Schröder quasigroups with a specified number of idempotents

AbstractSchröder quasigroups have been studied quite extensively over the years. Most of the attention has been given to idempotent models, which exist for all the feasible orders v, where v≡0,1(mod4) except for v=5,9. There is no Schröder quasigroup of order 5 and the known Schröder quasigroup of order 9 contains 6 non-idempotent elements. It is known that the number of non-idempotent elements in a Schröder quasigroup must be even and at least four. In this paper, we investigate the existence of Schröder quasigroups of order v with a specified number k of idempotent elements, briefly denoted by SQ(v,k). The necessary conditions for the existence of SQ(v,k) are v≡0,1(mod4), 0≤k≤v, k≠v−2, and v−k is even. We show that these conditions are also sufficient for all the feasible values of v and k with few definite exceptions and a handful of possible exceptions. Our investigation relies on the construction of holey Schröder designs (HSDs) of certain types. Specifically, we have established that there exists an HSD of type 4nu1 for u=1,9, and 12 and n≥max{(u+2)/2,4}. In the process, we are able to provide constructions for a very large variety of non-idempotent Schröder quasigroups of order v, all of which correspond to v2×4 orthogonal arrays that have the Klein 4-group as conjugate invariant subgroup

On Multiplication Groups of Quasigroups

Quasigroups are algebraic structures in which divisibility is always defined. In this thesis we investigate quasigroups using a group-theoretic approach. We first construct a family of quasigroups which behave in a group-like fashion. We then focus on the multiplication groups of quasigroups, which have first appeared in the work of A. A. Albert. These permutation groups allow us to study quasigroups using group theory. We also explore how certain natural operations on quasigroups affect the associated multiplication groups. Along the way we take the time and special care to pose specific questions that may lead to further work in the near future