Varieties of semigroups and varieties of completely regular semigroups closed for certain extensions
Let \(\mathcal V\) be any variety of universal algebras and let \(C_{\mathcal V}\) be the category whose members are the members of \(\mathcal V\) and whose morphisms are the surjective homomorphisms. For a given functor \(F : C_{\mathcal V} \to C_{\mathcal V}\), a subvariety \(\mathcal W\) of \(\mathcal V\) is said to be closed for \(F\) if for every \(A \in {\mathcal W}\) also \(F(A) \in {\mathcal W}\). Let \(L_ F\) be the partially ordered set of all subvarieties of \(\mathcal V\) which are closed for \(F\). The authors are interested in situations when \(L_ F\) is a complete sublattice of \(L({\mathcal V})\), the lattice of all subvarieties of \(\mathcal V\). They find the following -- very transparent -- sufficient condition for this case: for every \(A \in {\mathcal V}\) and for every family (\(\phi_ i, i\in I\)) of morphisms in \(C_{\mathcal V}\) with domain \(A\), \(\bigcap_{i \in I} \phi_ i \phi_ i^{-1} = \iota_ A\) implies \(\bigcap_{i \in I} (F \phi_ i)(F \phi_ i)^{- 1} = \iota_{F(A)}\). An appropriate analogue to this theorem holds for pseudovarieties as well. Though in the above situation \(F\) is an arbitrary functor, one should think of \(F(A)\) being obtained from \(A\) by a certain canonical construction. Two examples of such constructions are elaborated; however, the general theory will have many more applications. The first example is the functor \(B : C_{\mathcal S} \to C_{\mathcal S}\) where \(\mathcal S\) is the variety of all semigroups, and for any semigroup \(S\), \(B(S)\) denotes the well known Bruck extension of \(S\). This leads to a functor of the above kind since any surjective homomorphism \(\phi : S \to T\) may be canonically extended to a surjective homomorphism \(B(\phi) : B(S) \to B(T)\). The above mentioned theorem applies and so the set \(L_ B\) of all semigroup varieties closed for the Bruck extension forms a complete sublattice of the lattice of all semigroup varieties. Several properties of the lattice \(L_ B\) are investigated. First of all, the least element of \(L_ B\) is determined. This least variety closed for the Bruck extension turns out to be generated by the semigroup \(B^ \omega (1)\) which is obtained by -- roughly speaking -- application of the Bruck extension to the trivial semigroup 1 infinitely many times. The same process leads more generally for any variety \({\mathcal V} = \text{var }S\) generated by a semigroup \(S\) to the least variety \(B({\mathcal V})\) which is closed for the Bruck extension and contains \(\mathcal V\). It is further shown that the lattice \(L_ B\) contains an antichain of the cardinality of the continuum, and that any variety \(\mathcal V\) which is closed for \(B\) but does not coincide with \(\mathcal S\) is contained in a variety \(\mathcal W\) which is not closed for \(B\). In the second example the authors are concerned with the variety \({\mathcal C}{\mathcal R}\) of all completely regular semigroups. They use an old construction, due to the first named author [\textit{F. Pastijn}, Semigroup Forum 14, 247-263 (1977; Zbl 0363.20052)], to construct a functor \(I : C_{{\mathcal C} {\mathcal R}} \to C_{{\mathcal C} {\mathcal R}}\) which assigns to each completely regular semigroup \(S\) an idempotent generated extension \(I(S)\) of \(S\). This functor again satisfies the above mentioned condition so that \(L_ I\) turns out to be a complete sublattice of \(L({\mathcal C} {\mathcal R})\). Several interesting connections between the lattice \(L_ I\) and two complete congruences \(C\) and \(L\) on \(L({\mathcal C} {\mathcal R})\) are found. (Here \(C\) and \(L\) are defined by \({\mathcal V} C {\mathcal W}\) if \(\mathcal V\) and \(\mathcal W\) contain the same idempotent generated members, and \({\mathcal V} L {\mathcal W}\) iff \(\mathcal V\) and \(\mathcal W\) contain the same monoids).
scientific article; zbMATH DE number 554148