arXiv subjects
Publications and source records attributed to Temitope Gbolahan Jaiyeola.
A Code loop on a binary linear code that is doubly even with a factor set is shown to be a central loop, conjugacy closed loop, Burn loop and extra loop. General forms of the identities that define the factor set of a code are deduced.
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.
LC-loops, RC-loops and C-loops are collectively called central loops. It is shown that an LC(RC)-loop is a left(right) universal loop. But an LC(RC)-loop is a universal loop if and only if it is a right(left) universal loop. It is observed that not all RC-loops or LC-loops or C-loops are universal loops. But if an RC-loop(LC-loop, C-loop) is universal, then it is a right Bol loop(left Bol loop, Moufang loop) respectively. If a loop and its right or left isotope are commutative then the loop is a C-loop if and only if its right or left isotope is a C-loop. If a C-loop is central square and its right or left isotope is an alternative central square loop, then the latter is a C-loop. Necessary and sufficient condition for an LC-loop(RC-loop) to be a left(right)G-loop is established. Consequently, necessary and sufficient conditions for an LC-loop, and an RC-loop to be a G-loop are established. A necessary and sufficient condition for a C-loop to be a G-loop is established.
The present study further strenghtens the use of the Keedwell CIPQ against attack on a system. This is done as follows. The holomorphic structure of AIPQs(AIPLs) and CIPQs(CIPLs) are investigated. Necessary and sufficient conditions for the holomorph of a quasigroup(loop) to be an AIPQ(AIPL) or CIPQ(CIPL) are established. It is shown that if the holomorph of a quasigroup(loop) is a AIPQ(AIPL) or CIPQ(CIPL), then the holomorph is isomorphic to the quasigroup(loop). Hence, the holomorph of a quasigroup(loop) is an AIPQ(AIPL) or CIPQ(CIPL) if and only if its automorphism group is trivial and the quasigroup(loop) is a AIPQ(AIPL) or CIPQ(CIPL). Furthermore, it is discovered that if the holomorph of a quasigroup(loop) is a CIPQ(CIPL), then the quasigroup(loop) is a flexible unipotent CIPQ(flexible CIPL of exponent 2). By constructing two isotopic quasigroups(loops) $U$ and $V$ such that their automorphism groups are not trivial, it is shown that $U$ is a AIPQ or CIPQ(AIPL or CIPL) if and only if $V$ is a AIPQ or CIPQ(AIPL or CIPL). Explanations and procedures are given on how these CIPQs can be used to double encrypt information.
This study digs out some new algebraic properties of an Osborn loop that will help in the future to unveil the mystery behind the middle inner mappings $T_{(x)}$ of an Osborn loop. These new algebraic properties, will open our eyes more to the study of Osborn loops like CC-loops which has received a tremendious attention in this $21^\textrm{st}$ and VD-loops whose study is yet to be explored. In this study, some algebraic properties of non-WIP Osborn loops have been investigated in a broad manner. Huthnance was able to deduce some algebraic properties of Osborn loops with the WIP i.e universal weak WIPLs. So this work exempts the WIP. Two new loop identities, namely left self inverse property loop(LSIPL) identity and right self inverse property loop(RSLPL) are introduced for the first time and it is shown that in an Osborn loop, they are equivalent. A CC-loop is shown to be power associative if and only if it is a RSLPL or LSIPL. Among the few identities that have been established for Osborn loops, one of them is recognized and recommended for cryptography in a similar spirit in which the cross inverse property has been used by Keedwell following the fact that it was observed that Osborn loops that do not have the LSIP or RSIP or 3-PAPL or weaker forms of inverse property, power associativity and diassociativity to mention a few, will have cycles(even long ones). These identity is called an Osborn cryptographic identity(or just a cryptographic identity).
A Smarandache quasigroup(loop) is shown to be universal if all its f,g-principal isotopes are Smarandache f,g-principal isotopes. Also, weak Smarandache loops of Bol-Moufang type such as Smarandache: left(right) Bol, Moufang and extra loops are shown to be universal if all their f,g-principal isotopes are Smarandache f,g-principal isotopes. Conversely, it is shown that if these weak Smarandache loops of Bol-Moufang type are universal, then some autotopisms are true in the weak Smarandache sub-loops of the weak Smarandache loops of Bol-Moufang type relative to some Smarandache elements. Futhermore, a Smarandache left(right) inverse property loop in which all its f,g-principal isotopes are Smarandache f,g-principal isotopes is shown to be universal if and only if it is a Smarandache left(right) Bol loop in which all its f,g-principal isotopes are Smarandache f,g-principal isotopes. Also, it is established that a Smarandache inverse property loop in which all its f,g-principal isotopes are Smarandache f,g-principal isotopes is universal if and only if it is a Smarandache Moufang loop in which all its f,g-principal isotopes are Smarandache f,g-principal isotopes. Hence, some of the autotopisms earlier mentioned are found to be true in the Smarandache sub-loops of universal Smarandache: left(right) inverse property loops and inverse property loops.
The idea of left(right) palindromic permutations(LPPs,RPPs) and left(right) generalized Smarandache palindromic permutations(LGSPPs,RGSPPs) are introduced in symmetric groups S_n of degree n. It is shown that in S_n, there exist a LPP and a RPP and they are unique(this fact is demonstrated using S_2 and S_3). The dihedral group D_n is shown to be generated by a RGSPP and a LGSPP(this is observed to be true in S_3) but the geometric interpretations of a RGSPP and a LGSPP are found not to be rotation and reflection respectively. In S_3, each permutation is at least a RGSPP or a LGSPP. There are 4 RGSPPs and 4 LGSPPs in S_3, while 2 permutations are both RGSPPs and LGSPPs. A permutation in S_n is shown to be a LPP or RPP(LGSPP or RGSPP) if and only if its inverse is a LPP or RPP(LGSPP or RGSPP) respectively. Problems for future studies are raised.
The existence of A$_ρ$-loops, A$_λ$-loops and A$_μ$-loops that are neither extra loops nor CC-loops such that any two of their inner mappings $R(x,y),L(x,y)$ and $T(x)$ commute while the other one is of order 2 is shown.
Every quasigroup $(L,\cdot)$ belongs to a set of 6 quasigroups, called parastrophes denoted by $(L,π_i)$, $i\in \{1,2,3,4,5,6\}$. It is shown that $(L,π_i)$ is a Smarandache quasigroup with associative subquasigroup $(S,π_i) \forall i\in \{1,2,3,4,5,6\}$ if and only if for any of some four $j\in \{1,2,3,4,5,6\}$, $(S,π_j)$ is an isotope of $(S,π_i)$ or $(S,π_k)$ for one $k\in \{1,2,3,4,5,6\}$ such that $i\ne j\ne k$. Hence, $(L,π_i)$ is a Smarandache quasigroup with associative subquasigroup $(S,π_i) \forall i\in \{1,2,3,4,5,6\}$ if and only if any of the six Khalil conditions is true for any of some four of $(S,π_i)$.
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.
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.
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.
If two loops are isomorphic, then it is shown that their holomorphs are also isomorphic. Conversely, it is shown that if their holomorphs are isomorphic, then the loops are isotopic. It is shown that a loop is a Smarandache loop if and only if its holomorph is a Smarandache loop. This statement is also shown to be true for some weak Smarandache loops(inverse property, weak inverse property) but false for others(conjugacy closed, Bol, central, extra, Burn, A-, homogeneous) except if their holomorphs are nuclear or central. A necessary and sufficient condition for the Nuclear-holomorph of a Smarandache Bol loop to be a Smarandache Bruck loop is shown. Whence, it is found also to be a Smarandache Kikkawa loop if in addition the loop is a Smarandache A-loop with a centrum holomorph. Under this same necessary and sufficient condition, the Central-holomorph of a Smarandache A-loop is shown to be a Smarandache K-loop.