On some types of Galois connections arising in the theory of partial algebras
by
Peter Burmeister
Darmstadt University of Technology

On some types of Galois connections arising in the theory of partial algebras Galois connections arise quite often in Universal Algebra, but in connection with partial algebras they become even more important. We want to stress here three main areas of their influence in this theory.

(i) The wealth of concepts of identities for partial algebras - which mostly can be considered as specially formed ECE-equations - has been a big source of hard research in connection with trying to describe intrinsically the rsulting closed sets/classes on the syntactic as well as on the semantic side of the corresponding Galois connections based on the relation of validity of (special kinds of) first order formulas in partial algebras of some given type.
(ii) It has turned out that most of the important properties of homomorphisms between partial algebras can be described either by the reflection of some elementary implication, usually of some existence equation, or as the partner of such a class in a factorization structure, and these factorization structures are corresponding classes (in Formal Concept Analysis one would say ``concepts'') of a Galois connection defined by the so-called (unique) diagonal-fill-in property.
(iii) Thus, having so many interesting ``basic'' properties of homomorphisms around - and possibly some additional ones not describable in the sense of (ii) - it is interesting to investigate their interplay. Here a good tool is the so-called attribute exploration of Formal Concept Analysis, which relatively easily allows the classification of their combinations. And it is well known that from the point of view of Pure Mathematics Formal Concept Analysis is more or less just the theory of Galois connections.

Date received: March 12, 2001