Inverse Properties in Neutrosophic Triplet Loop and Their Application to Cryptography

Temitope Jaiyeola,Florentin Smarandache

doi:10.3390/a11030032

Abstract

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.

Highlights

Keedwell [45], Keedwell and Shcherbacov [46,47,48,49], Jaiyéo.lá [50,51,52,53,54,55], and Jaiyéo.lá and Adéníran [56] are of great significance in the study of quasigroups and loop with the weak inverse property (WIP), automorphic inverse property (AIP), cross inverse property (CIP), their generalizations (i.e., m-inverse loops and quasigroups, (r,s,t)-inverse quasigroups) and applications to cryptography
left cross inverse property neutrosophic triplet loop (LCIPNTL) Assume that the message to be transmitted can be represented as a single element x ∈ X
right inverse property neutrosophic triplet loop (RIPNTL) Assume that the message to be transmitted can be represented as a single element x ∈ X

Summary

Generalized Group

A generalized group is an algebraic structure which has a deep physical background in the unified gauge theory and has direct relation with isotopies. It was known that generalized groups are tools for constructions in unified geometric theory and electroweak theory. Some of the structures and properties of generalized groups have been studied by Vagner [2], Molaei [3], [4], Mehrabi, Molaei, and Oloomi [5], Agboola [6], Adeniran et al [7], and Fatehi and Molaei [8]. A generalized group G is a non-empty set admitting a binary operation called multiplication, subject to the set of rules given below. Right) cancellation law), ( L, ·) is called a left If a groupoid ( L, ·) is both a left quasigroup and right quasigroup, it is called a quasigroup. A generalized loop is the pair ( G, ·) where G is a non-empty set and “·” a binary operation such that the following are true.

Neutrosophic Triplet Group

Main Results

Application to Cryptography