Monotone and cone preserving mappings on posets

TL;DR

This paper explores various types of order-preserving mappings on posets, characterizing when these mappings coincide and analyzing their properties in different poset structures such as chains, semilattices, and products.

Contribution

It introduces new characterizations of mappings on posets and identifies conditions under which these mappings are equivalent or coincide, extending to complex poset constructions.

Findings

Monotone mappings coincide with semilattice homomorphisms only on chains.

Characterization of equivalence relations induced by strongly monotone mappings.

Quotients of posets by these relations are themselves posets.

Abstract

We define several sorts of mappings on a poset like monotone, strictly monotone, upper cone preserving and variants of these. Our aim is to characterize posets in which some of these mappings coincide. We define special mappings determined by two elements and investigate when these are strictly monotone or upper cone preserving. If the considered poset is a semilattice then its monotone mappings coincide with semilattice homomorphisms if and only if the poset is a chain. Similarly, we study posets which need not be semilattices but whose upper cones have a minimal element. We extend this investigation to posets that are direct products of chains or an ordinal sum of an antichain and a finite chain. We characterize equivalence relations induced by strongly monotone mappings and show that the quotient set of a poset by such an equivalence relation is a poset again.