

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 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 inverse chain {w_{n1}} >
{w_{n2}} >
… >
{w_{2}} >
{w_{1}} >
{w_{0}},
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 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 stronger structure {w_{1}}. 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, nonassociative semirings, bisemirings, nearrings, nonassociative nearring, binearrings, 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, nearrings, and semirings in automaton
theory, in error correcting codes, in the construction of Ssubbiautomaton, in
social and economic research
can be found in the below ebooks. International Conference on Smarandache Algebraic Structures, December 1719, 2004, Loyola College, Madras, Chennai  600 034 Tamil Nadu, India.
Program:
1) Smarandache type
strong groupoids, semigroups, rings, fields;
2) Smarandache type
strong kmodules, vector spaces, linear algebra, fuzzy algebra. Organizer: Dr. M. Mary John, Head of Department of Mathematics Book series: Article:


