Homomorphisms, Congruences, and Galois Connections

Abstract

It is clear that abstract algebraic properties of an algebra do not change if the elements and the operations of the algebra get other names. One can describe this fact mathematically using the concept isomorphism or isomorphic mapping. A generalization of this concept is homomorphism (or homomorphic mapping), with which one can describe similarities of algebras. Congruences and factor algebras are aids to determine homomorphisms.

A Galois connection is a pair of mappings (σ, τ) between two power sets ℬ(A) and ℬ(B), where the mappings τ and σ are antitone and extensive (see Section 4.4). With the aid of these mappings, one can define hull operators on A and on B. Then, these operators define closed set systems ($\subseteq$ ℬ(A) or $\subseteq$ ℬ(B), respectively). Part of the usefulness of such Galois connections resides in the possibility of drawing conclusions about a closed set system on the basis of information about the other system.

The Galois connection between function algebras and relation algebras, which we study in Chapter 2 of Part II, is an important aid in the solution of the completeness problem of the many-valued logic (see Part II, Chapter 5 and 6).