About Generalized One-to-One Mappings between Sets of Order Homomorphisms

Structural properties of finite posets $R$ and $S$ are studied which enforce H(P,R) \leq \# H(P,S)$ for every finite poset $P$, where $H(P,Q)$ is the set of order homomorphisms from to $Q$. The concept of the strong Hom-scheme is introduced. In the case of existence, a strong Hom-scheme from $R$ to $S$ defines a one-to-one mapping $\rho_P : H(P,R) \rightarrow H(P,S)$ for every poset $P \in P_r$, where $P_r$ is a representation system of the non-isomorphic finite posets. By postulating regularity conditions for the way, how a strong Hom-scheme maps the elements of $H(P,R)$ to the elements of $H(P,S)$, the strong I-scheme from $R$ to $S$ is defined. The existence of a strong I-scheme from $R$ to $S$ turns out to be equivalent to the existence of a one-to-one homomorphism $\epsilon$ between the so-called EV-systems of $R$ and $S$, where $\epsilon$ has to fulfill an additional condition. Methods are developed which allow - in many cases - for given finite posets $R$ and $S$, the proof of the relation $\# H(P,R) \leq \# H(P,S)$ for every finite poset $P$ or the refutation of this relation.