21 research outputs found
Cheban loops
Left Cheban loops are loops that satisfy the identity x(xy.z) = yx.xz. Right
Cheban loops satisfy the mirror identity {(z.yx)x = zx.xy}. Loops that are both
left and right Cheban are called Cheban loops. Cheban loops can also be
characterized as those loops that satisfy the identity x(xy.z) = (y.zx)x. These
loops were introduced in Cheban, A. M. Loops with identities of length four and
of rank three. II. (Russian) General algebra and discrete geometry, pp.
117-120, 164, "Shtiintsa", Kishinev, 1980. Here we initiate a study of their
structural properties. Left Cheban loops are left conjugacy closed. Cheban
loops are weak inverse property, power associative, conjugacy closed loops;
they are centrally nilpotent of class at most two.Comment: 6 page
On autotopies and automorphisms of n-ary linear quasigroups
In this article we study structure of autotopies,
automorphisms, autotopy groups and automorphism groups of nary linear quasigroups.
We find a connection between automorphism groups of some
special kinds of n-ary quasigroups (idempotent quasigroups, loops)
and some isotopes of these quasigroups. In binary case we find
more detailed connections between automorphism group of a loop
and automorphism group of some its isotope. We prove that every
finite medial n-ary quasigroup of order greater than 2 has a nonidentity automorphism group.
We apply obtained results to give some information on automorphism groups of n-ary quasigroups that correspond to the ISSN
code, the EAN code and the UPC code