Admissibility of Semigroup Structures On Continua
R. J. Koch, Alexander Doniphan Wallace · Transactions of the American Mathematical Society · 1958
The purpose of this note is to investigate the structure of certain compact connected semigroups 5 which satisfy S2 = S.If X is any continuum then a continuous associative multiplication may be introduced by (i) xy = x all x, y(EX or (ii) xy=y all x, y€zX.We shall give certain instances in which, due to the topological structure of S, the multiplication must be of this trivial kind.The central ideas of the paper spring in one way or another from dense connected sets.For example, 5 being a continuum (S will always denote a Hausdorff topological semigroup, or mob), S2 = S if and only if each dense ideal is connected.We will show, for example, (Theorem 6) that if 5 is irreducible between its minimal ideal K and some other set and if S2 -S then S must have a unit, S is commutative, and S/K (in the metric case) is an arc.In a related result (Theorem 5) it will be proved that maximal ideals are composants.Because of the relative novelty of the subject and the present lack of structure theorems we go into considerable detail in considering some examples.We follow the terminology of [l, p. 14].In particular K denotes the minimal ideal of S, if such exists, otherwise i£= D, and E denotes the set of idempotents.If 5 is compact it is known (see e.g.[14]) that K and E are nonempty and that K is completely simple.If S is connected and K is nonempty, then K is connected.For aG5, J(a) = a\JSa^JaSVlSaS