Galois connection for multiple-output operations

TL;DR

This paper generalizes the classical Galois connection between polymorphisms and invariants to multi-output functions, using invariants in partially ordered monoids, with applications to reversible computing.

Contribution

It introduces a duality framework for classes of multi-valued, partial functions using monoid-valued invariants, extending universal algebra concepts.

Findings

Established a Galois connection for multi-output functions.

Unified the treatment of permutations and other multi-valued functions.

Applicable to reversible computing problems.

Abstract

It is a classical result from universal algebra that the notions of polymorphisms and invariants provide a Galois connection between suitably closed classes (clones) of finitary operations $f\colon B^n\to B$, and classes (coclones) of relations $r\subseteq B^k$. We will present a generalization of this duality to classes of (multi-valued, partial) functions $f\colon B^n\to B^m$, employing invariants valued in partially ordered monoids instead of relations. In particular, our set-up encompasses the case of permutations $f\colon B^n\to B^n$, motivated by problems in reversible computing.