Abstract

TEMITOPE GBOLAHAN, JAIYEOLA. On Middle Universal Weak and Cross Inverse Property Loops with Equal Length of Inverse Cycles. Rev.colomb.mat. [online]. 2010, vol.44, n.2, pp.79-89. ISSN 0034-7426.

This study presents a special type of middle isotopism under which the weak inverse property (WIP) is isotopic invariant in loops. A sufficient condition for a WIPL that is specially isotopic to a loop to be isomorphic to the loop isotope is established. It is shown that under this special type of middle isotopism, whenever n is a positive even integer, a finite WIPL has an inverse cycle of length n if and only if its isotope is a finite WIPL with an inverse cycle of length n. But, when n is an odd positive integer and a loop (or its isotope) is a finite WIPL with only e and inverse cycles of length n, then its isotope (or the loop) is a finite WIPL with only e and inverse cycles of length n if and only if they are isomorphic. Hence, both are isomorphic CIPLs. Explanations and procedures are given on how these results can be used to apply CIPLs to cryptography.

Keywords : Cross inverse property loops (CIPLs); Weak inverse property loops (WIPLs); Inverse cycles.

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