On the Continuous Cancellative Semigroups on a Real Interval and on a Circle and Some Symmetry Issues

Symmetry

Mariusz Bajger¹  Janusz Brzdęk²  andEliza Jabłońska³  El-sayed El-hady⁴ 

[1] College of Science and Engineering, Flinders University, Adelaide, SA 5042, Australia;Faculty of Applied Mathematics, AGH University of Science and Technology, Mickiewicza 30, 30-059 Kraków, Poland;Institute of Mathematics, Pedagogical University of Cracow, Podchorążych 2, 30-084 Kraków, Poland;Mathematics Department, College of Science, Jouf University, Sakaka P.O. Box 2014, Saudi Arabia;

关键词: continuous binary operation;    semigroup;    cancellative operation;    unit circle;    real interval;    homomorphism;   

DOI  :  10.3390/sym12121974

来源: DOAJ

【 摘 要 】

Let S denote the unit circle on the complex plane and $☆:S^2\to S$ 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 ☆: $x☆y=F(F^{-1}\left(x\right)·F^{-1}\left(y\right))$ for $x,y\in S$. 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.

【 授权许可】

Unknown