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Mathematics 2019, 7(4), 336;

On Pre-Commutative Algebras

Department of Mathematics, Research Institute of Natural Sciences, Hanyang University, Seoul 04763, Korea
Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487-0350, USA
Department of Mathematics Education, Dongguk University, Seoul 04620, Korea
Correspondences: [email protected]; Tel.: +82-10-3783-3410 or +82-2-2260-3410
Author to whom correspondence should be addressed.
Correspondences: [email protected]; Tel.: 82-10-3783-3410 or 82-2-2260-3410
Received: 24 January 2019 / Revised: 8 March 2019 / Accepted: 4 April 2019 / Published: 8 April 2019
(This article belongs to the Special Issue General Algebraic Structures)
PDF [212 KB, uploaded 8 April 2019]


In this paper, we introduce the notions of generalized commutative laws in algebras, and investigate their relations by using Smarandache disjointness. Moreover, we show that every pre-commutative B C K -algebra is bounded.
Keywords: abelian; (left-, right-) pre-commutative; commutative; d/BCK-algebra; Smarandache disjoint abelian; (left-, right-) pre-commutative; commutative; d/BCK-algebra; Smarandache disjoint
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).

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Kim, H.S.; Neggers, J.; Ahn, S.S. On Pre-Commutative Algebras. Mathematics 2019, 7, 336.

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