Idempotent-generated semigroups and pseudovarieties

Several works have been dedicated to idempotent-generated semigroups, from the well-known result from Howie (1966), using full transformations semigroups, that any finite semigroup embeds into a finite regular idempotent-generated semigroup, to the work of Pastijn (1977) and the presentation of Petrich (2004), using Rees matrix semigroups, that proves the analogous result for completely simple semigroups. In a joint work with Jorge Almeida, in the attempt to characterize the pseudovarieties which are generated by their idempotent-generated semigroups, we use profinite techniques to prove that many familiar pseudovarieties have this property, namely, the pseudovarieties J, R, L and DA. In this talk, I will revisit some historical results and I will present our contribution for the referred characterization.