# Calculation Rules and Cancellation Rules for Strong Hom-Schemes

## TL;DR

This paper investigates the structural conditions of finite posets that influence the number of homomorphisms between them, focusing on how these relations behave under order operations and cancellation rules.

## Contribution

It analyzes the compatibility of hom-set inequalities with order arithmetic operations and establishes cancellation rules for strong hom-schemes.

## Key findings

- Identifies conditions under which hom-set inequalities are preserved under order operations.
- Establishes cancellation rules for strong hom-schemes.
- Provides insights into the structure of finite posets affecting homomorphism counts.

## Abstract

Let ${\cal H}(A,B)$ denote the set of homomorphisms from the poset $A$ to the poset $B$. In previous studies, the author has started to analyze what it is in the structure of finite posets $R$ and $S$ that results in $# {\cal H}(P,R) \leq # {\cal H}(P,S)$ for every finite poset $P$, if additional regularity conditions are imposed. In the present paper, it is examined if this relation (with or without regularity conditions) is compatible with the operations of order arithmetic and if cancellation rules hold.