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John Olushola Adeniran

Publications and source records attributed to John Olushola Adeniran.

9 recordsLinked to original sources
arXiv2020

On Another Two Cryptographic Identities In Universal Osborn Loops

In this study, by establishing an identity for universal Osborn loops, two other identities(of degrees 4 and 6) are deduced from it and they are recognized and recommended for cryptography in a similar spirit in which the cross inverse property(of degree 2) has been used by Keedwell following the fact that it was observed that universal Osborn loops that do not have the 3-power associative property or weaker forms of; inverse property, power associativity and diassociativity to mention a few, will have cycles(even long ones). These identities are found to be cryptographic in nature for universal Osborn loops and thereby called cryptographic identities. They were also found applicable to security patterns, arrangements and networks which the CIP may not be applicable to.

math.GR↗
arXiv2017

Holomorphy of Osborn loops

Let $(L,\cdot)$ be any loop and let $A(L)$ be a group of automorphisms of $(L,\cdot)$ such that $α$ and $ϕ$ are elements of $A(L)$. It is shown that, for all $x,y,z\in L$, the $A(L)$-holomorph $(H,\circ)=H(L)$ of $(L,\cdot)$ is an Osborn loop if and only if $xα(yz\cdot xϕ^{-1})= xα(yx^λ\cdot x) \cdot zxϕ^{-1}$. Furthermore, it is shown that for all $x\in L$, $H(L)$ is an Osborn loop if and only if $(L,\cdot)$ is an Osborn loop, $(xα\cdot x^ρ)x=xα$, $x(x^λ\cdot xϕ^{-1})=xϕ^{-1}$ and every pair of automorphisms in $A(L)$ is nuclear (i.e. $xα\cdot x^ρ,x^λ\cdot xϕ\in N(L,\cdot )$). It is shown that if $H(L)$ is an Osborn loop, then $A(L,\cdot)= \mathcal{P}(L,\cdot)\capΛ(L,\cdot)\capΦ(L,\cdot)\capΨ(L,\cdot)$ and for any $α\in A(L)$, $α= L_{eπ}=R^{-1}_{e\varrho}$ for some $π\in Φ(L,\cdot)$ and some $\varrho\in Ψ(L,\cdot)$. Some commutative diagrams are deduced by considering isomorphisms among the various groups of regular bijections (whose intersection is $A(L)$) and the nucleus of $(L,\cdot)$.

math.GR↗
arXiv2010

Not Every Osborn Loop Is Universal

Michael Kinyon's 2005 open problem, based on the universality of Osborn loops is solved. It is shown that not every Osborn loop is universal.

math.GM↗
arXiv2008

On Some Autotopisms Of Non-Steiner Central Loops

An algebraic process for the construction of an autotopism for a non-Steiner C-loop is described and this is demonstrated with an example using a known finite C-loop. In every C-loop, two of its parastrophes are not equivalent(equal) it, if and only if both the first and second components of the constructed autotopism and its inverse autotopism are not equal to the identity map. Hence, none of the other three parastrophes is equivalent(equal) to the C-loop. It is proved that the set of autotopisms that prevent a C-loop from being a Steiner loop forms a Steiner triple system.

math.GM↗
arXiv2008

Algebraic properties of some varieties of central loops

Isotopes of C-loops with unique non-identity squares are shown to be both C-loops and A-loops. The relationship between C-loops and Steiner loops is further studied. Central loops with the weak and cross inverse properties are also investigated. C-loops are found to be Osborn loops if every element in them are squares.

math.GM↗
arXiv2007

On the Derivatives of Central Loops

The right(left) derivative, $a^{-1},e-$ and $e,a^{-1}-$ isotopes of a C-loop are shown to be C-loops. Furthermore, for a central loop $(L,F)$, it is shown that $\big\{F,F^{a^{-1}},F_{a^{-1},e}\big\}$ and $\big\{F,F_{a^{-1}},F_{e,a^{-1}}\big\}$ are systems of isotopic C-loops that obey a form of generalized distributive law. Quasigroup isotopes $(L,\otimes)$ and $(L,\ominus)$ of a loop $(L,θ)$ and its parastrophe $(L,θ^*)$ respectively are proved to be isotopic if either $(L,\otimes)$ or $(L,\ominus )$ is commutative. If $(L,θ)$ is a C-loop, then it is shown that $\big\{(L,θ),(L,θ^*),(L,\otimes),(L,\oplus)\big\}$ is a system of isotopic C-quasigroup under the above mentioned condition. It is shown that C-loops are isotopic to some finite indecomposable groups of the classes ${\cal D}_i,i=1,2,3,4,5$ and that the center of such C-loops have a rank of 1,2 or 3.

math.GM↗
arXiv2007

On Isotopic Characterization of Central Loops

The representation sets of central loops are investigated and the results obtained are used to construct a finite C-loop. It is shown that for certain types of isotopisms, the central identities are isotopic invariant.

math.GM↗
arXiv2007

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

math.GR↗