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.
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