205 research outputs found
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}
Code loops in dimension at most 8
Code loops are certain Moufang -loops constructed from doubly even binary
codes that play an important role in the construction of local subgroups of
sporadic groups. More precisely, code loops are central extensions of the group
of order by an elementary abelian -group in the variety of loops
such that their squaring map, commutator map and associator map are related by
combinatorial polarization and the associator map is a trilinear alternating
form.
Using existing classifications of trilinear alternating forms over the field
of elements, we enumerate code loops of dimension
(equivalently, of order ) up to isomorphism. There are
code loops of order , and of order , and of order
3-nets realizing a diassociative loop in a projective plane
A \textit{-net} of order is a finite incidence structure consisting of
points and three pairwise disjoint classes of lines, each of size , such
that every point incident with two lines from distinct classes is incident with
exactly one line from each of the three classes. The current interest around
-nets (embedded) in a projective plane , defined over a field
of characteristic , arose from algebraic geometry. It is not difficult to
find -nets in as far as . However, only a few infinite
families of -nets in are known to exist whenever , or .
Under this condition, the known families are characterized as the only -nets
in which can be coordinatized by a group. In this paper we deal with
-nets in which can be coordinatized by a diassociative loop
but not by a group. We prove two structural theorems on . As a corollary, if
is commutative then every non-trivial element of has the same order,
and has exponent or . We also discuss the existence problem for such
-nets
Interval Algebraic Bistructures
This book has four chapters. In the first chapter interval bistructures
(biinterval structures) such as interval bisemigroup, interval bigroupoid,
interval bigroup and interval biloops are introduced. Throughout this book we
work only with the intervals of the form [0, a] where a \in Zn or Z+ \cup {0}
or R+ \cup {0} or Q+ \cup {0} unless otherwise specified. Also interval
bistructures of the form interval loop-group, interval groupgroupoid so on are
introduced and studied. In chapter two n-interval structures are introduced.
n-interval groupoids, n-interval semigroups, n-interval loops and so on are
introduced and analysed. Using these notions n-interval mixed algebraic
structure are defined and described. Some probable applications are discussed.
Only in due course of time several applications would be evolved by researchers
as per their need. The final chapter suggests around 295 problems of which some
are simple exercises, some are difficult and some of them are research
problems.Comment: 208 page
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