ABSTRACT. This paper is concerned with three aspects of the study of topological versions of the translational hull of a topological semigroup. These include topological properties, applications to the general theory of topological semigroups, and techniques for computing the translational hull. The central result of this paper is that if S is a compact reductive topological semigroup and its translational hull Sl(S) is given the topology of continuous convergence Motivation for the study of the translational huU has primarily been its appUcations to the theory of ideal extensions. For this reason, it appears to the authors of this paper that any coherent theory of ideal extensions of topological semigroups would be based on a topological version of the translational hull. Results of a brief effort in this direction appear in [2], where a topology is assigned to the translational huU. A study of the special case of a compact semilattice is presented in [1].In this paper we expand the knowledge of topological versions of the translational huU of a topological semigroup in terms of topological properties, applications to the general theory of semigroups, and techniques for computing the translational huU of a given topological semigroup. Whenever feasible, we present results in the more general setting of semitopological semigroups.Our prime objective in the first section of this paper is to find a reasonable
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