On the continuous cancellative semigroups on a real interval and on a circle and some symmetry issues

Let S denote the unit circle on the complex plane and ⋆: S² → S be a continuous binary, associative and cancellative operation. From some already known results, it can be deduced that the semigroup (S, ⋆) is isomorphic to the group (S, ·); thus, it is a group, where · is the usual multiplication of complex numbers. However, an elementary construction of such isomorphism has not been published so far. We present an elementary construction of all such continuous isomorphisms F from (S, ·) into (S, ⋆) and obtain, in this way, the following description of operation ⋆: x ⋆ y = F(F⁻¹ (x) · F⁻¹ (y)) for x, y ∈ S. We also provide some applications of that result and underline some symmetry issues, which arise between the consequences of it and of the analogous outcome for the real interval and which concern functional equations. In particular, we show how to use the result in the descriptions of the continuous flows and minimal homeomorphisms on S.