On Bruck-Reilly extensions of rectangular bands, zero semigroups and free monoids. (Q2569289)

Let \(M\) be a monoid and \(\theta\colon M\to M\) an endomorphism. \(\text{BR}(M,\theta)\) denotes the Bruck-Reilly extension of \(M\) determined by \(\theta\). \(\text{BR}(M,\theta)\) is a monoid defined on the set \(\mathbb{N}_0\times M\times\mathbb{N}_0\) by means of the operation \((m,a,n)(p,b,q)=(m-n+t,a\theta^tb\theta^t,p-q+t)\), where \(\mathbb{N}_0\) is the set of non-negative integers, \(t=\max(n,p)\) and \(\theta^0\) is the identity map on \(M\). It is proved that if \(\text{BR}(S^1,\theta)\) is finitely presented then \(S\) is finitely generated, where \(S\) is one of the semigroups mentioned in the title of the paper, and \(S^1\) is the monoid obtained from \(S\) by adjoining a unit.