A Smarandache Strong Structure on a set S means a structure on S that has a proper subset P with a stronger structure.

By proper subset of a set S, we mean a subset P of S, different from the empty set, from the original set S, and from the idempotent elements if any.

   In any field, a Smarandache strong n-structure on a set S means a structure {w₀} on S such that there exists a chain of proper subsets P_n-1 < P_n-2 < … < P₂ < P₁ < S, where '<' means 'included in', whose corresponding structures verify the inverse chain {w_n-1} > {w_n-2} > … > {w₂} > {w₁} > {w₀}, where '>' signifies 'strictly stronger' (i.e., structure satisfying more axioms).

   And by structure on S we mean the strongest possible structure {w} on S under the given operation(s).

   As a particular case, a Smarandache strong 2-algebraic structure (two levels only of structures in algebra) on a set S, is a structure {w₀} on S such that there exists a proper subset P of S, which is embedded with a stronger structure {w₁}.

For example, a Smarandache strong semigroup is a semigroup that has a proper subset which is a group.

Also, a Smarandache strong ring is a ring that has a proper subset which is a field.

   Properties of Smarandache strong semigroups, groupoids, loops, bigroupoids, biloops, rings, birings, vector spaces, semirings, semivector spaces, non-associative semirings, bisemirings, near-rings, non-associative  near-ring, binear-rings, fuzzy algebra and linear algebra are presented in the below books together with examples, solved and unsolved problems, and theorems.

   Also, applications of Smarandache strong groupoids, near-rings, and semirings in automaton theory, in error correcting codes, in the construction of S-sub-biautomaton, in social and economic research can be found in the below e-books.  

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