Alexander duality for monomial ideals associated with isotone maps between posets

TL;DR

This paper characterizes when the Alexander dual of a monomial ideal associated with isotone maps between finite posets coincides with a similar ideal with the roles of the posets reversed.

Contribution

It provides a complete classification of pairs of finite posets for which the Alexander dual of $L(P,Q)$ equals $L(Q,P)$ up to index switching.

Findings

Identifies all poset pairs with the duality property

Establishes conditions for the ideal equality

Enhances understanding of monomial ideals from poset maps

Abstract

For a pair $(P,Q)$ of finite posets the generators of the ideal $L(P,Q)$ correspond bijectively to the isotone maps from $P$ to $Q$. In this note we determine all pairs $(P,Q)$ for which the Alexander dual of $L(P,Q)$ coincides with $L(Q,P)$, up to a switch of the indices.