Inverse-cycles in weak-inverse loops

Abstract:

In a loop let J be the map taking every element into its right inverse. If the order of J is n, then $\{ x,xJ, \ldots ,x{J^{n - 1}}\}$ is called an inverse-cycle of length n. If a weak-inverse loop consists only of the unit element and m inverse-cycles of length n, then n divides 2m.