Groupoids with Primitive Normal and Additive Theories

In this work we investigate some groupoids with primitive normal and additive theories. We prove that the theory of a semigroup is primitive normal iff this semigroup is an inflation of rectangular band of the abelian groups and the product of its idempotents is an idempotent; the theory of semigroups is additive iff this semigroup is an abelian group. We show that for the theory of a finite quasigroup the notions of primitive normality, additivity and abelianty are equivalent. We prove that the theory of a groupoid with an identity is primitive normal iff this theory is additive, which is equivalent to a groupoid to be an abelian group.