A Smarandache Weak Structure on a set S means a structure on S that has a proper subset P with a weaker 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 weak n-structure on a set S means a structure {w0} on S such that there exists a chain of proper subsets Pn-1 < Pn-2 < < P2 < P1 < S, where '>' means 'included in', whose corresponding structures verify the chain {wn-1} < {wn-2} < < {w2} < {w1} < {w0}, where '<' signifies 'strictly weaker' (i.e., structure satisfying less axioms).

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

   As a particular case, a Smarandache weak 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 weaker structure {w1}.

For example, a Smarandache weak monoid is a monoid that has a proper subset which is a semigroup.

Also, a Smarandache weak ring is a ring that has a proper subset which is a near-ring.  

Book:
bullet Smarandache Special Definite Algebraic Structures, by W. B. Vasantha Kandasamy

See also:
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Smarandache Strong Structures

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Smarandache Strong-Weak Structures

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