Abstract
In this paper, by utilizing the concept of a neutrosophic extended triplet (NET), we define the neutrosophic image, neutrosophic inverse-image, neutrosophic kernel, and the NET subgroup. The notion of the neutrosophic triplet coset and its relation with the classical coset are defined and the properties of the neutrosophic triplet cosets are given. Furthermore, the neutrosophic triplet normal subgroups, and neutrosophic triplet quotient groups are studied.
Highlights
Neutrosophy was first introduced by Smarandache (Smarandache, 1999, 2003) as a branch of philosophy, which studied the origin, nature, and scope of neutralities, as well as their interactions with different ideational spectra: (A) is an idea, proposition, theory, event, concept, or entity; anti(A) is the opposite of (A); and means neither (A) nor anti(A), that is, the neutrality in between the two extremes
The neutrosophic triplets were first introduced by Florentin Smarandache and Mumtaz Ali [4,5,6,7,8,9,10], in 2014–2016
Let f: N → H be a neutro-homomorphism from a neutrosophic extended triplet group N to a nti(x)H, since anti(x)H= x= ∗(xneut(x)H
Summary
Neutrosophy was first introduced by Smarandache (Smarandache, 1999, 2003) as a branch of philosophy, which studied the origin, nature, and scope of neutralities, as well as their interactions with different ideational spectra: (A) is an idea, proposition, theory, event, concept, or entity; anti(A) is the opposite of (A); and (neut-A) means neither (A) nor anti(A), that is, the neutrality in between the two extremes. Aneutrosophic extended triplet, introduced by Smarandache [7,20] in 2016, is defined as the neutral of x (denoted by eneut( x) and called “extended neutral”), which is equal to the classical algebraic unitary element (if any). A neutrosophic extended triplet is a neutrosophic triplet, as defined in Definition 1, where neutral of x (denoted by eneut( x) and called extended neutral) is equal to the classical algebraic f(b)the. X ∗ eneut( x) = eneut( x) ∗ x = x, which can be equal to or different from the classical algebraic unitary element, if any, and x ∗ e anti( x) = e anti( x) ∗ x = eneut( x). 2triplet coset phic triplet normal subgroup of N, xH(anti[x])=H, ⩝ x ∈ N ⇒ xH(anti[x])x
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