Decomposition techniques for finite semigroups, using categories. II
The paper is devoted to a systematic study of the decomposition of morphisms \(\theta\) : \(S\to T\) of finite semigroups of the form \(\theta =\phi \pi\) where \(\phi\) : S\(<V_ n\wr (V_{n-1}\wr...(V_ 1\wr T)...)\) is a division, each \(V_ i\) is monoid, \(\wr\) represents either a semidirect product or a 2-sided semidirect product, and \(\pi\) : \(V_ n\wr (V_{n-1}..(V_ 1\wr T)...)\to T\) is the canonical projection. The basic ingredients in this study are: (1) the classification of maximal proper surmorphisms of finite semigroups described in part I of the paper [J. Pure Appl. Algebra 62, 269-284 (1989; see the preceding review Zbl 0684.20054)]; (2) the semidirect product decomposition-division theorems involving the derived category obtained by \textit{B. Tilson} [J. Pure Appl. Algebra 48, 83-198 (1987; Zbl 0627.20031)] and (3) its counterpart for the 2-sided semidirect product in terms of the kernel category by \textit{J. Rhodes} and \textit{B. Tilson} [J. Pure Appl. Algebra 62, 227-268 (1989)]; (4) the Rhodes and related expansions. Several examples of applications are included, such as decompositions associated with aperiodic, LG and LI-morphisms. The authors also give a couple of results which allow them to distinguish the decomposing powers of the 2-sided semidirect product and the combination of semidirect and reverse semidirect products.
scientific article; zbMATH DE number 4120426
scientific article