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A syntactical proof of locality of DA. (English) Zbl 0858.20052

With every pseudovariety \({\mathbf V}\) of (finite) monoids two pseudovarieties of (finite) categories are naturally associated: (i) \({\mathbf g}{\mathbf V}\), generated by the monoids in \({\mathbf V}\), and (ii) \({\mathbf l}{\mathbf V}\), consisting of the finite categories whose “loop” monoids belong to \({\mathbf V}\); \({\mathbf V}\) is “local” if these category pseudovarieties coincide. These concepts were introduced by B. Tilson [J. Pure Appl. Algebra 48, 83-198 (1987; Zbl 0627.20031)], motivated directly by a desire to study membership in semidirect products of pseudovarieties of monoids (and of semigroups), and indirectly by the established correspondence between varieties of languages and pseudovarieties of finite monoids (and between certain operations on languages and semidirect products of monoids). Various examples from that paper and others show that local pseudovarieties have particularly pleasant behaviour in relation to semidirect products.
Syntactic methods have previously been employed to prove that certain varieties of monoids are local (in the analogous sense). In the current paper, the author’s theory of profinite monoids (see [Finite Semigroups and Universal Algebra (World Scientific, Singapore, 1994; Zbl 0844.20039)], where the term “monoids of implicit operations” was used instead) is the basis for a syntactic proof of locality of certain pseudovarieties of monoids. Denote by \({\mathbf D}{\mathbf O}\) the class of finite monoids such that every regular \({\mathcal D}\)-class is an orthodox semigroup (and so a rectangular group). It is shown that for any pseudovariety \({\mathbf H}\) of groups, the pseudovariety consisting of those members of \({\mathbf D}{\mathbf O}\) all of whose subgroups belong to \({\mathbf H}\) is local. In particular, the pseudovariety \({\mathbf D}{\mathbf A}\) is local. This class consists of the finite monoids all of whose regular \({\mathcal D}\)-classes are rectangular bands, and is associated with an important class of languages. Its locality was claimed in an earlier paper, by another author, but that claim was much disputed.

MSC:

20M07 Varieties and pseudovarieties of semigroups
20M35 Semigroups in automata theory, linguistics, etc.
18B40 Groupoids, semigroupoids, semigroups, groups (viewed as categories)
20M05 Free semigroups, generators and relations, word problems
08C15 Quasivarieties
20L05 Groupoids (i.e. small categories in which all morphisms are isomorphisms)
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