**
Neutrosophic Transdisciplinarity**

**
(Multi-Space & Multi-Structure)**

**
**Florentin
Smarandache, UNM-Gallup, USA

**A)
****Definition:**

__Neutrosophic____ Transdisciplinarity__ means
to find common features to uncommon entities,

**i.e.,
for vague, imprecise, not-clear-boundary entity <A> one has: **

**<A>
∩ <nonA> ≠ Ø (empty set),**

**or even more <A> ∩ <antiA> ≠ Ø,**

**similarly <A> ∩ <neutA> ≠
Ø and <antiA> ∩ <neutA> ≠ Ø, **

**up to <A> ∩ <neutA> ∩
<antiA> ≠ Ø;**

**where <nonA> means what is not A, and <antiA>
means the opposite of <A>.**

There exists a __Principle of
Attraction__ not only between the opposites <A> and <antiA>

(as in dialectics),

**but also between
them and their neutralities <neutA> related to
them,**

**since <neutA> contributes to the Completeness of Knowledge.**

<neutA> means neither <A> nor <antiA>, but in between;

<neutA> is included in <nonA>.

** **

As part of Neutrosophic Transdisciplinarity we have:

**B) Multi-Structure and Multi-Space:**

**B1****) Multi-Concentric-Structure:**

**Let
S1 and S2
****be two distinct structures, induced by the ensemble of laws L,
which verify**

**the ensembles of axioms A1 and A2 respectively, such that A1
****is strictly included in A2.**

**One
says that the set M, endowed with the properties:**

**a)
M has an S1-structure;**

b) there is a proper subset P (different from the empty set Ø, from the unitary element, from

**the idempotent element if any with
respect to S _{2}, and from the whole set M)
of the initial set M, **

**which**** has an S _{2}-structure;**

**c)
M doesn't have an S _{2}-structure; is called
a 2-concentric-structure. **

**We
can generalize it to an n-concentric-structure, for n ≥ 2 (even
infinite-concentric-structure). **

(By
default, *1-concentric structure* on a
set M means only one structure on M and on its

**proper**** subsets.)**

**An **
**n-concentric-structure**** on a set S
means a weak structure {w(0)} on S **

**such**** that there
exists a chain of proper subsets **

**P(****n-1) < P(n-2) < … < P(2)
< P(1) < S, **

**where**** '<' means
'included in', **

**whose**** corresponding
structures verify the inverse chain **

**{w(n-1)} > {w(n-2)} > … >
{w(2)} > {w(1)} > {w(0)},**

**where**** '>'
signifies 'strictly stronger' (i.e., structure satisfying more axioms).**

**For
example:**

** Say a groupoid D,
which contains a proper subset S which is a semigroup,
which**

in its turn contains a proper subset M which is a monoid, which contains a proper subset NG

which is a non-commutative group, which contains a proper subset CG which is a commutative

**
group, where D includes S, which includes M, which
includes NG, which includes CG.**

**[This
is a 5-concentric-structure.]**

**B2****) ** **Multi-Space****: **

**Let
S _{1}, S_{2}, ..., S_{n} be n
structures on respectively the **

**sets**** M _{1}, M_{2},
..., M_{n}, where n ≥ 2 (n may even be
infinite). **

The
structures S_{i}, i = 1, 2,
…, n, may not necessarily be distinct two by two;

**each**** structure S _{i}
may be or not n_{i}-concentric, for n_{i} ≥ 1.**

**And
the sets M _{i}, i = 1, 2,
…, n, may not necessarily be disjoint, **

**also**** some sets M _{i}
may be equal to or included in other sets M_{j},
j = 1, 2, …, n.**

**We
define the Multi-Space M as a union of the previous sets: **

**M
= M _{1}
\/
M_{2}
\/
…
\/ M_{n}, hence we have n (different or not,**

**overlapping**** or not)
structures on M.**

**A
multi-space is a space with many structures that may overlap, **

**or some structures may include
others or may be equal, or the structures may **

**interact and influence
each other as in our everyday life.**

Therefore, a region (in particular a point) which belong to the intersection

**of 1 ≤ k ≤
n sets M _{i }may have k different structures in the same time. And**

**here it is the
difficulty and beauty of the a multi-space and its overlapping**

**multi-structures.**

**{We
thus may have ** **<R> ≠
<R>****, i.e. a region R different from itself, since **

R could be endowed with different structures simultaneously.}

**For
example we can construct a geometric multi-space formed by the union of**

**three distinct
subspaces: an Euclidean subspace, a Hyperbolic subspace, **

**and**** an Elliptic subspace.**

**As
particular cases when all M _{i} sets have the same type of structure,
we can define the Multi-Group (or n-group; for example; bigroup,
tri-group, etc., when all sets M_{i} are groups), Multi-Ring (or
n-ring, for example biring, tri-ring, etc. when all
sets M_{i} are rings), Multi-Field (n-field), Multi-Lattice
(n-lattice), Multi-Algebra (n-algebra), Multi-Module (n-module), and so on -
which may be generalized to Infinite-Structure-Space (when all sets have the
same type of structure), etc.**

**Conclusion.**

**The
multi-space comes from reality, it is not artificial, because our reality is
not homogeneous, but has many spaces with different structures.**

**A multi-space means a combination of
any spaces (may be all of the same dimensions, or of different dimensions – it doesn’t
matter).**

For example, a Smarandache geometry (SG) is a
combination of geometrical (manifold or pseudo-manifold, etc.) spaces, while
the multi-space is a combination of any (algebraic, geometric, analytical,
physics, chemistry, etc.) space.

So, the multi-space can be interdisciplinary, i.e. math and physics spaces, or
math and biology and chemistry spaces, etc.

Therefore, an SG is a particular case of a multi-space.

Similarly, a Smarandache algebraic structure is also a particular case of a multi-space.

This multi-space is a combination of spaces on the horizontal way, but also on
the vertical way (if needed for certain applications).

On the horizontal way means a simple union of spaces (that may overlap or not,
may have the same dimension or not, may have metrics or not, the metrics if any
may be the same or different, etc.).

On the vertical way means more spaces overlapping in the same time, every one
different or not.

The multi-space is really very general because it tries to model our reality. The parallel universes are particular cases
of the multi-space too.

So, they are multi-dimensional (they can have some dimensions on the horizontal
way, and other dimensions on the vertical way, etc.).

**The multi-space and multi-structure
form together a Theory of Everything. It can be used,
for example, in the Unified Field Theory that tries to unite the gravitational,
electromagnetic, weak, and strong interactions (in physics).**

Reference:

F. Smarandache, "Mixed Non-Euclidean Geometries", Arhivele Statului, Filiala Vâlcea,

Rm. Vâlcea, 1969.