The author studies the Smarandache Fuzzy Algebra, which, like its predecessor Fuzzy Algebra, arose from the need to define structures that were more compatible with the real world where the grey areas mattered, not only black or white.

In any human field, a Smarandache n-structure on a set S means a weak structure {w0} on S such that there exists a chain of proper subsets Pn–1 included in Pn–2 included in … included in P2 included in P1 included in S whose corresponding structures verify the chain {wn–1} > {wn–2} > … > {w2} > {w1} > {w0}, where ‘>’ signifies ‘strictly stronger’ (i.e. structure satisfying more axioms). This book is referring to a Smarandache 2-algebraic structure (two levels only of structures in algebra) on a set S, i.e. a weak structure {w0} on S such that there exists a proper subset P of S, which is embedded with a stronger structure {w1}.

Properties of Smarandache fuzzy semigroups, groupoids, loops, bigroupoids, biloops, non-associative rings, birings, vector spaces, semirings, semivector spaces, non-associative semirings, bisemirings, near-rings, non-associative near-rings, and binear-rings are presented in the second part of this book together with examples, solved and unsolved problems, and theorems.

Also applications of Smarandache groupoids, near-rings, and semi-rings in automaton theory, in error-correcting codes, and in the construction of S-sub-biautomaton can be found in the last chapter.

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