The variety generated by tournaments. (English) Zbl 0939.08003

The authors continue their investigations on the variety \({\mathbf T}\) generated by tournaments as binary idempotent commutative algebras (for \(a\to b\) write \(ab= a\)). In the first paper [Discrete Math. 211, No. 1-3, 243-248 (2000)], they proved that the equations of tournaments are not finitely based. In this paper they investigate the following question: Is every subdirectly irreducible algebra in this variety a tournament? They ask a more general question, hoping that it might be easier to answer: Let \({\mathbf T}^3\) denote the variety defined by all three-variable equations satisfied by tournaments. They conjecture that every subdirectly irreducible algebra of this variety is either a tournament or isomorphic to some given algebras (\({\mathbf J}_3\) or \({\mathbf M}_n\) for natural \(n>2\), these latter ones play an important rôle in the first paper, as well). At the end of the paper, they prove that \({\mathbf T}\) is inherently non-finitely-generated.
Unfortunately, right at the beginning of the paper there are some inaccuracies: The elements of \({\mathbf J}_3\) should be \(\{a,u_0, u_1,u_2,u_3\}\). At the same place one should add that it is meant to be an idempotent commutative groupoid, where \(x\to y\) means \(xy= x\), and it satisfies the identities listed in Theorem 4. Incidentally, in this theorem instead of “five” it should read “four”.
The paper is an essential contribution to the algebraic theory of tournaments.
Reviewer: E.Fried (Budapest)


08B20 Free algebras
05C20 Directed graphs (digraphs), tournaments
08B26 Subdirect products and subdirect irreducibility