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