Coset decomposition of positively self-conjugate inverse submonoids of polycyclic monoids (Q1864455)

A concrete realisation of the polycyclic monoid \(P(X)\) is introduced; \(P(X)\) is isomorphic to a certain inverse submonoid of the symmetric inverse monoid on \(Y^Z\), where \(Y=X\) if \(|X|\geq 2\), \(Y=X\cup\{t\}\) if \(|X|=1\). For every relation \(R\) on the free monoid on \(X\) a certain subset \(P(R)\) of \(P(X)\) is defined as is the notion of (right) cosets with respect to \(P(R)\). Using these ingredients, the author characterizes groups, periodic monoids, and right cancellative monoids in terms of the coset decomposition of so-called positively self-conjugate inverse submonoids of polycyclic monoids.