Galois Connections for Generalized Functions and Relational Constraints

TL;DR

This paper explores Galois connections between multivalued functions from A^n to the power set of B and relational constraints, providing a framework that generalizes previous work on total and partial functions.

Contribution

It introduces a comprehensive Galois theory for multivalued functions and constraints, extending existing frameworks to more general function types.

Findings

Characterization of Galois closed sets of multivalued functions

Decomposition of Galois operators for these functions

Necessary and sufficient conditions for closure properties

Abstract

In this paper we focus on functions of the form $A^n\rightarrow \mathcal{P}(B)$, for possibly different arbitrary non-empty sets $A$ and $B$, and where $\mathcal{P}(B)$ denotes the set of all subsets of $B$. These mappings are called \emph{multivalued functions}, and they generalize total and partial functions. We study Galois connections between these generalized functions and ordered pairs $(R,S)$ of relations on $A$ and $B$, respectively, called \emph{constraints}. We describe the Galois closed sets, and decompose the associated Galois operators, by means of necessary and sufficient conditions which specialize, in the total single-valued case, to those given in the author's previous work [M. Couceiro, S. Foldes. On closed sets of relational constraints and classes of functions closed under variable substitutions, Algebra Universalis 54 (2005) 149-165].