Isotonian Algebras

TL;DR

This paper studies isotonian algebras, a class of toric rings associated with pairs of finite posets, determining their Krull dimension and conditions for normality and quadratic Gr"obner bases.

Contribution

It introduces isotonian algebras as a generalization of Hibi rings and analyzes their algebraic properties such as dimension, normality, and Gr"obner bases.

Findings

Krull dimension of isotonian algebras determined

Normality established for certain poset classes

Quadratic Gr"obner basis exists in specific cases

Abstract

To a pair $P$ and $Q$ of finite posets we attach the toric ring $K[P,Q]$ whose generators are in bijection to the isotone maps from $P$ to $Q$. This class of algebras, called isotonian, are natural generalizations of the so-called Hibi rings. We determine the Krull dimension of these algebras and for particular classes of posets $P$ and $Q$ we show that $K[P,Q]$ is normal and that their defining ideal admits a quadratic Gr\"obner basis.