In any domain 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 $i \ne i_0$ , 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. Digital Library of Science:
2. W. B. Vasantha Kandasamy, Smarandache Algebraic Structures, book series: (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.