On the homomorphism order of labeled posets
Leonard Kwuida, Erkko Lehtonen

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.
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