Toric partial orders

TL;DR

This paper introduces toric partial orders, a new combinatorial structure related to regions of graphic toric hyperplane arrangements, extending many features of ordinary partial orders.

Contribution

It defines toric partial orders and develops their combinatorial properties, providing a toric analogue to classical partial order concepts.

Findings

Toric posets correspond to regions of graphic toric hyperplane arrangements.

Established toric analogues for chains, antichains, transitivity, and linear extensions.

Connected toric posets to finite posets via minimal-to-maximal element conversions.

Abstract

We define toric partial orders, corresponding to regions of graphic toric hyperplane arrangements, just as ordinary partial orders correspond to regions of graphic hyperplane arrangements. Combinatorially, toric posets correspond to finite posets under the equivalence relation generated by converting minimal elements into maximal elements, or sources into sinks. We derive toric analogues for several features of ordinary partial orders, such as chains, antichains, transitivity, Hasse diagrams, linear extensions, and total orders.