On a Pair of Universal Weak Inverse Property Loops

TL;DR

This paper introduces the ${ m T}$ condition to analyze pairs of isotopic loops, showing that under this condition, weak inverse property loops (WIPLs) are characterized by isomorphism and specific element properties.

Contribution

It establishes a new ${ m T}$ condition framework for studying isotopic loops and characterizes when WIPLs are isomorphic under this condition.

Findings

WIPLs in a pair are isomorphic under the ${ m T}$ condition.

Translation elements in certain loops are alternative, flexible, and central.

Necessary and sufficient conditions for isomorphism of WIPLs with weak ${ m T}$ condition.

Abstract

A new condition called ${\cal T}$ condition is introduced for the first time and used to study a pair of isotopic loops. Under this condition, a loop in the pair is a WIPL if and only if the other loop is a WIPL. Furthermore, such WIPLs are isomorphic. The translation elements $f$ and $g$ of a CIPL with the ${\cal T}$ condition(such that its $f,g$-isotope is an automorphic inverse property loop) are found to be alternative, flexible, centrum and equal elements. A necessary and sufficient condition for a pair WIPLs with a weak ${\cal T}$ condition to be isomorphic is shown. A CIPL and an isomorph are observed to have this weak ${\cal T}$ condition.