Strong G-schemes and strict homomorphisms

TL;DR

This paper introduces a new G-scheme relation for finite posets based on order homomorphisms, establishing equivalences with strict homomorphism counts and providing methods for poset construction.

Contribution

It defines the G-scheme relation for finite posets, proves its equivalence with strict homomorphism counts, and offers a finite inspection condition and construction methods.

Findings

G-scheme relation is equivalent to strict homomorphism count inequalities

Equality of strict homomorphism counts characterizes poset isomorphism

Provides a finite method to verify the G-scheme relation and construct related posets

Abstract

Let $\mathfrak{P}_r$ 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 $R \sqsubseteq_G S$ 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 $R \sqsubseteq_G S$ is equivalent to $\# {\cal S}(P,R) \leq \# {\cal S}(P,S)$ 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 $R \sqsubseteq_G S$ 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 $P + Q \sqsubseteq_G A + T$, where $A$ is a convex subposet of $P$.