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IMI/Publicaţii/QRS/Ediţii/QRS v.10, n.1 (10), 2003/

Quasigroups constructed from cycle systems

Authors: Lindner C. C.

Abstract

An m-cycle system of order n is a pair (S;C), where C is a collection of edge disjoint m-cycles which partitions the edge set of the complete undirected graph Kn with vertex set S. If the m-cycle system (S;C) has the additional property that every pair of vertices a != b are joined by a path of length 2 (and therefore exactly one) in an m-cycle of C, then (S;C) is said to be 2-perfect. Now given an m-cycle system (S;C) we can define a binary operation “o” on S by a o a = a and if a != b, a o b = c and b o a = d if and only if the cycle (…, d, a, b, c,… ) c C. This is called the Standard Construction and it is well known that the groupoid (S;o) s a quasigroup (which can be considered to be the “multiplicative” part of a universal algebra quasigroup (S,o, \,/) if and only if (S;C) is 2-perfect. The class of 2-perfect m-cycle systems is said to be equationally defined if and only if there exists a variety of universal algebra quasigroups V such that the finite members of V are precisely all universal algebra quasigroups whose multiplicative parts can be constructed from 2-perfect m-cycle systems using the Standard Construction. This paper gives a survey of results showing that 2-perfect m-cycle systems can be equationally defined for m = 3; 5, and 7 only. Similar results are obtained for m-perfect (2m + 1)-cycle systems using the Opposite Vertex Construction (too detailed to go into here). We conclude with a summary of similar results (without details) for 2-perfect and m-perfect directed cycle systems.

Lindner C. C.
Department of Discrete and Statistical Sciences,
Auburn University,
Auburn,
Alabama 36849,
USA.
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