  The Whitehead asphericity problem, regarded as a problem of combinatorial
group theory, asks whether any subpresentation of an aspherical group
presentation is also aspherical. This is a long standing open problem which has
attracted a lot of attention. Related to it, throughout the years there have
been given several useful characterizations of asphericity which are either
combinatorial or topological in nature. The aim of this paper is two fold.
First, it brings in methods from semigroup theory to give a new combinatorial
characterization of asphericity in terms of what we define here to be the weak
dominion of a submonoid of a monoid, and uses this to give a sufficient and
necessary condition under which a subpresentation of an aspherical group
presentation is aspherical.
