Smarandache n-structure

Smarandache n-structure

In any of knowledge, a

Smarandache

n

-

structure

, for

n

⩾

2

, 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

whose corresponding structures

satisfy

the

inverse

inclusion

chain

w

n

-

1

≻

w

n

-

2

≻

…

≻

w

2

≻

w

1

≻

w

0

, where

≻

signifies strictly stronger (i.e., structure satisfying more axioms).

By

proper subset

one understands a subset different from the

empty set

, from the

idempotent

if any, and from the whole set.

Now one defines the

weak structure

:

Let

A

be a set,

B

a proper subset of it,

ϕ

an

operation

on

A

, and

a

1

,

a

2

,

…

,

a

k

,

a

k

+

1

,

…

,

a

k

+

m

be

k

+

m

independent axioms, where

k

,

m

⩾

1

.

If the operation

ϕ

on the set

A

satisfies the axioms

a

1

,

a

2

,

…

,

a

k

and does not satisfy the axioms

a

k

+

1

,

…

,

a

k

+

m

, while on the subset

B

the operation

ϕ

satisfies the axioms

a

1

,

a

2

,

…

,

a

k

,

a

k

+

1

,

…

,

a

k

+

m

, one says that structure

w

A

=

(

A

,

ϕ

)

is

weaker

than structure

w

B

=

(

B

,

ϕ

)

and one writes

w

A

≺

w

B

, or one says that

w

B

is

stronger

than structure

w

A

and one writes

w

B

≻

w

A

.

But if

ϕ

satisfies the same axioms on

A

as on

B

one says that structures

w

A

and

w

B

are equal and one writes

w

A

=

w

B

.

When

ϕ

satisfies the same axioms or less axioms on

A

than on

B

one says that structures

w

A

is

weaker than or equal

to structure

w

B

and one writes

w

A

⪯

w

B

, or

w

B

is

stronger than or equal

to

w

B

and one writes

w

B

⪰

w

A

.

For example a

semigroup

is a structure weaker than a group structure.

This definition can be extended to structures with many operations

(

A

,

ϕ

1

,

ϕ

2

,

…

,

ϕ

r

)

for

r

⩾

2

. Thus, let

A

be a set and

B

a proper subset of it.

a) If

(

A

,

ϕ

i

)

⪯

(

B

,

ϕ

i

)

for all

1

≤

i

≤

r

, then

(

A

,

ϕ

1

,

ϕ

2

,

…

,

ϕ

r

)

⪯

(

B

,

ϕ

1

,

ϕ

2

,

…

,

ϕ

r

)

.

b) If

∃

i

0

∈

{

1

,

2

,

…

,

r

}

such that

(

A

,

ϕ

i

0

)

≺

(

B

,

ϕ

i

0

)

and

(

A

,

ϕ

i

)

⪯

(

B

,

ϕ

i

)

for all

i

≠

i

0

, then

(

A

,

ϕ

1

,

ϕ

2

,

…

,

ϕ

r

)

≺

(

B

,

ϕ

1

,

ϕ

2

,

…

,

ϕ

r

)

.

In this case, for two operations, a ring is a structure weaker than a field structure.

This definition comprises large classes of structures, some more important than others.

As a particular case, in

abstract algebra

, a

Smarandache 2-algebraic structure

(two levels only of structures in

algebra

) on a set

S

, is a weak

algebraic structure

w

0

on

S

such that there exists a proper subset

P

of

S

, which is embedded with a stronger algebraic structure

w

1

.

For example: a

Smarandache semigroup

is a semigroup (different from a group) which has a proper subset that is a group.

Other examples: a

Smarandache

groupoid

of

first order

is a groupoid (different from a semigroup) which has a proper subset that is a semigroup, while a

Smarandache groupoid of

second order

is a groupoid (different from a semigroup) which has a proper subset that is a group. And so on.

References:

1.

http://www.gallup.unm.edu/ smarandache/eBooks-otherformats.htm

Digital Library of Science:

2. W. B. Vasantha Kandasamy,

Smarandache Algebraic Structures

, book : (Vol. I: Groupoids; Vol. II: Semigroups; Vol. III:

Semirings

,

Semifields

, and Semivector Spaces; Vol. IV: Loops; Vol. V: Rings; Vol. VI:

Near-rings

; Vol. VII: Non-associative Rings; Vol. VIII: Bialgebraic Structures; Vol. IX: Fuzzy Algebra; Vol. X: Linear Algebra), Am. Res. Press & Bookman, Martinsville, 2002-2003.

Title

Smarandache n-structure

Canonical name

SmarandacheNstructure

Date of creation

2013-03-22 14:18:30

Last modified on

2013-03-22 14:18:30

Owner

jonnathan (5141)

Last modified by

jonnathan (5141)

Numerical id

29

Author

jonnathan (5141)

Entry type

Definition

Classification

msc 08A05

Related topic

FlorentinSmarandache

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