2016/09/08 by Ying-Ying Feng, Feng, Ying-Ying, Asawer Al-Aadhami +7
Biochemistry, Genetics and Molecular Biology · Computer Science · Mathematics · #Chemical Synthesis and Analysis #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Rings and Algebras (math.RA) #semigroups and automata theory
paper · doi:10.48550/arxiv.1609.02441
openalex publication_date 2016/09/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For a monoid M and a subsemigroup S of the full transformation semigroup Tn, the wreath product M\wr S is defined to be the semidirect product Mn\rtimes S, with the coordinatewise action of S on Mn. The full wreath product M\wr Tn is isomorphic to the endomorphism monoid of the free M-act on n generators. Here, we are particularly interested in the case that S=Singn is the singular part of Tn, consisting of all non-invertible transformations. Our main results are presentations for M\wr Singn in terms of certain natural generating sets, and we prove these via general results on semidirect products and wreath products. We re-prove a classical result of Bulman-Fleming that M\wr Singn is idempotent generated if and only if the set M/L of L-classes of M forms a chain under the usual ordering of L-classes, and we give a presentation for M\wr Singn in terms of idempotent generators for such a monoid M. Among other results, we also give estimates for the minimal size of a generating set for M\wr Singn, as well as exact values in some cases (including the case that M is finite and M/L is a chain, in which case we also calculate the minimal size of an idempotent generating set). As an application of our results, we obtain a presentation (with idempotent generators) for the idempotent generated subsemigroup of the endomorphism monoid of a uniform partition of a finite set.