Abelian quasigroups and T-quasigroups
Let A be a universal algebra. The centre Z(A) of A is the congruence consisting of all pairs (a,b) Î A2 such that, for each (n+1)-ary term operation t and each [u], [v] Î An, t(a, [u] ) = t(b, [v] ).
If Z(A) = A2, then A is said to be abelian.
A quasigroup (Q, · , \ , / ) is said to be a T-quasigroup if there are an abelian group (Q,+), c Î Q and
a, b Î Aut (Q,+) such that x·y = ax + by + c for all x,y Î Q.
It is shown that a quasigroup is abelian iff it is a T-quasigroup. The proof is based on known results on congruences of algebras.
(Raffaele Scapellato, MR 96e:20099)
|Belyavskaya G.B., Tabarov A.H. |
One-sided T-quasigroups and irreducible balanced identities
Let (Q,·) be a quasigroup such that for all x,y Î Q, xy=j(x) + a(y) or xy=a(x) + j(y), respectively, where (Q,+) is a certain group with automorphism j, and a: Q® Q is a bijection of Q onto itself. Then (Q,·) is said to be a left or a right linear quasigroup over the group (Q,+), respectively. Moreover, if the group (Q,+) is abelian, then (Q,·) is said to be a left or right T-quasigroup, respectively.
In this paper two main results concerning left and right T-quasigroups are proved.
First, it is shown that all of the primitive left or right T-quasigroups, respectively, form a variety that can be characterized by two identities (Theorem 1).
Second, it is shown how the primitive left or right T-quasigroups, respectively, can be characterized using irreducible balanced identities (Theorems 3 and 4).
Finally, some consequences of these results are discussed and a number of subvarieties of the varieties of left or right T-quasigroups, respectively, are described in the final section of this paper.
(Jaroslav Libicher, MR 96c:20126)
Transversals in groups. Elementary properties
A transversal in a group G of (the author uses "to" here) a subgroup H is a complete set of coset representatives for G/H. The binary operation of the group G induces on any transversal T in G a natural binary operation that makes T a quasigroup.
This paper contains a number of technical results concerning such quasigroups on transversals, concluding with conditions on the group and subgroup that are equivalent to the existence of a transversal of the subgroup whose quasigroup is a loop.
(Chris A. Rowley, MR 96e:20003)
Sharply double transitive sets of permutations and loop transversals in Sn
The work is devoted to the investigation of sharply k-transitive sets of permutations which are a natural generalization of sharply k-transitive groups. Its main result is the establishment of the connection between such notions as sharply k-transitive sets of permutations, sharply k-transitive loops of permutations (introduced by F. Bonetti, G. Lunardon and K. Strambach) and loop transversals.
( Zbl. 951:20501 )
The loop Q(·) is called a G-loop if the operations (·)a and a(·) are isomorphic to the operation (·), where (·)a = (·)(La,1,La) and a(·) = (·)(1,Ra,Ra) , a Î Q. It is called an Osborn loop if (·)a = Ia(·), where Ia = a-1, a Î Q, and an i-loop if xy\((xy)·u)v = u(v·(yx))/yx, x,y,u,v Î Q.
The author proves that
(1) a loop Q(·) with (·)I-1x = Ix(·) for every x Î Q is a G-loop,
(2) an Osborn loop Q(·) in which xx Î N (kernel of Q) for any x Î Q is a G-loop,
(3) every i-loop is a G-loop.
These results are a continuation of earlier work of the author.
(Elena Brozikova, MR 96e:20099)
Loops with universal elasticity
A loop is said to satisfy the law of elasticity if (xy)x = x(yx) for all elements x,y of the loop. An identity is called universal for a loop Q if it holds in that loop and in each of its principal isotopes. In this article the author investigates properties of loops for which the law of elasticity is universal, as well as the connections of this class of loops with some well-known classes of loops such as Bol and Moufang loops.
The main results obtained are the following:
If Q is a loop with universal elasticity, then
(a) Q is power-associative,
(b) the left and right nuclei are equal,
(c) all three nuclei of Q coincide iff each element of the middle nucleus is a Bol element
(an element a Î Q is called a Bol element if a(x(ay)) = (a(xa))y for all x,y in Q),
(d) the following properties are equivalent in Q: right inverse property, left inverse property, right
alternative property, left alternative property.
( Karl H. Robinson, MR 96d:20067 )
|Sokhatskyj F., Syvakivskyj P.|
On linear isotopes of cyclic groups
A description of all cyclic group n-ary linear isotopes is found to within isomorphism. Some results on their automorphism group and endomorphism semigroup are given.
( Zbl. 951:20510 )
A common form for autotopies of n-ary groups with the inverse property
A quasigroup Q(A) of arity n is an n-ary function A on the set Q such that, when all components xi of A(x1, ¼ , xn) except one are fixed, one obtains a bijection on Q. This is called an n-group [resp. n-IP-quasigroup (n-quasigroup with the inverse property)] if certain generalizations of the associativity [and inverse] properties of binary groups hold.
An autotopy of the quasigroup Q(A) is an (n+1)-tuple (a1, . . . , an+1) of bijections on Q such that
an+1-1A(a1(x1), . . . , an(xn)) = A(x1, . . . , xn).
The paper shows that for an n-IP-group, the components ai of such an autotopy are strongly related to the so-called quasiautomorphism an+1.
( Bernhard von Stengel, MR 96c:20132 )