The variety generated by tournaments
Jaroslav Ježek, Petar Marković, Miklós Maróti, Ralph McKenzie · Repository of the Academy's Library (Library of the Hungarian Academy of Sciences) · 1999
By a tournament we mean a directed graph (F, ->) such that whenever x, y are two distinct elements of T, then precisely one of the two cases, either x -> y or y -> x, takes place.There is a one-to-one correspondence between tournaments and commutative groupoids satisfying ab e {a, b] for all a and b: set ab = a if and only if a -> b.This makes it possible to identify tournaments with their corresponding groupoids and employ algebraic methods for their investigation.So, an equivalent definition is: A tournament is a commutative groupoid, every subset of which is a subgroupoid.For two elements a and ft of a tournament, we set a -> b if and only if ab = a.The aim of this paper is to investigate the the variety of groupoids generated by tournaments.This variety will be denoted by T. We have started the investigation in our previous paper [9], in which it is proved that the variety is not finitely based.Here we will find a four-element base for the three-variable equations of T, and proceed to investigate subdirectly irreducible algebras in T. Our main effort will be focused on an attempt to find a positive solution to a conjecture, which has several equivalent formulations: Conjecture 1. (1) Every subdirectly irreducible algebra in T is a tournament.(2) Every finite, subdirectly irreducible algebra in T is a tournament.(3) If A is a subalgebra of a direct product of finitely many finite tournaments, then every subdirectly irreducible homomorphic image of A is a tournament.(4) T is the same as the quasi-variety T q generated by tournaments.