Weak Inverse Property Loops and Some Isotopy-Isomorphy Properties

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 $\mathcal{T}$ condition, Artzy’s result that isotopic cross inverse property loops(CIPL) are isomorphic is proved for WIP loops.