A Mathematics Letter Lecture Note on Some Variety of Algebraic ?-Structures

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.