Galois connection for sets of operations closed under permutation, cylindrification and composition

TL;DR

This paper characterizes sets of operations on a set A that are closed under permutation, cylindrification, and composition using a Galois connection, revealing their uncountable nature for sets with at least two elements.

Contribution

It introduces a Galois connection framework to describe and analyze these operation sets and their duals, providing necessary and sufficient closure conditions.

Findings

Closed sets are characterized via Galois connection.

Closure systems are uncountable for sets with ≥2 elements.

Provides a dual description of these closed sets.

Abstract

We consider sets of operations on a set A that are closed under permutation of variables, addition of dummy variables and composition. We describe these closed sets in terms of a Galois connection between operations and systems of pointed multisets, and we also describe the closed sets of the dual objects by means of necessary and sufficient closure conditions. Moreover, we show that the corresponding closure systems are uncountable for every A with at least two elements.