Smarandache Semirings, Semifields, and Semivector Spaces

Generally, in any human field, a Smarandache Structure on a set A means a weak structure W on A such that there exists a proper subset B which is embedded with a stronger structure S.

By proper subset one understands a set included in A, different from the empty set, from the unit element if any, and from A.

These types of structures occur in our every day’s life, that’s why we study them in this book.

Thus, as three particular cases:

1) A Smarandache Semiring (or S-semiring) is a semiring A such that a proper subset B of A is a semifield (with respect to the same induced operation).

2) A Smarandache Semifield (or S-semifield) is a semifield A such that a proper subset B of A is a k-semi algebra, with respect to the same induced operations and an external operator.

3) A Smarandache Semivector Space (or S-semivector space) is a semivector space A (over a semifield B) which is a Smarandache semigroup (or S-semigroup).