Semidirectly closed pseudovarieties of locally trivial semigroups. (English) Zbl 0711.20032

Almeida, Jorge

Semigroup Forum 40, No. 3, 315-323 (1990).

Semidirect products of pseudovarieties of semigroups have previously been investigated by the author [J. Pure Appl. Algebra 60, No.2, 113-128 (1989; Zbl 0687.20053)]. In the present paper he considers the question of which pseudovarieties of semigroups are semidirectly closed. In the process, he determines all semidirectly closed varieties of semigroups and all semidirectly closed generalized varieties of nilpotent extensions of rectangular bands. Using the work of C. J. Ash [J. Algebra 92, 104-115 (1985; Zbl 0548.08007)] on the relationship between pseudovarieties and generalized varieties, he shows that the generalized variety generated by a semidirectly closed pseudovariety is also semidirectly closed. Finally he describes all semidirectly closed pseudovarieties of locally trivial semigroups, and shows that they form a complete distributive lattice.

Reviewer: G.Clarke

MSC:

20M07	Varieties and pseudovarieties of semigroups
08B15	Lattices of varieties
08C15	Quasivarieties
08B26	Subdirect products and subdirect irreducibility

Keywords:

Semidirect products; pseudovarieties of semigroups; semidirectly closed varieties of semigroups; generalized varieties; nilpotent extensions of rectangular bands; locally trivial semigroups; complete distributive lattice

Citations:

Zbl 0687.20053; Zbl 0548.08007

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