On Maximal Ideals of Compact Connected Topological Semigroups

Definition 1.1. A topological semigroup is a topological space S together with a continuous function m : S × S → S such that S is Hausdorff andm is associative. A subsemigroup of a semigroup S is a nonvoid set A ⊂ S such that A2 ⊂ A, and A is called a subgroup of S if it is a group with respect tom. An element e of a topological semigroup S is called an idempotent if e2 e. Similarly, an element e of S is called a left identity right identity if ea a ae a for all a ∈ S. An element of S is called an identity of S if it is both a left and a right identity of S. The set of all idempotents of S will be denoted by E throughout this paper. For each e ∈ E, letH e be the union of all subgroups of S containing e. It is shown in 3 thatH e is the maximal subgroup of S containing e.