Schroder 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 Schroder quasigroup of order 5 and the known Schroder quasigroup of order 9 contains 6 non-idempotent elements. It is known that the number of non-idempotent elements in a Schroder quasigroup must be even and at least four. In this paper, we investigate the existence of Schroder 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 Schroder designs (HSDs) of certain types. Specifically, we have established that there exists an HSD of type 4^nu^1 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 Schroder quasigroups of order v, all of which correspond to v^2x4 orthogonal arrays that have the Klein 4-group as conjugate invariant subgroup.
[1]
Lie Zhu,et al.
Existence of frame SOLS of type anb1
,
2002,
Discret. Math..
[2]
N. S. Mendelsohn.
SELF‐ORTHOGONAL WEISNER DESIGNS
,
1979
.
[3]
Ruizhong Wei,et al.
The existence of Schröder designs with equal-sized holes
,
1997,
Discret. Math..
[4]
Alexander Schrijver,et al.
Group divisible designs with block-size four
,
2006,
Discret. Math..
[5]
Richard M. Wilson,et al.
Constructions and Uses of Pairwise Balanced Designs
,
1975
.
[6]
F. E. Bennett,et al.
Incomplete idempotent Schröder quasigroups and related packing designs
,
1994
.
[7]
R. D. Baker.
Quasigroups and tactical systems
,
1978
.
[8]
N. S. Mendelsohn,et al.
On the construction of Schroeder quasigroups
,
1980,
Discret. Math..
[9]
Charles J. Colbourn,et al.
Edge-coloured designs with block size four
,
1988
.
[10]
Haim Hanani,et al.
Balanced incomplete block designs and related designs
,
1975,
Discret. Math..
[11]
Hantao Zhang,et al.
Combinatorial Designs by SAT Solvers
,
2021,
Handbook of Satisfiability.