Which self-maps appear as lattice endomorphisms?

TL;DR

This paper characterizes exactly when a self-map on a set can be viewed as a lattice endomorphism by establishing a necessary and sufficient condition for such a lattice structure to exist.

Contribution

It provides a complete criterion for when a self-map can be realized as a lattice endomorphism, bridging the gap between arbitrary maps and lattice theory.

Findings

Provides necessary and sufficient condition for a self-map to be a lattice endomorphism.

Characterizes the lattice structures compatible with given self-maps.

Advances understanding of the relationship between self-maps and lattice structures.

Abstract

Let f be a self-map of the set A. We give a necessary and sufficient condition for the existence of a lattice structure on A such that f becomes a lattice endomorphism with respect to this structure.