On the homomorphism order of labeled posets

TL;DR

This paper studies the structure of labeled posets with homomorphisms, showing their universality, lattice properties, and computational complexity of related decision problems, with applications to graph representations.

Contribution

It provides a new proof of the universality of the homomorphism order of k-posets and explores their lattice and complexity properties.

Findings

Homomorphism order of k-posets is universal and forms a distributive lattice.

Representation of directed graphs by k-posets is established.

Complexity results for decision problems in the homomorphism order are derived.

Abstract

Partially ordered sets labeled with k labels (k-posets) and their homomorphisms are examined. We give a representation of directed graphs by k-posets; this provides a new proof of the universality of the homomorphism order of k-posets. This universal order is a distributive lattice. We investigate some other properties, namely the infinite distributivity, the computation of infinite suprema and infima, and the complexity of certain decision problems involving the homomorphism order of k-posets. Sublattices are also examined.