Openly factorizable spaces and compact extensions of topological semigroups

We prove that the semigroup operation of a topological semigroup $S$ extends to a continuous semigroup operation on its the Stone-v{C}ech compactification $beta S$ provided $S$ is a pseudocompact openly factorizable space, which means that each map $f:Sto Y$ to a second countable space $Y$ can be written as the composition $f=gcirc p$ of an open map $p:Xto Z$ onto a second countable space $Z$ and a map $g:Zto Y$. We present a spectral characterization of openly factorizable spaces and establish some properties of such spaces.