NEUTROSOPHIC MULTISET STRUCTURES

The Neutrosophic Multisets and the Neutrosophic Multiset Algebraic Structures were introduced

by Florentin Smarandache [2, 3, 4] in 2016.

# 1. Definition of Neutrosophic Multiset

Let  be a universe of discourse, and .

A Neutrosophic Multiset  is a neutrosophic set where one or more elements are repeated

with the same neutrosophic components, or with different neutrosophic components.

### 2. Examples

is a neutrosophic set (not a neutrosophic multiset).

But

is a neutrosophic multiset, since the element a is repeated; we say that the element a has

neutrosophic multiplicity 2 with the same neutrosophic components.

While

is also a neutrosophic multiset, since the element a is repeated (it has neutrosophic

multiplicity 3), but with different neutrosophic components, since, for example, during

the time, the neutrosophic membership of an element may change.

If the element  is repeated  times keeping the same neutrosophic components ,

we say that a has multiplicity .

But if there is some change in the neutrosophic components of a, we say that a has the

neutrosophic multiplicity .

Therefore, we define in general the Neutrosophic Multiplicity Function:

where ,

and for any  one has

which means that a is repeated  times with the neutrosophic components ;

a is repeated  times with the neutrosophic components , ..., a is repeated

times with the neutrosophic components , ..., and so on.

Of course, all , and , for , with .

Also, all neutrosophic components are with respect to the set . Then, a neutrosophic

multiset A can be written as:

or .

Let's have:

Then: