On the join of two pseudovarieties

The aim of this lecture is to survey some recent developments in the theory of finite semigroups. More precisely, we shall consider the following problem about pseudovarieties of semigroups: given two pseudovarieties V and W, find a description of their join V ∨W (that is, of the pseudovariety they generate). This question is motivated by the theory of rational languages: it appears in a natural way when considering parallel operation of automata. The lattice of semigroup varieties (in Birkhoff’s sense) has been studied for a long time. In particular, it was proved that the join of two finitely based varieties might not be finitely based (see for instance Taylor [21]). Other important contributions in this area were given by Biryukov [11], Fennemore [12, 13] and Gerhard [14] who described the lattice of idempotent semigroup varieties, and by Polák [18] who described the lattice of varieties of completely regular semigroups. The problems appearing in the study of the lattice of pseudovarieties are analogous. Reiterman’s theorem [19] is the starting point of an equational theory for pseudovarieties: just as varieties are defined by identities, pseudovarieties are defined by pseudoidentities. Numerous algorithmic problems on pseudovarieties were proposed, for instance by Rhodes [20], Almeida [5] or Kharlampovich and Sapir [15]. Most of these problems are still open. Given a pseudovariety V, two important problems appear: