**ON MIDDLE UNIVERSAL WEAK AND CROSS INVERSE PROPERTY LOOPS WITH EQUAL LENGHT OF INVERES CYCLES**

*JAIYEOLA Temitope Gbolahan*

**Abstract: **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. Cross inverse
property loops(CIPLs) need not satisfy this sufficient condition. It
is shown that under this special type of middle isotopism, if $n$ is
a positive even integer, then a WIPL has an inverse cycle of length
$n$ if and only if its isotope is a WIPL with an inverse cycle of
length $n$. But, when $n$ is an odd positive integer. If a loop or
its isotope is a WIPL with only $e$ and inverse cycles of length
$n$, its isotope or the loop is a WIPL with only $e$ and inverse
cycles of length $n$ if and only if they are isomorphic. So, that
both are isomorphic CIPLs. Explanations and procedures are given on
how these results can be used to apply CIPLs to cryptography.

**Category / Keywords: **cross inverse property loops(CIPLs), weak inverse property(WIPLs), inverse cycles

**Publication Info: **Submitted for Publication

**Date: **received 4 Jun 2008

**Contact author: **tjayeola at oauife edu ng

**Available format(s): **PDF | BibTeX Citation

**Note: **Loops are weaker algebraic structures that generalize groups. Every group is a loop but not every loop is a group.

**Version: **20080610:130004 (All versions of this report)

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