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 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.

[1]  K. Auinger.  The congruence lattice of a strict regular semigroup , 1992 .

[2]  David M. Clark,et al.  Standard topological algebras: syntactic and principal congruences and profiniteness , 2005 .

[3]  X. Caicedo.  Finitely axiomatizable quasivarieties of graphs , 1995 .

[4]  M. Maróti,et al.  The variety generated by tournaments , 1999 .

[5]  Stanley Burris,et al.  A course in universal algebra , 1981, Graduate texts in mathematics.

[6]  Marcel Jackson,et al.  Flat algebras and the translation of universal Horn logic to equational logic , 2008, Journal of Symbolic Logic.

[7]  V. Gorbunov,et al.  Algebraic theory of quasivarieties , 1998 .

[8]  V. Gorbunov,et al.  Antivarieties and colour-families of graphs , 2001, algebra universalis.

[9]  Edmond W. H. Lee.  Combinatorial Rees-sushkevich Varieties are Finitely Based , 2008, Int. J. Algebra Comput..

[10]  David M. Clark,et al.  The axiomatizability of topological prevarieties , 2008 .

[11]  Jaroslav Nesetril,et al.  On classes of relations and graphs determined by subobjects and factorobjects , 1978, Discret. Math..

[12]  Ross Willard.  On Mckenzie's method , 1996 .

[13]  Ralph Freese,et al.  Universal Algebra and Lattice Theory , 1983 .

[14]  Ralph McKenzie,et al.  Interpreting Graph Colorability in Finite Semigroups , 2006, Int. J. Algebra Comput..

[15]  R. McKenzie,et al.  The variety generated by equivalence algebras , 2001 .

[16]  S. Margolis,et al.  Algorithmic problems for finite groups and finite 0-simple semigroups , 1997 .

[17]  S. V. Sizyi.  Quasivarieties of graphs , 1994 .

[18]  Jorge Almeida,et al.  Finite Semigroups and Universal Algebra , 1995 .

[19]  ON THE STRUCTURE OF THE LATTICE OF C OMBINATORIAL REES-SUSHKEVICH VARIETIES , 2007 .

[20]  A. V. Kravchenko,et al.  $$\mathcal{Q}$$ -Universal Quasivarieties of Graphs , 2002 .

[21]  J. Howie.  Fundamentals of semigroup theory , 1995 .

[22]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[23]  Belinda Trotta,et al.  Residual properties of reflexive, anti-symmetric digraphs , 2011 .

[24]  George F. McNulty,et al.  Inherently nonfinitely based finite algebras , 1983 .