Vol. 60, No. 2, 1975

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Construction functors for topological semigroups

Thomas Theodore Bowman

Vol. 60 (1975), No. 2, 27–36

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.

Mathematical Subject Classification 2000
Primary: 22A15
Secondary: 54A10
Received: 29 May 1973
Revised: 25 July 1974
Published: 1 October 1975
Thomas Theodore Bowman