No Finite–Infinite Antichain Duality in the Homomorphism Poset of Directed Graphs

Abstract

$\mathbb D$ denotes the homomorphism poset of finite directed graphs. An antichain duality is a pair $\left\langle{\mathcal F},{\mathcal D}\right\rangle$ of antichains of $\mathbb D$ such that $({{\mathcal F}\!\!\to}) \cup ({\to\!\!{\mathcal D}}) = \mathbb D$ is a partition. A generalized duality pair in$\mathbb{D}$ is an antichain duality $\left\langle{\mathcal F},{\mathcal D}\right\rangle$ with finite ${\mathcal F}$ and ${\mathcal D}.$ We give a simplified proof of the Foniok–Nešetřil–Tardif theorem for the special case $\mathbb D$, which gave full description of the generalized duality pairs in $\mathbb{D}$. Although there are many antichain dualities $\left\langle{\mathcal F},{\mathcal D}\right\rangle$ with infinite ${\mathcal D}$ and ${\mathcal F}$, we can show that there is no antichain duality $\left\langle{\mathcal F},{\mathcal D}\right\rangle$ with finite ${\mathcal F}$ and infinite ${\mathcal D}$.