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Extension of Toyoda's theorem on entropic groupoids

✍ Scribed by Vladimir Volenec

Publisher
John Wiley and Sons
Year
1981
Tongue
English
Weight
281 KB
Volume
102
Category
Article
ISSN
0025-584X

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✦ Synopsis

Extension of Toyoda's theorem on entropic groupoids

By VLADIMRC VOLENEC of Zagreb (Eingegangen am 8. 10. 1980) A groupoid (a, .) is said to be entropic iff for every a, b, c, d E G the equality (1) ab cd = acbd holds true. A well-known TOYODA'S theorem ([S], [a], [21 and [l], 1). 33) asserts that every entropic quasigroup can be obtained in a certain way from an abelian group. STBECKER ( [ 5 ] , Satz 2.1) extended the TOYODA'S result t o the case of entropic groupoids.

The main result of this paper is a weakening of hypothesis and a siniplification of the proof of the mentioned STRECKER'S theorem. This result will be stated as a theorem. Thc preliminaries for the proof of this theorem will be proved in Lenimas 1-7. These arc mainly the results froin [1]-[5] adapted for our purposes. Let (G, .) be EX groupoid and a E G any element. We define the mappings A, , , p,,:

element a is said to be left resp. right cancellalive element of the groupoid (a, *) iff A, , resp. pa is an injection and left resp. right regular element iif A, resp. is a bijection. Lernrna 1. (151) Let f E C be a left regular elemnt of a n entropic groupoid (G, -) such that ff L.S (c left cctncellative clement of this grmpoid. If f' E G i s such a n element thtrt f f ' = f ,

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