Definition:
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 included in A which is embedded with a stronger structure S.
These types of structures occur in our everyday life, that’s why we study them in this book.
Thus, as a particular case:
A Smarandache Semigroup is a semigroup A which has a proper subset B included in A that is a group (with respect to the same binary operation on A).