A Survey of Algebra of Tournaments

The present paper surveys the history of algebraic represetations of complete directed graphs, known in graph theory as tournaments, or equivalently, relational structures with a trichotomous binary relation. Essentially, two kinds of algebraizations of tournaments were studied in the literature: algebras with one binary operation (called groupoids of tournaments) and algebras with two binary operations (weakly associative lattices). Different properties of these algebras have been explored by many authors. Also, varieties generated by algebraic versions of tournaments attracted a certain interest in universal algebra, primarily such questions as equations satisfied by tournaments and finite base problems, congruences, simple and subdirectly irreducible algebras, homomorphisms and, especially, automorphism groups, decidability problems, etc. A selection of results concerning these problems are also presented in the current survey. AMS Math. Subject Classification (1991): 08A99, 05C20, 08B05