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A Class of Loops with the Isotopy-Isomorphy Property

Published online by Cambridge University Press:  20 November 2018

Eric L. Wilson
Eric L. Wilson*
Affiliation:
Vanderbilt University and Wittenberg University
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A loop L has the isotopy-isomorphy property provided each loop isotopic to L is isomorphic to L. A familiar problem is that of characterizing those loops having this property.

It is well known (1, p. 56) that the loop isotopes of (L, ·) are those loops L(a, b, *) defined by x * y = x/b·a\y for some a, b in L. In this paper we first show (Corollary to Theorem 1) that a loop L with identity element 1 has the isotopy-isomorphy property if L is isomorphic to 1,(1, x) and to L(x, 1) for each x in L. We then determine necessary and sufficient conditions (Theorems 2 and 3) for L to be isomorphic to these isotopes under translations (i.e. permutations of the form xv = cx or xv = xc for c fixed).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 18 , 1966 , pp. 589 - 592
Copyright
Copyright © Canadian Mathematical Society 1966

References

1. Bruck, R. H., A survey of binary systems (Berlin-Göttingen-Heidelberg, 1958).Google Scholar
2. Bruck, R. H., Some theorems on Moufang loops, Math. Z., 73 (1960), 59–78.Google Scholar
3. Bryant, B. F. and Schneider, Hans, Principal loop-isotopes of quasigroups, Can. J. Math., 18 (1966), 120–125.Google Scholar
4. Osborn, J. M., Loops with the weak inverse property, Pacific J. Math., 10 (1960), 295–304.Google Scholar