2019/08/19 by Frank a Campo, Campo, Frank a
Mathematics · #06A06 #06A07 #Advanced Topology and Set Theory #Algebraic Geometry and Number Theory #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #math.CO #msc:06A06 #msc:06A07
paper · pdf · doi:10.48550/arxiv.1908.06897
23 pages, 9 figures
arxiv created 2019/08/19 · openalex publication_date 2019/08/19 · arxiv updated 2019/08/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let \mathfrakPr be a representation system of the non-isomorphic finite posets, and let \cal H(P,Q) be the set of order homomorphisms from P to Q. For finite posets R and S, we write R \sqsubseteqG S iff, for every P ∈ \mathfrakPr, a one-to-one mapping ρP : \cal H(P,R) → \cal H(P,S) exists which fulfills a certain regularity condition. It is shown that R \sqsubseteqG S is equivalent to # \cal S(P,R) ≤ # \cal S(P,S) for every finite posets P, where \cal S(P,Q) is the set of strict order homomorphisms from P to Q. In consequence, # \cal S(P,R) = # \cal S(P,S) holds for every finite posets P iff R and S are isomorphic. A sufficient condition is derived for R \sqsubseteqG S which needs the inspection of a finite number of posets only. Additionally, a method is developed which facilitates for posets P + Q (direct sum) the construction of posets T with P + Q \sqsubseteqG A + T, where A is a convex subposet of P.