Abstract: Algebraic -structures represent a natural generalization of classical algebraic struc- tures. Results studied in semigroups are particular cases of those studied in -semigroups as every semigroup is a -semigroups but not vice-versa. This research paper is based on the introduction and initiation of rectangular -semigroups, quasi-rectangular -semigroups, total -semigroups, viable -semigroups and idempotent -semigroups. Among lots of results, we prove that a rect- angular -semigroup is the direct product of a left singular and a right singular -semigroups. Moreover, this product is unique up to isomorphism.
Keywords: -semigroup, rectangular bands, rectangular -semigroup, total -semigroup, viable -semigroup, quasi-rectangular -semigroup, singular -semigroup, -ideal