# Smarandache n-structure

In any of knowledge, a *Smarandache* $n$-*structure ^{}*, for $n\u2a7e2$, on a set $S$ means a weak structure ${w}_{0}$ on $S$ such that there exists a chain of proper subsets

^{}${P}_{n-1}\subset {P}_{n-2}\subset \mathrm{\cdots}\subset {P}_{2}\subset {P}_{1}\subset S$ whose corresponding structures satisfy the inverse

^{}inclusion chain ${w}_{n-1}\succ {w}_{n-2}\succ \mathrm{\dots}\succ {w}_{2}\succ {w}_{1}\succ {w}_{0}$, 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, $\varphi $ an operation^{} on $A$, and ${a}_{1},{a}_{2},\mathrm{\dots},{a}_{k},{a}_{k+1},\mathrm{\dots},{a}_{k+m}$ be $k+m$ independent axioms, where $k,m\u2a7e1$.

If the operation $\varphi $ on the set $A$ satisfies the axioms ${a}_{1},{a}_{2},\mathrm{\dots},{a}_{k}$ and does not satisfy the axioms ${a}_{k+1},\mathrm{\dots},{a}_{k+m}$, while on the subset $B$ the operation $\varphi $ satisfies the axioms ${a}_{1},{a}_{2},\mathrm{\dots},{a}_{k},{a}_{k+1},\mathrm{\dots},{a}_{k+m}$, one says that structure ${w}_{A}=(A,\varphi )$ is *weaker* than structure ${w}_{B}=(B,\varphi )$ 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 $\varphi $ 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 $\varphi $ 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 ${w}_{A}\u2aaf{w}_{B}$, or ${w}_{B}$ is *stronger than or equal* to ${w}_{B}$ and one writes ${w}_{B}\u2ab0{w}_{A}$.

For example a semigroup is a structure weaker than a group structure.

This definition can be extended to structures with many operations $(A,{\varphi}_{1},{\varphi}_{2},\mathrm{\dots},{\varphi}_{r})$ for $r\u2a7e2$. Thus, let $A$ be a set and $B$ a proper subset of it.

a) If $(A,{\varphi}_{i})\u2aaf(B,{\varphi}_{i})$ for all $1\le i\le r$, then $(A,{\varphi}_{1},{\varphi}_{2},\mathrm{\dots},{\varphi}_{r})\u2aaf(B,{\varphi}_{1},{\varphi}_{2},\mathrm{\dots},{\varphi}_{r})$.

b) If $\exists {i}_{0}\in \{1,2,\mathrm{\dots},r\}$ such that $(A,{\varphi}_{{i}_{0}})\prec (B,{\varphi}_{{i}_{0}})$ and $(A,{\varphi}_{i})\u2aaf(B,{\varphi}_{i})$ for all $i\ne {i}_{0}$, then $(A,{\varphi}_{1},{\varphi}_{2},\mathrm{\dots},{\varphi}_{r})\prec (B,{\varphi}_{1},{\varphi}_{2},\mathrm{\dots},{\varphi}_{r})$.

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 |
---|---|

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 |