This class, called the class of quasicylindric algebras, arises as an effect of algebraizing the partial elementary logic. The algebraization follows from a general method applicable (among other ways) to the investigation of connections between Boolean algebras and classical propositional calculus or between cylindric algebras and classical elementary logic. For constructing the class of quasicylindric algebras we apply elementary metalogical notions, such as the notion of first-order language or first-order theory. The new class of algebras is discussed with respect to algebraic properties. In particular, an algebraic characterization of quasicylindric algebras and some connections between these algebras and cylindric algebras are briefly described. Moreover, it is shown how the new class may be used in an algebraic proof of Craig’s interpolation property and of Beth’s definability property for partial elementary logic.Keywordsfirst-order languagealgebra of formulasfirst-order theoryrelational systemcylindric algebra