Smarandache n-structure

In any of knowledge, a Smarandache n-structureMathworldPlanetmath, for n2, on a set S means a weak structure w0 on S such that there exists a chain of proper subsetsMathworldPlanetmathPlanetmath Pn-1Pn-2P2P1S whose corresponding structures satisfy the inversePlanetmathPlanetmathPlanetmathPlanetmath inclusion chain wn-1wn-2w2w1w0, where signifies strictly stronger (i.e., structure satisfying more axioms).

By proper subset one understands a subset different from the empty setMathworldPlanetmath, from the idempotentMathworldPlanetmath 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 operationMathworldPlanetmath on A, and a1,a2,,ak,ak+1,,ak+m be k+m independent axioms, where k,m1.
If the operation ϕ on the set A satisfies the axioms a1,a2,,ak and does not satisfy the axioms ak+1,,ak+m, while on the subset B the operation ϕ satisfies the axioms a1,a2,,ak,ak+1,,ak+m, one says that structure wA=(A,ϕ) is weaker than structure wB=(B,ϕ) and one writes wAwB, or one says that wB is stronger than structure wA and one writes wBwA.
But if ϕ satisfies the same axioms on A as on B one says that structures wA and wB are equal and one writes wA=wB.
When ϕ satisfies the same axioms or less axioms on A than on B one says that structures wA is weaker than or equal to structure wB and one writes wAwB, or wB is stronger than or equal to wB and one writes wBwA.
For example a semigroup is a structure weaker than a group structure.

This definition can be extended to structures with many operations (A,ϕ1,ϕ2,,ϕr) for r2. Thus, let A be a set and B a proper subset of it.
a) If (A,ϕi)(B,ϕi) for all 1ir, then (A,ϕ1,ϕ2,,ϕr)(B,ϕ1,ϕ2,,ϕr).
b) If i0{1,2,,r} such that (A,ϕi0)(B,ϕi0) and (A,ϕi)(B,ϕi) for all ii0, then (A,ϕ1,ϕ2,,ϕr)(B,ϕ1,ϕ2,,ϕr).
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 algebraMathworldPlanetmathPlanetmath) on a set S, is a weak algebraic structurePlanetmathPlanetmath w0 on S such that there exists a proper subset P of S, which is embedded with a stronger algebraic structure w1.
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.

1. smarandache/eBooks-otherformats.htmDigital Library of Science:
2. W. B. Vasantha Kandasamy, Smarandache Algebraic Structures, book : (Vol. I: Groupoids; Vol. II: Semigroups; Vol. III: SemiringsMathworldPlanetmath, 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