Construction functors for topological semigroups
Thomas T. Bowman · Pacific Journal of Mathematics · 1975
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.