Weak Inverse Property Loops and Some Isotopy-Isomorphy Properties

TL;DR

This paper explores new isotopy-isomorphy conditions for weak inverse property loops, establishing when isotopic loops are also isomorphic, and extends Artzy's results to this class of loops.

Contribution

It introduces two new isotopy-isomorphy conditions for WIPLs, one characterizing and the other sufficient for isotopy-isomorphy, and proves isomorphism under the ${ m T}$ condition.

Findings

One condition characterizes isotopy-isomorphy in WIPLs.

The other condition is sufficient for isotopy-isomorphy.

Isotopic cross inverse property loops are isomorphic under the ${ m T}$ condition.

Abstract

Two distinct isotopy-isomorphy conditions, different from those of J. M. Osborn and Wilson's condition, for a weak inverse property loop(WIPL) are shown. Only one of them characterizes isotopy-isomorphy in WIPLs while the other is just a sufficient condition for isotopy-isomorphy. Under the sufficient condition called the ${\cal T}$ condition, Artzy's result that isotopic cross inverse property loops(CIPL) are isomorphic is proved for WIP loops.