{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:26:10Z","timestamp":1760239570027,"version":"build-2065373602"},"reference-count":31,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2020,11,29]],"date-time":"2020-11-29T00:00:00Z","timestamp":1606608000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Let S denote the unit circle on the complex plane and \u2605:S2\u2192S be a continuous binary, associative and cancellative operation. From some already known results, it can be deduced that the semigroup (S,\u2605) is isomorphic to the group (S,\u00b7); thus, it is a group, where \u00b7 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,\u00b7) into (S,\u2605) and obtain, in this way, the following description of operation \u2605: x\u2605y=F(F\u22121(x)\u00b7F\u22121(y)) for x,y\u2208S. 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.<\/jats:p>","DOI":"10.3390\/sym12121974","type":"journal-article","created":{"date-parts":[[2020,11,29]],"date-time":"2020-11-29T21:55:31Z","timestamp":1606686931000},"page":"1974","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["On the Continuous Cancellative Semigroups on a Real Interval and on a Circle and Some Symmetry Issues"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1866-2451","authenticated-orcid":false,"given":"Mariusz","family":"Bajger","sequence":"first","affiliation":[{"name":"College of Science and Engineering, Flinders University, Adelaide, SA 5042, Australia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6529-4885","authenticated-orcid":false,"given":"Janusz","family":"Brzd\u0119k","sequence":"additional","affiliation":[{"name":"Faculty of Applied Mathematics, AGH University of Science and Technology, Mickiewicza 30, 30-059 Krak\u00f3w, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4955-0842","authenticated-orcid":false,"given":"El-sayed","family":"El-hady","sequence":"additional","affiliation":[{"name":"Mathematics Department, College of Science, Jouf University, Sakaka 72388, Saudi Arabia"},{"name":"Basic Science Department, Faculty of Computers and Informatics, Suez Canal University, Ismailia 41522, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0347-0214","authenticated-orcid":false,"given":"Eliza","family":"Jab\u0142o\u0144ska","sequence":"additional","affiliation":[{"name":"Institute of Mathematics, Pedagogical University of Cracow, Podchor\u0105\u017cych 2, 30-084 Krak\u00f3w, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,11,29]]},"reference":[{"key":"ref_1","first-page":"59","article-title":"Sur les operations definies pour nombres r\u00e9els","volume":"76","year":"1948","journal-title":"Bull. 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