Letterplace and co-letterplace ideals of posets

TL;DR

This paper introduces letterplace and co-letterplace ideals associated with posets, unifying various classes of monomial ideals and connecting them to well-known algebraic structures.

Contribution

It defines new classes of ideals linked to posets and demonstrates how they encompass many existing monomial ideals through regular sequence reductions.

Findings

Unifies multiple monomial ideal classes under a common framework.

Shows how to derive known ideals via regular sequences from the new ideals.

Provides a systematic approach to study algebraic properties of these ideals.

Abstract

To a natural number $n$, a finite partially ordered set $P$ and a poset ideal ${\mathcal J}$ in the poset $Hom(P,[n])$ of isotonian maps from $P$ to the chain on $n$ elements, we associate two monomial ideals, the letterplace ideal $L(n,P;{\mathcal J})$ and the co-letterplace ideal $L(P,n;{\mathcal J})$. These ideals give a unified understanding of a number of ideals studied in monomial ideal theory in recent years. By cutting down these ideals by regular sequences of variable differences we obtain: multichain ideals and generalized Hibi type ideals, initial ideals of determinantal ideals, strongly stable ideals, $d$-partite $d$-uniform ideals, Ferrers ideals, edge ideals of cointerval $d$-hypergraphs, and uniform face ideals.