Belyavskaya G.B. Abelian quasigroups and Tquasigroups 
Let A be a universal algebra. The centre Z(A) of A is the congruence consisting of all pairs (a,b) Î A^{2} such that, for each (n+1)ary term operation t and each [u], [v] Î A^{n}, t(a, [u] ) = t(b, [v] ).
If Z(A) = A^{2}, then A is said to be abelian.
A quasigroup (Q, · , \ , / ) is said to be a Tquasigroup 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 Tquasigroup. The proof is based on known results on congruences of algebras.
(Raffaele Scapellato, MR 96e:20099)
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Belyavskaya G.B., Tabarov A.H. Onesided Tquasigroups 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 Tquasigroup, respectively.
In this paper two main results concerning left and right Tquasigroups are proved.
First, it is shown that all of the primitive left or right Tquasigroups, respectively, form a variety that can be characterized by two identities (Theorem 1).
Second, it is shown how the primitive left or right Tquasigroups, 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 Tquasigroups, respectively, are described in the final section of this paper.
(Jaroslav Libicher, MR 96c:20126)
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Kuznetsov E.A. 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)
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Kuznetsov E.A. Sharply double transitive sets of permutations and loop transversals in S_{n} 
The work is devoted to the investigation of sharply ktransitive sets of permutations which are a natural generalization of sharply ktransitive groups. Its main result is the establishment of the connection between such notions as sharply ktransitive sets of permutations, sharply ktransitive loops of permutations (introduced by F. Bonetti, G. Lunardon and K. Strambach) and loop transversals. ( Zbl. 951:20501 )
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Basarab A.S. Osborn's Gloops 
The loop Q(·) is called a Gloop 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 iloop 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}1_{x} = _{Ix}(·) for every x Î Q is a Gloop,
(2) an Osborn loop Q(·) in which xx Î N (kernel of Q) for any x Î Q is a Gloop,
(3) every iloop is a Gloop.
These results are a continuation of earlier work of the author.
(Elena Brozikova, MR 96e:20099)
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Syrbu P.N. 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 wellknown 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 powerassociative,
(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 )
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Sokhatskyj F., Syvakivskyj P. On linear isotopes of cyclic groups 
A description of all cyclic group nary linear isotopes is found to within isomorphism. Some results on their automorphism group and endomorphism semigroup are given.
( Zbl. 951:20510 )
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Ursu L.A. A common form for autotopies of nary groups with the inverse property 
A quasigroup Q(A) of arity n is an nary function A on the set Q such that, when all components x_{i} of A(x_{1}, ¼ , x_{n}) except one are fixed, one obtains a bijection on Q. This is called an ngroup [resp. nIPquasigroup (nquasigroup 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 (a_{1}, . . . , a_{n+1}) of bijections on Q such that a_{n+1}^{1}A(a_{1}(x_{1}), . . . , a_{n}(x_{n})) = A(x_{1}, . . . , x_{n}).
The paper shows that for an nIPgroup, the components a_{i} of such an autotopy are strongly related to the socalled quasiautomorphism a_{n+1}.
( Bernhard von Stengel, MR 96c:20132 )
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