Cheban loops

TL;DR

This paper introduces Cheban loops, a class of loops satisfying specific identities, and explores their structural properties, including conjugacy closure, nilpotency, and inverse properties, expanding understanding of their algebraic structure.

Contribution

The paper initiates the study of Cheban loops, characterizing their properties and establishing their place within loop theory, including conjugacy closure and nilpotency.

Findings

Cheban loops are conjugacy closed.

They are weak inverse property loops.

They are centrally nilpotent of class at most two.

Abstract

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.