On logical systems with implications and theories of algebras

Abstract

In paper [8] we concerned ourselves with logical systems with implications and paper [9] was devoted to theories of algebras with relations. The object of the present paper is an investigation of connections existing between logical systems and theories of algebras. Consequently, the present paper may be considered as part 3 of a more exhaustive work, its first two parts being papers [8] and [9]. These three papers are devoted to the same general problem, that is the problem of settling the connections existing between logical systems and mathematical theories. In this work, we first define the relation of "corresponding35 (which occurs between theories of algebras and logical systems) and we study the connections occurring between this relation and the equivalence relation of logical systems or the equivalence relation of theories of algebras. Then we concern ourselves with the problem what properties of logical systems become transposed on the theories of algebras which correspond to these lo? gical systems and vice versa. It follows from Theorem 5 of [9] that classes of models of equivalent theories of algebras only consist of models such that for every model Sffix belonging to one class there exists a model 2Ji2 belonging to the other class such that there exist definable extensions 9Kf and 2Jt* of models 9ft x and 9Jl2, respectively, which are identical. Classes of models of equivalent theories, although may be virtually different, are however similar to the extent of possessing a common class consisting of definable extensions of models of both the first and the second class. The class of all models of a theory T of algebras will be denoted by M (T). Let sft(T) be the algebra of terms of a theory T. Algebra of terms plays a special role in the class of models of theory of algebras. In [9] we proved that two non-degenerated1 theories of algebras such that for their algebras of terms there exist definable exten? sions which are isomorphic, are equivalent in the sense given in [9]. If algebra sft(T) is free in M (T), then the following set a (T) = {[t]t; t is an individual variable}2 is a set of free generators for sft(T). For every logical system with implication there exists its algebra of Lindenbaum. A similar role that is played by algebras of terms for the theories of algebras, is played