Compact semigroups irreducibly connected between two idempotents

The simplest and most familiar compact connected space which can be provided with the structure of a (topological) group is the simple closed curve C (topological circle). Perhaps the most natural example of a compact connected (topological) semigroup is the closed unit interval I with the usual multiplication. The space C admits the unique structure of a group, whereas simple examples show (see Examples 1, 2, 3) that the space I admits many structures of a semigroup. Such multiplications need not be abelian, may admit both nilpotents and idempotents, and may not have a zero element. In this note we initiate the analysis of the semigroup structures with which the space I may be provided. While our theorems are much more general, the following corollary will give a fair picture of our results. Suppose that I admits such a multiplication that its end points play the natural roles of zero and unit. If there are no other idempotents and no nilpotent elements except zero, then the multiplication must be that of the real numbers. The major step in the proof is that of showing that unique qth roots exist for any positive dyadic rational q. It is with pleasure that we acknowledge the helpful suggestions and advice of A. D. Wallace in the preparation of this paper. We define a mob to be a Hausdorff space together with a continuous, associative multiplication. A clan is a compact connected mob with unit. If S is a mob, a set TCS is a left (right) ideal if T,F and STC T(TSC T). A two-sided ideal is both a left and right ideal. Using Clifford's terminology 1, we shall denote the minimal twosided ideal of a mob S by K and the set of idempotents of S by E. By a zero element, we mean an element 0, such that Ox =0 = xO for all xCS. We define an element s to be nilpotent if Sn=O, for some positive integer n. A connected space S is irreducibly connected between two points a and b if no proper connected subset of S contains both a and b. In such a space, every point different from a and b is a cut point, separating the space into exactly two components 6. We can intro-