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*.

File translated from T

On 4 Apr 2002, 15:40.