

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 nstructure on a set S means a structure {w_{0}}
on S such that there exists a chain of proper subsets P_{n1 }
<
P_{n2 }
< …
<
P_{2 }< P_{1
}<
S, where '>' means 'included in', whose corresponding structures verify the chain {w_{n1}} <
{w_{n2}} <
… <
{w_{2}} <
{w_{1}} <
{w_{0}},
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 2algebraic structure (two levels only of structures in algebra) on a set S, is a structure {w_{0}} on S such that there exists a proper subset P of S, which is embedded with a weaker structure {w_{1}}. 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 nearring. Book:
