In any of knowledge, a Smarandache -structure, for , on a set means a weak structure on such that there exists a chain of proper subsets whose corresponding structures satisfy the inverse inclusion chain , where signifies strictly stronger (i.e., structure satisfying more axioms).
Now one defines the weak structure:
Let be a set, a proper subset of it, an operation on , and be independent axioms, where .
If the operation on the set satisfies the axioms and does not satisfy the axioms , while on the subset the operation satisfies the axioms , one says that structure is weaker than structure and one writes , or one says that is stronger than structure and one writes .
But if satisfies the same axioms on as on one says that structures and are equal and one writes .
When satisfies the same axioms or less axioms on than on one says that structures is weaker than or equal to structure and one writes , or is stronger than or equal to and one writes .
For example a semigroup is a structure weaker than a group structure.
This definition can be extended to structures with many operations for . Thus, let be a set and a proper subset of it.
a) If for all , then .
b) If such that and for all , then .
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 , is a weak algebraic structure on such that there exists a proper subset of , which is embedded with a stronger algebraic structure .
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. http://www.gallup.unm.edu/ smarandache/eBooks-otherformats.htmDigital 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.
|Date of creation||2013-03-22 14:18:30|
|Last modified on||2013-03-22 14:18:30|
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