Abstract

We study groupoids with primitively normal and additive theories. We prove that the theory of a semigroup is primitively normal if and only if the semigroup is an inflation of rectangular band of Abelian groups and the product of its idempotents is an idempotent. the theory of a semigroup is additive if and only if the semigroup is an Abelian group. We show that for theories of finite quasigroups the notions of primitive normality, additivity, and Abelianess are equivalent. We prove that theory of a groupoid with unity is primitively normal if and only if this theory is additive, which is equivalent to the fact that a groupoid is an Abelian group.

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