Quasiregular regressive transformation semigroups. (Q1881610)

Let \(X\) be a poset and \(S\) a (possibly partial) transformation semigroup on \(X\). The author says that \(S\) is regressive (elsewhere in the literature ``decreasing'') if for all \(\alpha\) in \(S\) and \(x\) in \(\text{Dom\,}\alpha\) we have \(x\alpha\leq x\); \(S\) is almost identical if \(x\alpha=x\) for all but finitely many values of \(x\in\text{Dom\,}\alpha\). In this paper, a semigroup is called quasiregular if some power of each element is regular (elsewhere in the literature this condition is ``eventual regularity'' or ``\(\pi\)-regularity''). It is proved that if \(S\) is the regressive partial transformation semigroup on \(X\), the full regressive almost identical transformation semigroup on \(X\), or the regressive almost identical partial one-to-one transformation semigroup on \(X\), then \(S\) is quasiregular if and only if there is a finite upper bound on the lengths of all chains in \(X\).