Strong G-schemes and strict homomorphisms

Let ${\mathfrak{P}}_r$ be a representation system of the non-isomorphic finite posets, and let $\mathcal{H}(P,Q)$ be the set of order homomorphisms from $P$ to $Q$. For finite posets $R$ and $S$, we write $R{⊑}_{G}S$ iff, for every $P\in {\mathfrak{P}}_r$, a one-to-one mapping $\rho_P : {\cal H}(P,R) \rightarrow {\cal H}(P,S)$ exists which fulfills a certain regularity condition. It is shown that $R{⊑}_{G}S$ is equivalent to $# {\cal S}(P,R) \leq # {\cal S}(P,S)$foreveryfiniteposets$P$,where${\cal S}(P,Q)$isthesetofstrictorderhomomorphismsfrom$P$to$Q$. In consequence, $\mathrm{\#}\mathcal{S}(P,R)=\mathrm{\#}\mathcal{S}(P,S)$ holds for every finite posets $P$ iff $R$ and $S$ are isomorphic. A sufficient condition is derived for $R{⊑}_{G}S$ 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 $P+Q{⊑}_{G}A+T$, where $A$ is a convex subposet of $P$.