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Temitope Gbolahan Jaiyeola

Publications and source records attributed to Temitope Gbolahan Jaiyeola.

At least 19 recordsLinked to original sources
arXiv2022

Parastrophes and Cosets of Soft Quasigroups

This paper introduced the concept of soft quasigroup, its parastrophes, soft nuclei, left (right) coset, distributive soft quasigroups and normal soft quasigroups. Necessary and sufficient conditions for a soft set over a quasigroup (loop) to be a soft quasigroup (loop) were established. It was proved that a soft set over a group is a soft group if and only if it is a soft loop or either of two of its parastrophes is a soft groupoid. For a finite quasigroup, it was shown that the orders (arithmetic and geometric means) of the soft quasigroup over it and its parastrophes are equal. It was also proved that if a soft quasigroup is distributive, then all its parastrophes are distributive, idempotent and flexible soft quasigroups. For a distributive soft quasigroup, it was shown that its left and right cosets form families of distributive soft quasigroups that are isomorphic. If in addition, a soft quasigroup is normal, then its left and right cosets forms families of normal soft quasigroups. On another hand, it was found that if a soft quasigroup is a normal and distributive soft quasigroup, then its left (right) quotient is a family of commutative distributive quasigroups which have a 1-1 correspondence with the left (right) coset of the soft quasigroup.

math.GR↗
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↗
arXiv2019

On Holomorphy of Fenyves BCI-Algebras

Fenyves BCI-algebras are BCI-algebras that satisfy the Bol-Moufang identities. In this paper, the holomorphy of BCI-algebras are studied. It is shown that whenever a loop and its holomorph are BCI-algebras, the former is p-semisimple if and only if the latter is p-semisimple. Whenever a loop and its holomorph are BCI-algebras, it is established that the former is a BCK-algebra if and only if the latter has a BCK-subalgebra. Moreover, the holomorphy of the associative and some non-associative Fenyves BCI-algebras are also studied.

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↗
arXiv2014

Varieties of groupoids and quasigroups generated by linear-bivariate polynomials over ring Z_n

Some varieties of groupoids and quasigroups generated by linear-bivariate polynomials $P(x,y)=a+bx+cy$ over the ring $\mathbb{Z}_n$ are studied. Necessary and sufficient conditions for such groupoids and quasigroups to obey identities which involve one, two, three (e.g. Bol-Moufang type) and four variables w.r.t. $a$, $b$ and $c$ are established. Necessary and sufficient conditions for such groupoids and quasigroups to obey some inverse properties w.r.t. $a$, $b$ and $c$ are also established. This class of groupoids and quasigroups are found to belong to some varieties of groupoids and quasigroups such as medial groupoid(quasigroup), F-quasigroup, semi automorphic inverse property groupoid(quasigroup) and automorphic inverse property groupoid(quasigroup).

math.GR↗
arXiv2014

Some simplicial complexes of universal Osborn loops

A loop is shown to be a universal Osborn loop if and only if it has a particular simplicial complex. A loop is shown to be a universal Osborn loop and obeys two new identities if and only if it has another particular simplicial complex. A universal Osborn loop and four of its isotopes are shown to form a rectangular pyramid in a $3$-dimensional space.

math.GR↗
arXiv2010

On a Pair of Universal Weak Inverse Property Loops

A new condition called ${\cal T}$ condition is introduced for the first time and used to study a pair of isotopic loops. Under this condition, a loop in the pair is a WIPL if and only if the other loop is a WIPL. Furthermore, such WIPLs are isomorphic. The translation elements $f$ and $g$ of a CIPL with the ${\cal T}$ condition(such that its $f,g$-isotope is an automorphic inverse property loop) are found to be alternative, flexible, centrum and equal elements. A necessary and sufficient condition for a pair WIPLs with a weak ${\cal T}$ condition to be isomorphic is shown. A CIPL and an isomorph are observed to have this weak ${\cal T}$ condition.

math.GM↗
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↗
arXiv2010

Basic Properties Of Second Smarandache Bol Loops

The pair $(G_H,\cdot)$ is called a special loop if $(G,\cdot)$ is a loop with an arbitrary subloop $(H,\cdot)$. A special loop $(G_H,\cdot)$ is called a second Smarandache Bol loop(S$_{2^{\tiny\textrm{nd}}}$BL) if and only if it obeys the second Smarandache Bol identity $(xs\cdot z)s=x(sz\cdot s)$ for all $x,z$ in $G$ and $s$ in $H$. The popularly known and well studied class of loops called Bol loops fall into this class and so S$_{2^{\tiny\textrm{nd}}}$BLs generalize Bol loops. The basic properties of S$_{2^{\tiny\textrm{nd}}}$BLs are studied. These properties are all Smarandache in nature. The results in this work generalize the basic properties of Bol loops, found in the Ph.D. thesis of D. A. Robinson. Some questions for further studies are raised.

math.GM↗
arXiv2009

New Identities In Universal Osborn Loops

A question associated with the 2005 open problem of Michael Kinyon (Is every Osborn loop universal?), is answered. Two nice identities that characterize universal (left and right universal) Osborn loops are established. Numerous new identities are established for universal (left and right universal) Osborn loops like CC-loops, VD-loops and universal weak inverse property loops. Particularly, Moufang loops are discovered to obey the new identity $[y(x^{-1}u)\cdot u^{-1}](xu)=[y(xu)\cdot u^{-1}](x^{-1}u)$ surprisingly. For the first time, new loop properties that are weaker forms of well known loop properties like inverse property, power associativity and diassociativity are introduced and studied in universal (left and right universal) Osborn loops. Some of them are found to be necessary and sufficient conditions for a universal Osborn to be 3 power associative. For instance, four of them are found to be new necessary and sufficient conditions for a CC-loop to be power associative. A conjugacy closed loop is shown to be diassociative if and only if it is power associative and has a weak form of diassociativity.

math.GM↗
arXiv2009

Some necessary and sufficient conditions for parastrophic invariance of the associative law in quasigroups

Every quasigroup $(S,\cdot)$ belongs to a set of 6 quasigroups, called parastrophes denoted by $(S,π_i)$, $i\in \{1,2,3,4,5,6\}$. It is shown that isotopy-isomorphy is a necessary and sufficient condition for any two distinct quasigroups $(S,π_i)$ and $(S,π_j)$, $i,j\in \{1,2,3,4,5,6\}$ to be parastrophic invariant relative to the associative law. In addition, a necessary and sufficient condition for any two distinct quasigroups $(S,π_i)$ and $(S,π_j)$, $i,j\in \{1,2,3,4,5,6\}$ to be parastrophic invariance under the associative law is either if the $π_i$-parastrophe of $H$ is equivalent to the $π_i$-parastrophe of the holomorph of the $π_i$-parastrophe of $S$ or if the $π_i$-parastrophe of $H$ is equivalent to the $π_k$-parastrophe of the $π_i$-parastrophe of the holomorph of the $π_i$-parastrophe of $S$, for a particular $k\in \{1,2,3,4,5,6\}$.

math.GM↗
arXiv2009

A Double Cryptography Using The Smarandache Keedwell Cross Inverse Quasigroup

The present study further strengthens the use of the Keedwell CIPQ against attack on a system by the use of the Smarandache Keedwell CIPQ for cryptography in a similar spirit in which the cross inverse property has been used by Keedwell. This is done as follows. By constructing two S-isotopic S-quasigroups(loops) $U$ and $V$ such that their Smarandache automorphism groups are not trivial, it is shown that $U$ is a SCIPQ(SCIPL) if and only if $V$ is a SCIPQ(SCIPL). Explanations and procedures are given on how these SCIPQs can be used to double encrypt information.

math.GM↗
arXiv2008

Smarandache Isotopy Theory Of Smarandache: Quasigroups And Loops

The concept of Smarandache isotopy is introduced and its study is explored for Smarandache: groupoids, quasigroups and loops just like the study of isotopy theory was carried out for groupoids, quasigroups and loops. The exploration includes: Smarandache; isotopy and isomorphy classes, Smarandache $f,g$ principal isotopes and G-Smarandache loops.

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

A Pair of Smarandachely Isotopic Quasigroups and Loops of the Same Variety

The isotopic invariance or universality of types and varieties of quasigroups and loops described by one or more equivalent identities has been of interest to researchers in loop theory in the recent past. A variety of quasigroups(loops) that are not universal have been found to be isotopic invariant relative to a special type of isotopism or the other. Presently, there are two outstanding open problems on universality of loops: semi automorphic inverse property loops(1999) and Osborn loops(2005). Smarandache isotopism(S-isotopism) was originally introduced by Vasantha Kandasamy in 2002. But in this work, the concept is re-restructured in order to make it more explorable. As a result of this, the theory of Smarandache isotopy inherits the open problems as highlighted above for isotopy. In this short note, the question 'Under what type of S-isotopism will a pair of S-quasigroups(S-loops) form any variety?' is answered by presenting a pair of specially S-isotopic S-quasigroups(loops) that both belong to the same variety of S-quasigroups(S-loops). This is important because pairs of specially S-isotopic S-quasigroups(e.g Smarandache cross inverse property quasigroups) that are of the same variety are useful for applications(e.g cryptography).

math.GM↗
arXiv2008

An holomorphic study of Smarandache automorphic and cross inverse ploperty loops

By studying the holomorphic structure of automorphic inverse property quasigroups and loops[AIPQ and (AIPL)] and cross inverse property quasigroups and loops[CIPQ and (CIPL)], it is established that the holomorph of a loop is a Smarandache; AIPL, CIPL, K-loop, Bruck-loop or Kikkawa-loop if and only if its Smarandache automorphism group is trivial and the loop is itself is a Smarandache; AIPL, CIPL, K-loop, Bruck-loop or Kikkawa-loop.

math.GM↗
arXiv2008

On Smarandache Bryant Schneider group of a Smarandache loop

The concept of Smarandache Bryant Schneider Group of a Smarandache loop is introduced. Relationship(s) between the Bryant Schneider Group and the Smarandache Bryant Schneider Group of an S-loop are discovered and the later is found to be useful in finding Smarandache isotopy-isomorphy condition(s) in S-loops just like the formal is useful in finding isotopy-isomorphy condition(s) in loops. Some properties of the Bryant Schneider Group of a loop are shown to be true for the Smarandache Bryant Schneider Group of a Smarandache loop. Some interesting and useful cardinality formulas are also established for a type of finite Smarandache loop.

math.GM↗