Structure of the semigroup of semigroup extensions

Abstract:

Let $B$ denote a compact semigroup with identity and $G$ a compact abelian group. Let $\operatorname {Ext} (B,G)$ denote the semigroup of extensions of $G$ by $B$. We show that $\operatorname {Ext} (B,G)$ is always a union of groups. We show that it is a semilattice whenever $B$ is. In case $B$ is also an abelian inverse semigroup with its subspace of idempotent elements totally disconnected, we obtain a determination of the maximal groups of a commutative version of $\operatorname {Ext} (B,G)$ in terms of the extension functor of discrete abelian groups.