On a Pair of Universal Weak Inverse Property Loops

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A new condition called 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 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 T condition to be isomorphic is shown. A CIPL and an isomorph are observed to have this weak T condition.