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 Vestn. Novosib. Gos. Univ., Ser. Mat. Mekh. Inform., 2011, Volume 11, Issue 4, Pages 107–116 (Mi vngu104)

Groupoids with Primitive Normal and Additive Theories

N. V. Trikashnaya

Institute of Mathematics and Computer Sciences, Far Eastern State University

Abstract: 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.

Keywords: primitive normal theory, additive theory, semigroup, quasigroup.

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English version:
Journal of Mathematical Sciences, 2013, 195:6, 857–864

UDC: 510.8:512.57

Citation: N. V. Trikashnaya, “Groupoids with Primitive Normal and Additive Theories”, Vestn. Novosib. Gos. Univ., Ser. Mat. Mekh. Inform., 11:4 (2011), 107–116; J. Math. Sci., 195:6 (2013), 857–864

Citation in format AMSBIB
\Bibitem{Tri11} \by N.~V.~Trikashnaya \paper Groupoids with Primitive Normal and Additive Theories \jour Vestn. Novosib. Gos. Univ., Ser. Mat. Mekh. Inform. \yr 2011 \vol 11 \issue 4 \pages 107--116 \mathnet{http://mi.mathnet.ru/vngu104} \transl \jour J. Math. Sci. \yr 2013 \vol 195 \issue 6 \pages 857--864 \crossref{https://doi.org/10.1007/s10958-013-1624-y}