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). The theory of loops (groups without associativity), though researched by several mathematicians has not found a sound expression, for books, be it research level or otherwise, solely dealing with the properties of loops are absent. This is in marked contrast with group theory where books are abundantly available for all levels: as graduate texts and as advanced research books.