Finite generation and presentability of wreath products of monoids.
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)].
scientific article; zbMATH DE number 1975847
scientific article