On the pre-ordering of automorphic loops and moufang loops

Abstract

A set L endowed with three operations: multiplication, right division, and left division, denoted by ·, /. \, is called a loop if (L, ·) is a groupoid with unit, 1 ∈ L, and x/y · y = xy/y = y · (y\x) = y\(yx) = x for all x, y of L. A loop L is called partially ordered if L is endowed with a partial order (L,≤) such that if x ≤ y then xz ≤ yz, x/z ≤ y/z, zx ≤ zy and z\x ≤ z\y. If ≤ is actually a total order then the loop L is said to be linearly ordered. A loop is said to be pre-ordered if any partial order can be extended to a linear order. In connection with the study of partially ordered nilpotent loops, the question naturally arises when these loupes are pre-ordered. This question was solved for nilpotent groups. E.P. Shimbareva [1] showed that a partially ordered abelian group can be pre-ordered up if it does not contain elements of nite order. However, in [2], A.I. Maltsev proved the theorem on the pre-ordered holds for nilpotent groups as well as for locally nilpotent groups without elements of nite order. In another way, this result was proved by A.H. Rhemtulla [3]