Semidirectly closed pseudovarieties of locally trivial semigroups
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 \textit{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.
semidirectly closed varieties of semigroups
nilpotent extensions of rectangular bands
scientific article; zbMATH DE number 4168951