The global of a pseudovariety of monoids is the pseudovariety of categories it generates when its elements are viewed as the monoids of edges in one-vertex categories. The global of a pseudovariety of monoids is useful in the computation of various operators involving the pseudovariety. The simplest case occurs when the global of the pseudovariety is characterized by properties of the local submonoids of its members, in which case the pseudovariety is said to be local. In this paper we consider a pseudovariety which intervenes in the study of hierarchies of concatenation of rational languages, namely the bilateral semidirect product of the pseudovarieties Sl of finite semilattices and J of finite J-trivial finite semigroups. We prove it is local. As a biproduct of our arguments, we also compute the global of the join of the pseudovarieties R and L respectively of finite R and L-trivial semigroups. The global of this join turns out to be defined by a single pseudoidentity on a two-vertex category which cannot be expressed locally.
Sponsored in part by the FCT approved projects POCTI 32817/99 and POCTI/MAT/37670/2001 in participation with the European Community Fund FEDER and by FCT through Centro de Matemática da Universidade do Porto. Also sponsored in part by FCT, the Faculdade de Ciências da Universidade do Porto, Programa Operacional Ciência, Tecnologia, Inovação do Quadro Comunitário de Apoio III, and by Caixa Geral de Depósitos.