115 research outputs found
Global Left Loop Structures on Spheres
On the unit sphere in a real Hilbert space , we
derive a binary operation such that is a
power-associative Kikkawa left loop with two-sided identity ,
i.e., it has the left inverse, automorphic inverse, and properties. The
operation is compatible with the symmetric space structure of
. is not a loop, and the right translations
which fail to be injective are easily characterized.
satisfies the left power alternative and left Bol identities ``almost
everywhere'' but not everywhere. Left translations are everywhere analytic;
right translations are analytic except at where they have a
nonremovable discontinuity. The orthogonal group is a
semidirect product of with its automorphism group (cf.
http://www.arxiv.org/abs/math.GR/9907085). The left loop structure of
gives some insight into spherical geometry.Comment: 18 pages, no figures, 10pt, LaTeX2e, uses amsart.cls & tcilatex.tex.
To appear in Comment. Math. Univ. Carolin. (special issue: Proceedings of
LOOPS99) Revised version: various fixes and improvements suggested by refere
Computing the sets of totally symmetric and totally conjugate orthogonal partial Latin squares by means of a SAT solver
Conjugacy and orthogonality of Latin squares have been widely studied in the literature not only for their theoretical interest in combinatorics, but also for their applications in distinct fields as experimental design, cryptography or code theory, amongst others. This paper deals with a series of binary constraints that characterize the sets of partial Latin squares of a given order for which their six conjugates either coincide or are all of them distinct and pairwise orthogonal. These constraints enable us to make use of a SAT solver to enumerate both sets. As an illustrative application, it is also exposed a method to construct totally symmetric partial Latin squares that gives rise,
under certain conditions, to new families of Lie partial quasigroup rings
Distributive and trimedial quasigroups of order 243
We enumerate three classes of non-medial quasigroups of order up to
isomorphism. There are non-medial trimedial quasigroups of order
(extending the work of Kepka, B\'en\'eteau and Lacaze), non-medial
distributive quasigroups of order (extending the work of Kepka and
N\v{e}mec), and non-medial distributive Mendelsohn quasigroups of order
(extending the work of Donovan, Griggs, McCourt, Opr\v{s}al and
Stanovsk\'y).
The enumeration technique is based on affine representations over commutative
Moufang loops, on properties of automorphism groups of commutative Moufang
loops, and on computer calculations with the \texttt{LOOPS} package in
\texttt{GAP}
Some new conjugate orthogonal Latin squares
AbstractWe present some new conjugate orthogonal Latin squares which are obtained from a direct method of construction of the starter-adder type. Combining these new constructions with earlier results of K. T. Phelps and the first author, it is shown that a (3, 2, 1)- (or (1, 3, 2)-) conjugate orthogonal Latin square of order v exists for all positive integers v â 2, 6. It is also shown that a (3, 2, 1)- (or (1, 3, 2)-) conjugate orthogonal idempotent Latin square of order v exists for all positive integers v â 2, 3, 6 with one possible exception v = 12, and this result can be used to enlarge the spectrum of a certain class of Mendelsohn designs and provide better results for problems on embedding
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Constructing and embedding mutually orthogonal Latin squares: reviewing both new and existing results
We review results for the embedding of orthogonal partial Latin squares in orthogonal Latin squares, comparing and contrasting these with results for embedding partial Latin squares in Latin squares. We also present a new construction that uses the existence of a set of mutually orthogonal Latin squares of order to construct a set of mutually orthogonal Latin squares of order
The set of autotopisms of partial Latin squares
Symmetries of a partial Latin square are determined by its autotopism group.
Analogously to the case of Latin squares, given an isotopism , the
cardinality of the set of partial Latin squares which
are invariant under only depends on the conjugacy class of the latter,
or, equivalently, on its cycle structure. In the current paper, the cycle
structures of the set of autotopisms of partial Latin squares are characterized
and several related properties studied. It is also seen that the cycle
structure of determines the possible sizes of the elements of
and the number of those partial Latin squares of this
set with a given size. Finally, it is generalized the traditional notion of
partial Latin square completable to a Latin square.Comment: 20 pages, 4 table
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 vâĄ0,1(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 vâĄ0,1(mod4), 0â€kâ€v, kâ vâ2, and vâk 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 nâ„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 v2Ă4 orthogonal arrays that have the Klein 4-group as conjugate invariant subgroup
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