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 Pn-1 < Pn-2 < < P2 < P1 < S, where '<' means 'included in', whose corresponding structures verify the inverse chain {wn-1} > {wn-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.  

International Conference on Smarandache Algebraic Structures, December 17-19, 2004, Loyola College, Madras, Chennai - 600 034 Tamil Nadu, India. 


1) Smarandache type strong groupoids, semigroups, rings, fields;

2) Smarandache type strong k-modules, vector spaces, linear algebra, fuzzy algebra.

Organizer: Dr. M. Mary John, Head of Department of Mathematics

Book series:
bulletNeutrosophic Rings, by W. B. Vasantha Kandasamy, F. Smarandache
bulletN-Algebraic Structures, by W. B. Vasantha Kandasamy, F. Smarandache
bulletIntroduction to N-Adaptive Fuzzy Models to Analyze Public Opinion on AIDS, by W. B. Vasantha Kandasamy, F. Smarandache
bulletSmarandache Algebraic Structures, W. B. Vasantha Kandasamy:    (Vol. I: GroupoidsVol. II: SemigroupsVol. III: Semirings, Semifields, and Semivector SpacesVol. IV: Loops; Vol. V: Rings; Vol. VI: Near-rings; Vol. VII: Non-associative Rings; Vol. VIII: Bialgebraic Structures; Vol. IX: Fuzzy Algebra; Vol. X: Linear Algebra)
bulletA Study of New Concepts in Smarandache Quasigroups and Loops, by Jaiyeola Temitope Gbolahan


R. Padilla's editing

See also:

Smarandache Weak Structures


Smarandache Strong-Weak Structures

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