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Open AccessArticle

On the Classification of Bol-Moufang Type of Some Varieties of Quasi Neutrosophic Triplet Loop (Fenyves BCI-Algebras)

1
Department of Mathematics, Faculty of Science, Obafemi Awolowo University, P.M.B. 13, Ile-Ife, Osun 220282, Nigeria
2
Department of Mathematics, College of Physical Sciences, Federal University of Agriculture, Abeokuta 110101, Nigeria
3
Department of Mathematics, Faculty of Science, Obafemi Awolowo University, P.M.B. 13, Ile-Ife, Osun 220282, Nigeria
4
Department of Mathematics, University of New Mexico, Gallup, NM 87301, USA
*
Author to whom correspondence should be addressed.
Symmetry 2018, 10(10), 427; https://doi.org/10.3390/sym10100427
Received: 4 September 2018 / Revised: 19 September 2018 / Accepted: 20 September 2018 / Published: 21 September 2018
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. View Full-Text
Keywords: quasigroup; loop; BCI-algebra; Bol-Moufang; quasi neutrosophic loops; Fenyves identities quasigroup; loop; BCI-algebra; Bol-Moufang; quasi neutrosophic loops; Fenyves identities
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MDPI and ACS Style

Jaíyéọlá, T.G.; Ilojide, E.; Olatinwo, M.O.; Smarandache, F. On the Classification of Bol-Moufang Type of Some Varieties of Quasi Neutrosophic Triplet Loop (Fenyves BCI-Algebras). Symmetry 2018, 10, 427.

AMA Style

Jaíyéọlá TG, Ilojide E, Olatinwo MO, Smarandache F. On the Classification of Bol-Moufang Type of Some Varieties of Quasi Neutrosophic Triplet Loop (Fenyves BCI-Algebras). Symmetry. 2018; 10(10):427.

Chicago/Turabian Style

Jaíyéọlá, Tèmítọ́pẹ́ G.; Ilojide, Emmanuel; Olatinwo, Memudu O.; Smarandache, Florentin. 2018. "On the Classification of Bol-Moufang Type of Some Varieties of Quasi Neutrosophic Triplet Loop (Fenyves BCI-Algebras)" Symmetry 10, no. 10: 427.

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