In the study of semigroups
there often exists a natural decomposition of a semigroup into a semilattice of
subsemigroups of known structure. The harder question is, given a semilattice of
semigroups to construct from these components a larger semigroup. If in addition,
the given semilattice and semigroups are topological, this paper studies the
construction of topological semigroups with emphasis on the functorial nature of the
construction. It is shown that a semilattice of topological monoids has a unique
minimal compatible topology. This is a new characterization of a topology that has
been widely used in the special case of compact semigroups, but in the general case it
does not necessarily give rise to a Hausdorff topology. It also lacks desirable
functorial properties. In §2, a topological construction is given which satisfies the
desired functorial properties. In §3, we restrict our attention to subcategories and
sufficient conditions for the constructed topologies to be Hausdorff. The constructed
topologies of §§2 and 3 were motivated by the topologies of dual semigroups in a
Pontryagin type duality.
