The Algebra of Adjacency Patterns: Rees Matrix Semigroups with Reversion

  • Marcel Jackson
  • Mikhail Volkov
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6300)


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.


Rees matrix semigroup unary semigroup identity unary semigroup variety graph universal Horn sentence universal Horn class variety membership problem finite basis problem 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Marcel Jackson
    • 1
  • Mikhail Volkov
    • 2
  1. 1.La Trobe UniversityVictoriaAustralia
  2. 2.Ural State UniversityEkaterinburgRussia

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