The Position of C-sets in Semigroups1

A continuum is a compact connected Hausdorff space. If X is a continuum, then CEX is a C-set if (i) C^X, (ii) for each subcontinuum A EX that meets C we have either A EC or CEA. Thus a composant of a solenoid is a C-set. A ?«0& is a Hausdorff space together with a continuous associative multiplication. A clan is a compact connected mob with (two-sided) unit, denoted by u. In what follows we assume that S is a mob. (*) Assume that S is compact. Let E be the set of idempotents of S and let K be the minimal ideal of S [4]. We know that each eEE is contained in a maximal subgroup H(e) of S and that no two such maximal subgroups intersect [8, Theorem l]. Moreover ([l] and [8, Theorem 3]),