✦ LIBER ✦

On the construction of cyclic quasigroups

✍ Scribed by Charles C. Lindner

Publisher: Elsevier Science
Year: 1973
Tongue: English
Weight: 920 KB
Volume: 6
Category: Article
ISSN: 0012-365X
DOI: 10.1016/0012-365x(73)90044-7

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✦ Synopsis

Let Fix, y) be the free groupoid on two generators x and y. Define an infinite class of words in F(x, y) by WO(X, y) = X, WI (x, y) = y and ~f+2t~, y) -Wi(X, y) wi+ 1(x, y). An identity of the form \v~~(x, y) = x is called a cyclic identity and ;f quasigroup satisfying a cyclic identity is called a c_~i+c quasigroup. The most extensively sfuriied cyclic quasigroups have been models of the identity y (X J) = x. The more general notion o j. cyclic quasigroups was introduced by N.S. Mendelsohn. In this paper a new construction fc r cyclic quasigroups is given. This construction is useful in constructing large numbers of ,lonisomorphic quasigroups satisfying a given cyclic identity or a consequence of a cyclic identity. The construction is based on a generalization of A. Sade's singular direct product of'quasigrotrps.

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