Smarandache Loops

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.

As an example:

A non-empty set L is said to form a loop, if on L is defined a binary operation called product, denoted by '·', such that:

1) For all a, b in L we have a · b in L (closure property);

2) There exists an element e in L such that a·e = e·a = a for all a in L (e is the identity element of L);

3) For every ordered pair (a, b) in L x L there exists a unique pair (x, y) in L such that ax = b and ya = b.

Whence:

A Smarandache Loop (or S-loop) is a loop L such that a proper subset M of L is a subgroup (with respect to the same induced operation).