The join of the pseudovarieties of idempotent semigroups and locally trivial semigroups

Abstract

This article solves a problem proposed by Almeida: the computation of the join of two well-known pseudovarieties of semigroups, namely the pseudovariety of bands and the pseudovariety of locally trivial semigroups. We use a method developed by Almeida, based on the theory of implicit operations.

Some pseudovariety joins involvinglocally trivial semigroups

“…The results obtained generalize a number of results already known, namely the computation involving the pseudovarieties: Com [2]; J of J -trivial semigroups [13]; B of idempotent semigroups [27]; Sl of semilattices [6,27]; subpseudovarieties of CR (for joins with N) [21].…”

“…Thus, this last corollary permits us to deduce, for instance, the following equalities which were first proved by Zeitoun [27],…”

“…Another remarkable example is the calculation of R ∨ L, the join of the pseudovarieties of R-trivial and L-trivial semigroups, by Almeida and Azevedo [7]. More recently, several other calculations and answers to decision problems were performed by Almeida, Azevedo, Trotter, Volkov, Weil, Zeitoun and the author [9,10,11,13,14,15,17,24,25,26,27]. This article solves, in particular, the problem of the calculation of the join LI ∨ V where V is any subpseudovariety of CR h m N. This result is extremely natural: we show that if V is defined by a set Σ of pseudoidentities and if a and b are symbols not appearing in Σ, then LI ∨ V is the subpseudovariety of CR h m N defined by the pseudoidentities of the form a ω xb ω = a ω yb ω where x = y is an element of Σ.…”