Medial division groupoids
Medial quasigroups of prime square order
We prove that, for any prime , there are precisely medial quasigroups of order , up to isomorphism.
Medial quasigroups of type
Our aim is to demonstrate how the apparatus of groupoid terms (on two variables) might be employed for studying properties of parallelism in the so called -quasigroups. We show that an incidence structure associated with a medial quasigroup of type , , is either an affine space of dimension at least three, or a desarguesian plane. Conversely, if we start either with an affine space of order and dimension , or with a desarguesian affine plane of order then there is a medial quasigroup of...
Minimally nonassociative Moufang loops with a unique nonidentity commutator are ring alternative
We investigate finite Moufang loops with a unique nonidentity commutator which are not associative, but all of whose proper subloops are associative. Curiously, perhaps, such loops turn out to be ``ring alternative'', in the sense that their loop rings are alternative rings.
Moufang loops arising from Zorn vector matrix algebras
In “A class of simple Moufang loops”, Proc. Amer. Math. Soc. 7 (1956), 471–482, Paige used the vector matrix construction over fields to produce simple Moufang loops. The purpose of this paper is to generalize the construction to the class of commutative rings, and examine the Moufang loops arising in this fashion. Specific attention is paid to the construction over the ring of integers modulo four.
Moufang loops of odd order .
Moufang loops of odd order with non-trivial nucleus
It has been proven by F. Leong and the first author (J. Algebra 190 (1997), 474–486) that all Moufang loops of order where and are odd primes, are associative if , and
Moufang loops of order 243
We present a computer-assisted determination of the 72 non-isomorphic, non-associative Moufang loops of order 243. Some of their properties and distinguishing features are discussed.
Moufang loops of order coprime to three that cyclically extend groups of dihedral type
This paper completely solves the isomorphism problem for Moufang loops where is a noncommutative group with cyclic subgroup of index two and , is cyclic, , and is finite of order coprime to three.
Moufang semidirect products of loops with groups and inverse property extensions
We investigate loops which can be written as the semidirect product of a loop and a group, and we provide a necessary and sufficient condition for such a loop to be Moufang. We also examine a class of loop extensions which arise as a result of a finite cyclic group acting as a group of semiautomorphisms on an inverse property loop. In particular, we consider closure properties of certain extensions similar to those as in [S. Gagola III, Cyclic extensions of Moufang loops induced by semiautomorphisms,...
Multiplication groups of free loops. I
Multiplication groups of free loops. II
Multiplication groups of quasigroups and loops I.
Multiplication groups of quasigroups and loops II.
Multiplication groups of quasigroups and loops III.
Multiplication groups of quasigroups and loops IV.
-T-quasigroup codes with one check symbol and their error detection capabilities
It is well known that there exist some types of the most frequent errors made by human operators during transmission of data which it is possible to detect using a code with one check symbol. We prove that there does not exist an -T-code that can detect all single, adjacent transposition, jump transposition, twin, jump twin and phonetic errors over an alphabet that contains 0 and 1. Systems that detect all single, adjacent transposition, jump transposition, twin, jump twin errors and almost all...
- арные лупы со свойствами обратимости с одним параметром обращения
- арные лупы со свойством обратимости
Near loop rings of Moufang loops.