Work of Ehresmann and
Schein shows that an inverse semigroup can be viewed as a groupoid with an order
structure; this approach was generalized by Nambooripad to apply to arbitrary
regular semigroups. This paper introduces the notion of an ordered 2-complex and
shows how to represent any ordered groupoid as the fundamental groupoid of an
ordered 2-complex. This approach then allows us to construct a standard 2-complex
for an inverse semigroup presentation.
Our primary applications are to calculating the maximal subgroups of an inverse
semigroup which, under our topological approach, turn out to be the fundamental
groups of the various connected components of the standard 2-complex. Our main
results generalize results of Haatja, Margolis, and Meakin giving a graph of groups
decomposition for the maximal subgroups of certain regular semigroup amalgams. We
also generalize a theorem of Hall by showing the strong embeddability of certain
regular semigroup amalgams as well as structural results of Nambooripad and Pastijn
on such amalgams.
