Jaiyeola Temitope Gbolahan
@oauife.edu.ng
Professor of Mathematics, Faculty of Science
Obafemi Awolowo University
RESEARCH, TEACHING, or OTHER INTERESTS
Algebra and Number Theory, Discrete Mathematics and Combinatorics
Scopus Publications
Scholar Citations
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Scopus Publications
Adewale Roland Tunde Sòlárìn, John Olusola Adéníran, Tèmítópé Gbóláhàn Jaiyéọlá, Abednego Orobosa Isere, and Yakub Tunde Oyebo
Springer International Publishing
Y. T. Oyebo, T. G. Jaiyéọlá, and J. O. Adéníran
Informa UK Limited
J. O. Adéníran, T. G. Jaiyéọlá, and K. A. Ìdòwú
Universidad Catolica del Norte - Chile
In this study, the notion of isotopy of generalized Bol loop is characterized. A loop isotope of a σ-generalized Bol loop is shown to be a σ’-generalized Bol loop if σ’ fixes its (isotope) identity element where σ’ is some conjugate of σ. A loop isotope of a σ-generalized Bol loop is shown to be a σ’-generalized Bol loop if and only if the image of the isotope’s identity element under σ’ is right nuclear (where σ’ is some conjugate of σ). It is shown that a generalized Bol loop can be constructed using a group and a subgroup of it. A right conjugacy closed σ-generalized Bol loop is shown to be a σ-generalized right central loop.
Olumuyiwa S. Asaolu, Temitope G. Jaiyeola, Mojisola R. Usikalu, Ezra Gayawan, Olubunmi Atolani, and Oluyomi S. Adeyemi
Elsevier BV
Temitope Gbolahan Jaiyeola, J.O. Olaleru, and A. Oyem
Institute of Mathematics, University of Zielona Gora, Poland
Abstract In this work, a soft set (F, A) was introduced over a quasigroup (Q,) and the study of finite soft quasigroup was carried out, motivated by the study of algebraic structures of soft sets. By introducing the order of a finite soft quasigroup, various inequality relationships that exist between the order of a finite quasigroup, the order of its soft quasigroup and the cardinality of its set of parameters were established. By introducing the arithmetic mean 𝒜ℱ(F, A) and geometric mean 𝒢ℱ(F, A) of a finite soft quasigroup (F, A), a sort of Lagrange’s Formula |(F, A)| = |A|𝒜ℱ(F, A) for finite soft quasigroup was gotten. Some of the inequalities gotten gave an upper bound for the order of a finite soft quasigroup in terms of the order of its quasigroup and cardinality of its set of parameters, and a lower bound for the order of the quasigroup in terms of the arithmetic mean of the finite soft quasigroup. A chain of inequalities called the Maclaurin’s inequality for any finite soft quasigroup (F, A)(Q,·) was shown to exist. A necessary and sufficient condition for a type of finite soft quasigroup to be extensible to a finite super soft quasigroup was established. This result is of practical use whenever a larger set of parameters is required. The results therein were illustrated with examples. Application to uniformity, equality and equity in distribution for social living is considered.
George Olufemi Olakunle and Tèmítòpé Gbóláhàn Jaíyéolá
Vladimir Andrunachievici Institute of Mathematics and Computer Science
In this work, we discovered a dozen of new loop identities we called identities of 'second Bol-Moufang type'. This was achieved by using a generalized and modified nuclear identification model originally introduced by Dr\\'{a}pal and Jedli\\u{c}ka. Among these twelve identities, eight of them were found to be distinct (from well known loop identities), among which two pairs axiomatize the weak inverse property power associative conjugacy closed (WIP PACC) loop. The four other new loop identities individually characterize the Moufang identities in loops. Thus, now we have eight loop identities that characterize Moufang loops. We also discovered two (equivalent) identities that describe two varieties of Buchsteiner loops. In all, only the extra identities which the Dr\\'{a}pal and Jedli\\u{c}ka nuclear identification model tracked down could not be tracked down by our own nuclear identification model. The dozen laws $\\{Q_i\\}_{i=1}^{12}$ induced by our nuclear identification form four cycles in the following sequential format: $\\big(Q_{4i-j}\\big)_{i=1}^3,~j=0,1,2,3,$ and also form six pairs of dual identities. With the help of twisted nuclear identification, we discovered six identities of lengths five that describe the abelian group variety and commutative Moufang loop variety (in each case). The second dozen identities $\\{Q_i^*\\}_{i=1}^{12}$ induced by our twisted nuclear identification were also found to form six pairs of dual identities. Some examples of loops of smallest order that obey non-Moufang laws (which do not necessarily imply the other) among the dozen laws $\\{Q_i\\}_{i=1}^{12}$ were found.
Tèmítòpé Gbóláhàn Jaíyéolá, Benard Osoba, and Anthony Oyem
Vladimir Andrunachievici Institute of Mathematics and Computer Science
In the recent past, Grecu and Syrbu (in no order of preference) have jointly and individually reported some results on isostrophy invariants of Bol loops. Also, the Bryant-Schneider group of a loop has been found important in the study of the isotopy-isomorphy of some varieties of loops (e.g. Bol loops, Moufang loops, Osborn loops). In this current work, the Bryant-Schneider group of a middle Bol loop was linked with some of the isostrophy-group invariance results of Grecu and Syrbu. In particular, it was shown that some subgroups of the Bryant-Schneider group of a middle Bol loop are equal (or isomorphic) to the automorphism and pseudo-aumorphism groups of its corresponding right (left) Bol loop. Some elements of the Bryant-Schneider group of a middle Bol loop were shown to induce automorphisms and middle pseudo-automorphisms. It was discovered that if a middle Bol loop is of exponent 2, then, its corresponding right (left) Bol loop is a left (right) G-loop.
O.O. George, J.O. Olaleru, J.O. Adénı́ran, and T.G. Jaiyéolá
Universidad de Extremadura - Servicio de Publicaciones
Let LWPC denote the identity (xy · x) · xz = x((yx · x)z), and RWPC the mirror identity. Phillips proved that a loop satisfies LWPC and RWPC if and only if it is a WIP PACC loop. Here, it is proved that a loop Q fulfils LWPC if and only if it is a left conjugacy closed (LCC) loop that fulfils the identity (xy · x)x = x(yx · x). Similarly, RWPC is equivalent to RCC and x(x · yx) = (x · xy)x. If a loop satisfies LWPC or RWPC, then it is power associative (PA). The smallest nonassociative LWPC-loop was found to be unique and of order 6 while there are exactly 6 nonassociative LWPC-loops of order 8 up to isomorphism. Methods of construction of nonassociative LWPC-loops were developed.
Temitope Jaiyéolá and Gideon Effiong
Universidad Catolica del Norte - Chile
A loop (Q; ·) is called a Basarab loop if the identities: (x · yxρ)(xz) = x · yz and (yx) · (xλz · x) = yz · x hold. It was shown that the left, right and middle nuclei of the Basarab loop coincide, and the nucleus of a Basarab loop is the set of elements x whose middle inner mapping Tx are automorphisms. The generators of the inner mapping group of a Basarab loop were refined in terms of one of the generators of the total inner mapping group of a Basarab loop. Necessary and su_cient condition(s) in terms of the inner mapping group (associators) for a loop to be a Basarab loop were established. It was discovered that in a Basarab loop: the mapping x ↦ Tx is an endomorphism if and only if the left (right) inner mapping is a left (right) regular mapping. It was established that a Basarab loop is a left and right automorphic loop and that the left and right inner mappings belong to its middle inner mapping group. A Basarab loop was shown to be an automorphic loop (A-loop) if and only if it is a middle automorphic loop (middle A-loop). Some interesting relations involving the generators of the total multiplication group and total inner mapping group of a Basarab loop were derived, and based on these, the generators of the total inner mapping group of a Basarab loop were finetuned. A Basarab loop was shown to be a totally automorphic loop (TA-loop) if and only if it is a commutative and exible loop. These aforementioned results were used to give a partial answer to a 2013 question and an ostensible solution to a 2015 problem in the case of Basarab loop.
Abednego Orobosa Isere, John Olúsolá Adéníran, and Temitópé Gbóláhán Jaiyéolá
Brno University of Technology
This work investigated some properties of Latin quandles that are applicable in cryptography. Four distinct cores of an Osborn loop (non-diassociative and non-power associative) were introduced and investigated. The necessary and sufficient conditions for these cores to be (i) (left) quandles (ii) involutory quandles (iii) quasi-Latin quandles and (iv) involutory quasi-Latin quandles were established. These conditions were judiciously used to build cipher algorithms for cryptography in some peculiar circumstances.
T. G. Jaiyéolá, S. P. David, and O. O. Oyebola
Universidad Catolica del Norte - Chile
A loop (Q, ·, \\, /) is called a middle Bol loop (MBL) if it obeys the identity x(yz\\x)=(x/z)(y\\x). To every MBL corresponds a right Bol loop (RBL) and a left Bol loop (LBL). In this paper, some new algebraic properties of a middle Bol loop are established in a different style. Some new methods of constructing a MBL by using a non-abelian group, the holomorph of a right Bol loop and a ring are described. Some equivalent necessary and sufficient conditions for a right (left) Bol loop to be a middle Bol loop are established. A RBL (MBL, LBL, MBL) is shown to be a MBL (RBL, MBL, LBL) if and only if it is a Moufang loop.
Xiaohong Zhang, Xuejiao Wang, Florentin Smarandache, Tèmítópé Gbóláhàn Jaíyéolá, and Tieyan Lian
Elsevier BV
Abstract Neutrosophic extended triplet group (NETG) is an interesting extension of the concept of classical group, which can be used to express general symmetry. This paper further studies the structural characterizations of NETG. First, some examples are given to show that some results in literature are false. Second, the differences between generalized groups and neutrosophic extended triplet groups are investigated in detail. Third, the notion of singular neutrosophic extended triplet group (SNETG) is introduced, and some homomorphism properties are discussed and a Lagrange-like theorem for finite SNETG is proved. Finally, the following important result is proved: a semigroup is a singular neutrosophic extended triplet group (SNETG) if and only if it is a generalized group.
Abednego O. Isere, J. O. Adéniran, and T. G. Jaiyéolá
Universidad Catolica del Norte - Chile
The smallest non-associative Osborn loop is of order 16. Attempts in the past to construct higher orders have been very difficult. In this paper, some examples of finite Osborn loops of order 4n, n = 4, 6, 8, 9, 12, 16 and 18 were presented. The orders of certain elements of the examples were considered. The nuclei of two of the examples were also obtained and these were used to establish the classification of these Osborn loops up to isomorphism. Moreover, the central properties of these examples were examined and were all found to be having a trivial center and no non-trivial normal subloop. Therefore, these examples of Osborn loops are simple Osborn loops.
O. Atolani, O.S. Adeyemi, F.O. Agunbiade, O.S. Asaolu, E. Gayawan, T.G. Jaiyeola, M.R. Usikalu, and E. I. Unuabonah
Sciendo
Tèmítópé Jaíyéolá and Florentin Smarandache
MDPI AG
This article is based on new developments on a neutrosophic triplet group (NTG) and applications earlier introduced in 2016 by Smarandache and Ali. NTG sprang up from neutrosophic triplet set X: a collection of triplets ( b , n e u t ( b ) , a n t i ( b ) ) for an b ∈ X that obeys certain axioms (existence of neutral(s) and opposite(s)). Some results that are true in classical groups were investigated in NTG and were shown to be either universally true in NTG or true in some peculiar types of NTG. Distinguishing features between an NTG and some other algebraic structures such as: generalized group (GG), quasigroup, loop and group were investigated. Some neutrosophic triplet subgroups (NTSGs) of a neutrosophic triplet group were studied. In particular, for any arbitrarily fixed a ∈ X , the subsets X a = { b ∈ X : n e u t ( b ) = n e u t ( a ) } and ker f a = { b ∈ X | f ( b ) = n e u t ( f ( a ) ) } of X, where f : X → Y is a neutrosophic triplet group homomorphism, were shown to be NTSG and normal NTSG, respectively. Both X a and ker f a were shown to be a-normal NTSGs and found to partition X. Consequently, a Lagrange-like formula was found for a finite NTG X ; | X | = ∑ a ∈ X [ X a : ker f a ] | ker f a | based on the fact that | ker f a | | | X a | . The first isomorphism theorem X / ker f ≅ Im f was established for NTGs. Using an arbitrary non-abelian NTG X and its NTSG X a , a Bol structure was constructed. Applications of the neutrosophic triplet set, and our results on NTG in relation to management and sports, are highlighted and discussed.
Temitope Jaiyeola and Florentin Smarandache
MDPI AG
This paper is the first study of the neutrosophic triplet loop (NTL) which was originally introduced by Floretin Smarandache. NTL originated from the neutrosophic triplet set X: a collection of triplets ( x , n e u t ( x ) , a n t i ( x ) ) for an x ∈ X which obeys some axioms (existence of neutral(s) and opposite(s)). NTL can be informally said to be a neutrosophic triplet group that is not associative. That is, a neutrosophic triplet group is an NTL that is associative. In this study, NTL with inverse properties such as: right inverse property (RIP), left inverse property (LIP), right cross inverse property (RCIP), left cross inverse property (LCIP), right weak inverse property (RWIP), left weak inverse property (LWIP), automorphic inverse property (AIP), and anti-automorphic inverse property are introduced and studied. The research was carried out with the following assumptions: the inverse property (IP) is the RIP and LIP, cross inverse property (CIP) is the RCIP and LCIP, weak inverse property (WIP) is the RWIP and LWIP. The algebraic properties of neutrality and opposite in the aforementioned inverse property NTLs were investigated, and they were found to share some properties with the neutrosophic triplet group. The following were established: (1) In a CIPNTL (IPNTL), RIP (RCIP) and LIP (LCIP) were equivalent; (2) In an RIPNTL (LIPNTL), the CIP was equivalent to commutativity; (3) In a commutative NTL, the RIP, LIP, RCIP, and LCIP were found to be equivalent; (4) In an NTL, IP implied anti-automorphic inverse property and WIP, RCIP implied AIP and RWIP, while LCIP implied AIP and LWIP; (5) An NTL has the IP (CIP) if and only if it has the WIP and anti-automorphic inverse property (AIP); (6) A CIPNTL or an IPNTL was a quasigroup; (7) An LWIPNTL (RWIPNTL) was a left (right) quasigroup. The algebraic behaviours of an element, its neutral and opposite in the associator and commutator of a CIPNTL or an IPNTL were investigated. It was shown that ( Z p , ∗ ) where x ∗ y = ( p − 1 ) ( x + y ) , for any prime p, is a non-associative commutative CIPNTL and IPNTL. The application of some of these varieties of inverse property NTLs to cryptography is discussed.
Tèmítọ́pẹ́ Jaíyéọlá, Emmanuel Ilojide, Memudu Olatinwo, and Florentin Smarandache
MDPI AG
In this paper, Bol-Moufang types of a particular quasi neutrosophic triplet loop (BCI-algebra), chritened Fenyves BCI-algebras are introduced and studied. 60 Fenyves BCI-algebras are introduced and classified. Amongst these 60 classes of algebras, 46 are found to be associative and 14 are found to be non-associative. The 46 associative algebras are shown to be Boolean groups. Moreover, necessary and sufficient conditions for 13 non-associative algebras to be associative are also obtained: p-semisimplicity is found to be necessary and sufficient for a F 3 , F 5 , F 42 and F 55 algebras to be associative while quasi-associativity is found to be necessary and sufficient for F 19 , F 52 , F 56 and F 59 algebras to be associative. Two pairs of the 14 non-associative algebras are found to be equivalent to associativity ( F 52 and F 55 , and F 55 and F 59 ). Every BCI-algebra is naturally an F 54 BCI-algebra. The work is concluded with recommendations based on comparison between the behaviour of identities of Bol-Moufang (Fenyves’ identities) in quasigroups and loops and their behaviour in BCI-algebra. It is concluded that results of this work are an initiation into the study of the classification of finite Fenyves’ quasi neutrosophic triplet loops (FQNTLs) just like various types of finite loops have been classified. This research work has opened a new area of research finding in BCI-algebras, vis-a-vis the emergence of 540 varieties of Bol-Moufang type quasi neutrosophic triplet loops. A ‘Cycle of Algebraic Structures’ which portrays this fact is provided.
T. G. Jaiyéọlá, A. A. Adeniregun, and M. A. Asiru
World Scientific Pub Co Pte Lt
A loop [Formula: see text] is called a FRUTE loop if it obeys the identity [Formula: see text]. Interestingly, a FRUTE loop is a Moufang loop but not necessarily an extra loop or a group (and vice versa). In this paper, algebraic properties of the left (right) regular representation set of a FRUTE loop are deduced. A FRUTE loop is shown to be universal and an [Formula: see text]-loop for all [Formula: see text]. A Moufang loop is shown to be a FRUTE loop if and only if it is nuclear cube if and only if it is an [Formula: see text]-loop. It is established that: the smallest, associative, non-commutative FRUTE loop is of order [Formula: see text] (the quaternion group [Formula: see text]); for any [Formula: see text], there exists at least a non-commutative group of order [Formula: see text] that is a FRUTE loop; there exists [Formula: see text]-groups of orders [Formula: see text] that are non-commutative FRUTE loops; there are no non-commutative groups that are FRUTE loops of the following range of orders [Formula: see text]; there are two non-associative FRUTE loops of order [Formula: see text] up to isomorphism and there are six non-isomorphic, non-associative FRUTE loops of order [Formula: see text]. It is noted that there exists a non-associative and non-commutative FRUTE loop of order [Formula: see text]. The study is concluded with some questions, conjectures and problem.
Temitope Gbolahan Jaiyeola, S. P. David, E. Ilojide and Y. T. Oyebo
A loop $(Q,cdot,backslash,/)$ is called a middle Bol loop if it obeys the identity $x(yzbackslash x)=(x/z)(ybackslash x)$.To every right (left) Bol loop corresponds a middle Bol loop via an isostrophism. In this paper, the structure of the holomorph of a middle Bol loop is explored. For some special types of automorphisms, the holomorph of a commutative loop is shown to be a commutative middle Bol loop if and only if the loop is a middle Bol loop and its automorphism group is abelian and a subgroup of both the group of middle regular mappings and the right multiplication group. It was found that commutativity (flexibility) is a necessary and sufficient condition for holomorphic invariance under the existing isostrophy between middle Bol loops and the corresponding right (left) Bol loops. The right combined holomorph of a middle Bol loop and its corresponding right (left) Bol loop was shown to be equal to the holomorph of the middle Bol loop if and only if the automorphism group is abelian and a subgroup of the multiplication group of the middle Bol loop. The obedience of an identity dependent on automorphisms was found to be a necessary and sufficient condition for the left combined holomorph of a middle Bol loop and its corresponding left Bol loop to be equal to the holomorph of the middle Bol loop.