About finite posets R and S with \# H(P,R) <= \# H(P,S) for every finite poset P

Frank a Campo

Published 2020-08-04Version 1

Finite posets $R$ and $S$ are studied with $\# {\cal H}(P,R) \leq \# {\cal H}(P,S)$ for every finite poset $P$, where ${\cal H}(P,Q)$ is the set of order homomorphisms from $P$ to $Q$. It is shown that under an additional regularity condition, $\# {\cal H}(P,R) \leq \# {\cal H}(P,S)$ for every finite poset $P$ is equivalent to $\# {\cal S}(P,R) \leq \# {\cal S}(P,S)$ for every finite poset $P$, where ${\cal S}(P,Q)$ is the set of strict order homomorphisms from $P$ to $Q$. A method is developed for the rearrangement of a finite poset $R$, resulting in a poset $S$ with $\# {\cal H}(P,R) \leq \# {\cal H}(P,S)$ for every finite poset $P$. The results are used in constructing pairs of posets $R$ and $S$ with this property.