An application of polycyclic monoids to rings (Q1375917)

The polycyclic inverse monoid \(P_\alpha\), where \(\alpha\) is countable, is the inverse monoid generated by the set \(\{p_i\mid 1\leq i\leq\alpha\}\) subject to the relations \(p_ip_j^{-1}=\delta_{ij}\). If \(\alpha=1\), this is the classical bicyclic monoid. It is shown that \(P_2\) contains a copy of \(P_n\) for every finite \(n\geq 2\). If \(P_n\) is a submonoid of (the multiplicative semigroup of) a ring \(R\) with identity and the zero of \(P_2\) is that of \(R\), then \(e=\sum^n_{i=1}p_i^{-1}p_i\) is an idempotent and \(eRe\simeq M_n(R)\). Moreover, if \(n=2\) and \(e=1\) then \(R\simeq M_k(R)\) for every \(k\geq 2\). Certain analogues of these results are discussed for semigroups of matrix type \({\mathcal M}(S,I,I,Q)\) over a monoid \(S\) containing \(P_\alpha\), \(\alpha\geq 2\).