Finite generation and presentability of wreath products of monoids.
type: Item
instance of: scholarly article
title: Finite generation and presentability of wreath products of monoids.
zbMATH Open document ID: 1040.20046
DOI: 10.1016/S0021-8693(03)00311-9
author: M. R. Thomson
author: E. F. Robertson
author: N. Ruškuc
published in: Journal of Algebra
publication date: 2003-09-07T00:00:00Z
review text: Let \(A\) and \(B\) be monoids and \(A^B\) the set of all functions \(f\colon B\to A\). Defining \(x(fg)=(xf)(xg)\) and \(x{^bf}=(xb)f\) for any \(x,b\in B\), \(f,g\in A^B\), the wreath product \(A\,\text{Wr}\,B\) is the set \(A^B\times B\) together with the operation: \((f,b)(g,c)=(f{^bg},bc)\). If \(A^{(B)}\) denotes the set of all \(f\in A^B\) with finite support (i.e., \(\{x\in B\mid xf\neq 1_A\}\) is finite) then the submonoid of \(A\,Wr\,B\) generated by \(A^{(B)}\times B\) is called the restricted wreath product of \(A\) by \(B\). The following two important theorems are proved: (1) Let \(A\) and \(B\) be monoids and let \(G\) be the group of units of \(B\); then the restricted wreath product of \(A\) by \(B\) is finitely generated if and only if both \(A\) and \(B\) are finitely generated, and either \(A\) is trivial, or \(B=VG\) for some finite subset \(V\) of \(B\). (2) The restricted wreath product of monoids \(A\) and \(B\) is finitely presented if and only if either \(A\) is finitely presented and \(B\) is finite, or \(A\) is trivial and \(B\) is finitely presented. This result is identical to that on groups found by \textit{G. Baumslag} [Math. Z. 79, 22--28 (1961; Zbl 0090.24402)].
Mathematics Subject Classification ID: 20M05
zbMATH DE Number: 1975847
zbMATH Keywords: finitely generated monoids
zbMATH Keywords: finitely presented monoids
zbMATH Keywords: wreath products
zbMATH Keywords: generators
zbMATH Keywords: presentations
zbMATH Keywords: groups of units
reviewed by: H. Mitsch
MaRDI profile type: Publication
cites work: Wreath products and finitely presented groups
cites work: Constructions and presentations for monoids
cites work: Q3853827
cites work: Q4840945
cites work: Presentations for subgroups of monoids
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description: scientific article; zbMATH DE number 1975847