Galois Connections for Partial Algebras

Abstract

In connection with partial algebras one has many more relevant polarities (i.e. Galois connections induced by binary relations) than in the case of total algebras. On one side there are many different subsets of the set of first order formulas, which one wants to use as a concept of identity in some special context, and where one is interested in the closure operators induced by restricting the validity of first order formulas to this special subset. On the other hand the polarity induced by the reflection of formulas by mappings allows us to keep track many interesting properties of homomorphisms between partial algebras, while others can be related to these via factorization systems — which can be considered as special pairs of corresponding closed classes (in Formal Concept Analysis one would call such pairs “formal concepts”) of the polarity induced by the (unique) diagonalfill-in property on the class of all homomorphisms. Moreover, having an interesting set of properties of homomorphisms, the relation “a homomorphism has a property” can be used to apply the method of attribute exploration from Formal Concept Analysis in order to elaborate a basis for all implications among these properties and on the other hand a small but “complete” set of counterexamples against all non-valid implications. In this note we want to describe some of these polarities or corresponding pairs of interest in them, and we shall present them in the context of many-sorted partial algebras, since this context seems to be less known. Moreover, we want to give an example of an attribute exploration as mentioned above.