Multisorted Boolean Clones Determined by Binary Relations up to Minion Homomorphisms

TL;DR

This paper characterizes a class of multisorted Boolean clones determined by binary relations using minion homomorphisms, introduces minion cores for canonical representation, and explores their structural properties.

Contribution

It provides a new classification of multisorted Boolean clones via minion homomorphisms and introduces minion cores as canonical representatives for clone equivalence classes.

Findings

Classifies clones determined by binary relations with small projections

Introduces minion cores for canonical clone representation

Analyzes the ordering of clones via minion homomorphisms

Abstract

We describe the ordering of a class of clones by minion homomorphisms, also known as minor preserving maps or height 1 clone homomorphisms. The class consists of all clones on finite sets determined by binary relations whose projections to both coordinates have at most two elements. This class can be alternatively described up to minion homomorphisms as the class of multisorted Boolean clones determined by binary relations. We also introduce and apply the concept of a minion core which provides canonical representatives for equivalence classes of clones, more generally minions, on finite sets.