The algebra of adjacency patterns: Rees matrix semigroups with reversion

TL;DR

This paper explores the deep connection between classes of directed graphs and algebraic structures called adjacency semigroups, revealing structural similarities and complexity results.

Contribution

It establishes a correspondence between universal Horn classes of directed graphs and varieties generated by adjacency semigroups, including new examples and complexity insights.

Findings

Lattice of subvarieties matches universal Horn classes

Identifies a limit variety of regular unary semigroups

Finite unary semigroups have NP-hard membership problems

Abstract

We establish a surprisingly close relationship between universal Horn classes of directed graphs and varieties generated by so-called adjacency semigroups which are Rees matrix semigroups over the trivial group with the unary operation of reversion. In particular, the lattice of subvarieties of the variety generated by adjacency semigroups that are regular unary semigroups is essentially the same as the lattice of universal Horn classes of reflexive directed graphs. A number of examples follow, including a limit variety of regular unary semigroups and finite unary semigroups with NP-hard variety membership problems.