# Smarandache n-structure

In any of knowledge, a Smarandache $n$-structure, for $n\geqslant 2$, on a set $S$ means a weak structure $w_{0}$ on $S$ such that there exists a chain of proper subsets $P_{n-1}\subset P_{n-2}\subset\cdots\subset P_{2}\subset P_{1}\subset S$ whose corresponding structures satisfy the inverse inclusion chain $w_{n-1}\succ w_{n-2}\succ\dots\succ w_{2}\succ w_{1}\succ w_{0}$, where $\succ$ 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, $\phi$ an operation on $A$, and $a_{1},a_{2},\ldots,a_{k},a_{k+1},\ldots,a_{k+m}$ be $k+m$ independent axioms, where $k,m\geqslant 1$.
If the operation $\phi$ on the set $A$ satisfies the axioms $a_{1},a_{2},\ldots,a_{k}$ and does not satisfy the axioms $a_{k+1},\ldots,a_{k+m}$, while on the subset $B$ the operation $\phi$ satisfies the axioms $a_{1},a_{2},\ldots,a_{k},a_{k+1},\ldots,a_{k+m}$, one says that structure $w_{A}=(A,\phi)$ is weaker than structure $w_{B}=(B,\phi)$ and one writes $w_{A}\prec w_{B}$, or one says that $w_{B}$ is stronger than structure $w_{A}$ and one writes $w_{B}\succ w_{A}$.
But if $\phi$ 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 $\phi$ 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 $w_{A}\preceq w_{B}$, or $w_{B}$ is stronger than or equal to $w_{B}$ and one writes $w_{B}\succeq w_{A}$.
For example a semigroup is a structure weaker than a group structure.

This definition can be extended to structures with many operations $(A,\phi_{1},\phi_{2},\ldots,\phi_{r})$ for $r\geqslant 2$. Thus, let $A$ be a set and $B$ a proper subset of it.
a) If $(A,\phi_{i})\preceq(B,\phi_{i})$ for all $1\leq i\leq r$, then $(A,\phi_{1},\phi_{2},\ldots,\phi_{r})\preceq(B,\phi_{1},\phi_{2},\ldots,\phi_{% r})$.
b) If $\exists i_{0}\in\{1,2,\ldots,r\}$ such that $(A,\phi_{i_{0}})\prec(B,\phi_{i_{0}})$ and $(A,\phi_{i})\preceq(B,\phi_{i})$ for all $i\neq i_{0}$, then $(A,\phi_{1},\phi_{2},\ldots,\phi_{r})\prec(B,\phi_{1},\phi_{2},\ldots,\phi_{r})$.
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.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.

Title Smarandache n-structure SmarandacheNstructure 2013-03-22 14:18:30 2013-03-22 14:18:30 jonnathan (5141) jonnathan (5141) 29 jonnathan (5141) Definition msc 08A05 FlorentinSmarandache