Date of Award
Doctor of Philosophy (PhD)
Professor A. Rosa
An n-ary quasigroup is a set together with an n-ary operation which is cancellative in every variable. To every permutation on n + 1 elements there is associated a conjugate quasigroup of the original quasigroup. The elements of both quasigroups are the same, but the conjugate n-ary operation is defined as follows. It acts on a permuted set of elements to produce a permutation of the results of the original operation on the unpermuted elements.
These conjugate quasigroups need not be distinct. The number of distinct such conjugates is called the conjugacy class number of the quasigroup. It has been shown that this number must always be a divisor of (n+1)!
In the case of ordinary quasigroups, it is known that for any order greater than or equal to four, there exists a quasigroup of that order having a specified number of distinct conjugates. An investigation of the conjugacy class number leads to a study of quasigroup identities. The existence of quasigroups satisfying certain identities has been widely investigated for ordinary quasigroups, but for higher dimensional quasigroups, much less is known.
We investigate the existence of ternary quasigroups having a given class number. In all but two cases, the question is completely answered. Ternary quasigroups, having six of the possible eight class numbers, are shown to exist of every order, expect for a small, finite number of low orders. In the remaining two cases, infinitely many quasigroups have been constructed with these conjugacy class numbers.
An investigation is begun of the existence of n-ary quasigroups with prescribed conjugacy class numbers. The problem is solved for two sets of classes and for n-ary quasigroups having sufficiently large orders.
A combination of methods is used throughout, varying from exact constructions, to "ad hoc" constructions for low orders and adaptations of block designs.
McLeish, Mary Elizabeth (Deutsch), "On the Number of Conjugates of Ternary Quasigroups" (1976). Open Access Dissertations and Theses. Paper 789.