Semigroup expansions using the derived category, kernel, and Malcev products
For each finite set \(A\), let \({\mathcal M}_A\) denote the category of \(A\)-generated monoids, together with \(A\)-preserving monoid morphisms. An ``expansion'' is a functor from \({\mathcal M}_A\) into itself, together with a natural transformation from the functor to the identity functor such that its arrows are surjective morphisms. The purpose of this paper is to provide several constructions of expansions, based on wreath and Malcev products, both for monoids and for semigroups. For simplicity and brevity, we focus on the wreath, monoidal expansion in this review. Let \(\mathcal V\) be a semigroup variety. Given an \(A\)-generated monoid \(S\) and the associated morphism \(\phi\) from the free monoid \(A^*\) on \(A\) to \(S\), we may associate the derived category \(D_\phi\) (for which the reader is referred to \textit{B. Tilson} [J. Pure Appl. Algebra 48, 83-198 (1987; Zbl 0627.20031)]), which in this situation is isomorphic to the free category on the Cayley graph of \(S\). Let \(D\) be the greatest quotient of \(D_\phi\) whose local monoids belong to \(\mathcal V\) and let \(D_{cd}\) be its consolidation -- the semigroup obtained in the natural way from the arrows of the category by setting undefined products, if any, equal to an adjoined zero. Now, apparently motivated by the ``Derived Category Theorem'' of Tilson [op. cit.], there is an \(A\)-generated subsemigronp \(S_{\mathcal V}\) of the wreath product of \(D_{cd}\) with \(S\) which, together with the projection \(\epsilon_S\) upon \(S\), serves as the basis for the expansion. It has the property that it is the ``largest'' monoid with the property that there is a morphism upon \(S\) in whose derived category the local monoids all belong to \(\mathcal V\). In addition, it preserves various finiteness properties. A related semigroup expansion is defined. It turns out that in the case where \(\mathcal V\) is the variety of one-element semigroups, this construction yields the well-known Karnofsky-Rhodes expansion, an extension of the classical Rhodes expansion. Various other cases are considered, along with the two-sided analogue and a variant based on the Malcev product.
scientific article; zbMATH DE number 1308323