Neutrosophic Transdisciplinarity

 

by Florentin Smarandache

 

 

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> ∩ <Non-A> ≠ Ø (empty set),

or even more <A> ∩ <Anti-A> ≠ Ø.

 

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 S₂, and from the whole set M) of the initial set M, which has an S₂-structure;

c) M doesn't have an S₂-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 S₁, S₂, ..., S_n be distinct two by two structures on respectively the

sets M₁, M₂, ..., M_k, 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 also be n_i-concentric, 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 = M₁ È M₂ È … È M_n, hence we have n (different) structures on M.

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

or some structures include others, or the structures may interact and

influence each other as in our everyday life.

 

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

three distinct subspaces: an Euclidean space, a Hyperbolic one, and an Elliptic one.

 

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.

 

{F. Smarandache, "Mixed Non-Euclidean Geometries", 1969.}