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The Generalized Equations of Bisymmetry Associativity and Transitivity on Quasigroups

Published online by Cambridge University Press:  20 November 2018

M. A. Taylor
M. A. Taylor*
Affiliation:
University of Waterloo, Waterloo, Ontario
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The generalized equations of bisymmetry, associativity and transitivity are, respectively,

  1. (1) (x1y)2(z3u) = (x4z)5(y6u)

  2. (2) (x1y)2z = x3(y4z)

  3. (3) (x1z)2(y3z) = x4y.

The numbers 1, 2, 3,…, 6 represent binary operations and x, y, z and u are taken freely from certain sets.

We shall be concerned with the cases in which x, y, z, and u are from the same set and each operation is a quasigroup operation. Under these conditions the solution of all three equations is known [1], [2]; equations (1) and (3) having been reduced to the form of (2) and a solution of (2) being given. We wish to present a new approach to these equations which we feel has the advantages that the equations may be resolved independently, the motivation behind the proof is clear, and the method lends itself to application on algebraic structures weaker than quasigroups. (Details of these generalizations will be given elsewhere.)

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 15 , Issue 1 , March 1972 , pp. 119 - 124
Copyright
Copyright © Canadian Mathematical Society 1972

References

1. Aczél, J., Quasigroups, nets and nomograms, Advances in Math. 1, Fasc. 3, 1965.Google Scholar
2. Aczél, J., Theory of functional equations and its applications, Academic Press, New York, 1966.Google Scholar
3. Aczél, J., Belousov, V. D., and M. Hossz?, Generalised associativity and bisymmetry on quasigroups, Acta Math. Acad. Sci. Hungar 11 (1960), 127-136.Google Scholar
4. Radó, F., équations fonctionnelles caractérisant les nomogrammes avec trois échelles rectilignes, Mathematica (Cluj) 1 (24), 1 (1959), 143-166.Google Scholar
5. Radó, F., Sur quelques équations fonctionnelles avec plusieures fonctions ? deux variables, Mathematica (Cluj) 1(24), 2 (1959), 321-339.Google Scholar