Galois theory for semiclones

TL;DR

This paper develops a Galois theory linking finitary operations with relation pairs, characterizing closed sets as locally closed subuniverses of semiclones and relation pair clones, with modifications for bounded arity.

Contribution

It introduces a novel Galois correspondence between operations and relation pairs, expanding the algebraic framework for semiclones and relation pair clones.

Findings

Characterization of Galois closed sets as locally closed subuniverses

Description of modified closure operators for bounded arity

Establishment of a Galois connection between operations and relation pairs

Abstract

We present a Galois theory connecting finitary operations with pairs of finitary relations one of which is contained in the other. The Galois closed sets on both sides are characterised as locally closed subuniverses of the full iterative function algebra (semiclones) and relation pair clones, respectively. Moreover, we describe the modified closure operators if only functions and relation pairs of a certain bounded arity, respectively, are considered.