Semigroups embeddable in hyperplane face monoids

TL;DR

This paper characterizes which finite semigroups can be embedded in hyperplane face monoids, showing the problem is polynomial time decidable and exploring algebraic properties of the associated quasivariety.

Contribution

It provides a characterization and decision procedure for embeddability of finite semigroups in hyperplane face monoids, and analyzes the algebraic complexity of the related quasivariety.

Findings

Embedding problem is polynomial time decidable.

The quasivariety is minimally non-finitely based.

Results extend to complex hyperplane semigroups.

Abstract

The left regular band structure on a hyperplane arrangement and its representation theory provide an important connection between semigroup theory and algebraic combinatorics. A finite semigroup embeds in a real hyperplane face monoid if and only if it is in the quasivariety generated by the monoid obtained by adjoining an identity to the two-element left zero semigroup. We prove that this quasivariety is on the one hand polynomial time decidable, and on the other minimally non-finitely based. A similar result is obtained for the semigroups embeddable in complex hyperplane semigroups.