Splitting finite antichains in the homomorphism order

TL;DR

This paper investigates the structure of finite maximal antichains in the homomorphism order of relational structures, identifying conditions for splitting, and exploring properties like looseness, extension, and cut-points within this order.

Contribution

It provides a structural condition for when finite maximal antichains split and analyzes the presence of non-splitting antichains and other properties in the homomorphism order.

Findings

Non-splitting antichains occur only at the bottom of the order.

A structural condition for splitting of finite maximal antichains is established.

Examines properties like looseness, extension, and cut-points in the homomorphism order.

Abstract

A structural condition is given for finite maximal antichains in the homomorphism order of relational structures to have the splitting property. It turns out that non-splitting antichains appear only at the bottom of the order. Moreover, we examine looseness and finite antichain extension property for some subclasses of the homomorphism poset. Finally, we take a look at cut-points in this order.