The Algebra of Adjacency Patterns: Rees Matrix Semigroups with Reversion

We establish a surprisingly close relationship between universal Horn classes of directed graphs and varieties generated by socalled 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. Introduction and overview The aim of this paper is to establish and to explore a new link between graph theory and algebra. Since graphs form a universal language of discrete mathematics, the idea to relate graphs and algebras appears to be natural, and several useful links of this kind can be found in the literature. We mean, for instance, the graph algebras of McNulty and Shallon [20], the closely related flat graph algebras [25], and “almost trivial” algebras investigated in [15, 16] amongst other places. While each of the approaches just mentioned has proved to be useful and has yielded interesting applications, none of them seem to share two important features of the present contribution. The two features can be called naturalness and surjectivity. Speaking about naturalness, we want to stress that the algebraic objects (adjacency semigroups) that we use here to interpret graphs have not been invented for this specific purpose. Indeed, adjacency semigroups belong to a well established class of unary semigroups1 that have been considered by many authors. We shall demonstrate how graph theory both sheds a new light on some previously known algebraic results and provides their extensions and generalizations. By surjectivity we mean that, on the level 2000 Mathematics Subject Classification. 20M07, 08C15, 05C15, 20M17.