Interval clans with nondegenerate kernel

Abstract

Introduction. The object of this paper is to characterize the clans (compact connected Hausdorff topological semigroups with an identity element) which are homeomorphic to a unit interval and which have a nondegenerate kernel (minimal two-sided ideal). The corresponding case when the kernel is degenerate has been characterized in a paper by H. Cohen and L. I. Wade [2] together with an earlier paper by Mostert and Shields [5]. In a topological semigroup T, K(T) or K denotes the kernel of T. The symbol u is reserved to denote an identity element. The term will mean a clan with zero which is homeomorphic to a unit interval and whose endpoints are its zero and identity element. In a standard thread T with identity element u and zero 0, for a, bCT, [a, b] will denote the interval from a to b, (or b to a) inclusive and a ? b will mean aC [0, b], with a x implies aRx (closed). We denote by Ra the set {x aRx }. The analog of Theorem I where R satisfies (2)' (a, b, cCX, aRb implies caRcb (left congruence) instead of (2) is also true. I would like to express my appreciation to Professor H. Cohen and Professor R. J. Koch for their assistance in the preparation of this paper.