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

Abstract

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 60Bol-Moufang identities. In this paper, the isotopy of BCI-algebras are studied. Neccessary andsufficient conditions for a groupoid isotope of a BCI-algebra to be a BCI-algebra are established. It is shown that ð‘-semisimplicity, quasi-associativity and BCK-algebra are invariant under isotopies which are determined by some regular permutation groups. Furthermore, the isotopy of both the 46associative and 14non-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 (ð¹ð‘–-algebras) are invariant under any isotopy:ð‘–∈{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 (ð¹ð‘–-algebras) are invariant under isotopies which are determined by some regular permutation groups: ð‘–∈{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 120particular types of the 540varieties of Fenyves quasi neutrosophic triplet loops (FQNTLs) which were recently discovered, wherein the 14non-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.