About the equivalent theories of algebras with relations

Abstract

equivalent theories. Next, in the paper just mentioned, the problem of axiomatisation of equivalent theories is being considered. Following the example of A. Tarski [6] logical theories as well as mathematical theories are being treated in [3] as sets of theses. In paper [2] dedicated to logical systems with implications, the problem of settling connections between some logical systems and some mathematical theories, namely theories of algebras, has been formulated. The aim of the present paper is a closer characterization of some properties of equivalent theories. These properties are important with regard to the problem for? mulated in [2]. This paper is, theorefore, both a continuation of studies started in [3] and a continuation of [2] since this paper and [2] are concerned with the same pro? blem. A further paper will be dedicated to a final solution of the problem which has been formulated in [2]. In that paper, connection between different theories of al? gebras (starting with simplest algebras, such as implicative semilattices (see [1]), and ending with more complex algebras, such as generalized cylindric algebras (see [4] and [5])) and logical systems will be discussed. It should be remarked that the equivalence notion of theories of algebras given in this paper is different from the equivalence notion of mathematical theories given in [3]. Equivalence in the sense given in the present paper implies equivalence in the sense given in [3]. The narrowing of the equivalence notion allows, however, the ob? taining of a number of theorems which will be applied with the solution of the above mentioned problem formulated in [2]. It appears, however, that all the examples of mathematical equivalent theories given in [3], which theories are theories of algebras, are also equivalent in the sense formulated in the present paper. In mathematical practice, we often have to deel with "pure" algebras, but also with algebras with certain relations, as for instance with semigroups, groups, rings and fields are ordered in a certain way. It appears that a natural generalization of con? siderations on theories of algebras with relations does not bring about any difficulties. That is why this more general case is being studied in this paper. A mathematical theory T will be called the theory of algebras with relations, if among relations of this theory there exists, or is definable, a relation being congruence