On the structure of algebras and homomorphisms

Abstract

Introduction. The subject matter of this paper belongs to the general theory of sets. The objectives here are the examinations of the structure of classes of sets which are closed under various finite and transfinite set operations and those transformations which preserve these operations. The major results of this paper are partial extensions of (a) the well known theory of a-algebras (e.g. [4]), (b) the work of E. Marczewski [1] on isomorphisms, (c) the work of R. Sikorski [2] on o-homomorphisms and (d) the work of A. Tarski [3] on fields of sets and set functions. Three classes of sets-the algebra, m-algebra and total algebra-as well as four transformations-the m-homomorphism, weak isomorphism, m-isomorphism, and total isomorphism-are studied. A class 3C of sets is an algebra if it is closed under finite set operations, i.e. addition of two sets and complementation. Analogically 3C is an m-algebra if X is closed under m-operations, i.e. complementations and addition of not more than m sets, where m is an arbitrary fixed cardinal number. X is a total algebra if it is closed under all operations, i.e. complementation and arbitrary addition. Two classes of sets, 3C and ?, are weakly isomorphic if 3C and ? considered as partially ordered by the relation of proper inclusion are similar. 3C and ? are m-isomorphic if they have the same properties from the point of view of m-operations; and, finally, X and ? are totally isomorphic if they have the same properties from the point of view of all operations on sets. The most important structure theorems concern the m-operations for mr_n&. The formulation of the corresponding finite and total structure theorems follow easily from the m-theorems and will be omitted except in special cases. The work is divided into two parts: (1) algebras of sets and (2) homomorphisms. In (1) the existence, composition and construction are treated; and, further, the relation between algebras and the natural set units is developed. In (2) are discussed some properties of m-additive, complementative transformations. Homomorphisms