On the Isotopy of some Varieties of Fenyves Quasi Neutrosophic Triplet Loop (Fenyves BCI-algebras)

• Main Authors: Temitope Gbolahan Jaiyéolá, Emmanuel Ilojide, Adisa Jamiu Saka, Kehinde Gabriel Ilori
• Format: Article
• Language: English
• Published: University of New Mexico 2020-01-01
• Series: Neutrosophic Sets and Systems
• Subjects: bci-algebra; quasi neutrosophic loops; fenyves identities; bol-moufang type
• Online Access: http://fs.unm.edu/NSS/IsotopyofsomeVarieties.pdf
• ISSN: 2331-6055; 2331-608X

Neutrosophy theory has found application in health sciences in recent years. There is the need to develop neutrosophic algebraic systems which are good and appropriate for studying and understanding the effects of diseases and their possible treatments. In order to achieve this, special types of quasi neutrosophic loops and their isotopy needed to be introduced for this purpose. Fenyves BCI-algebras are BCI-algebras (special types of quasi neutrosophic loops) that satisfy the 60 Bol-Moufang identities. In this paper, the isotopy of BCI-algebras are studied. Neccessary and sufficient conditions for a groupoid isotope of a BCI-algebra to be a BCI-algebra are established. It is shown that 𝑝-semisimplicity, quasi-associativity and BCK-algebra are invariant under isotopies which are determined by some regular permutation groups. Furthermore, the isotopy of both the 46 associative and 14 non-associative Fenyves BCI-algebras are also studied. It is shown that for BCI-alegbras, associativity is isotopic invariant. Hence, the following set of Fenyves BCI algebras (𝐹𝑖 -algebras) are invariant under any isotopy: 𝑖 ∈ {1,2,4,6,7,9,10,11,12,13,14,15,16,17,18,20,22,23,24 , 25,26,27,28,30,31,32,33,34,35,36,37,38,40,41,43,44,45,47,48,49,50,51,53,57,58,60}. It is shown that the following sets of non-associative Fenyves BCI algebras (𝐹𝑖 -algebras) are invariant under isotopies which are determined by some regular permutation groups: 𝑖 ∈ {3,5,8,19,21,29,39,42,46,52,55,56,59},{56},{8,19,29,39,46,59}. In conclusion, this is the isotopic study of 120 particular types of the 540 varieties of Fenyves quasi neutrosophic triplet loops (FQNTLs) which were recently discovered, wherein the 14 non-associative Fenyves BCI-algebras do not necessarily have the Iseki's conditions (S). Importantly, applying these results, the initial (old, sick or healthy) state of a person can be represented by a type of Fenyves BCI-algebra, while the Fenyves BCI-algebra isotope will represent the final (new, healthy or sick) state of the person as a result of the prescribed medical treatment, which the isotopism represents. The isotopism is a measure of the change from the old state of body condition to the new state.