On block commutative groupoids

Document Type : Original Article


1 Department of Mathematics, Daejin University, Pochen 11159, Korea

2 Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487-0350, U.S.A.

3 Department of Mathematics, Research Institute of Natural Sciences, Hanyang University, Seoul 04763, Korea


In this paper, we introduce the notion of a block commutativity in several groupoids, and show that the class of block commutative groupoids and the class of d/BCK-algebras are Smarandache disjoint. The block commutativity in linear/quadratic groupoids is investigated, and we prove that every group is a normal groupoid. Moreover, we discuss block n-commutative groupoids and block ranks.


[1] P.J. Allen, H.S. Kim, J. Neggers, Smarandache disjoint in BCK/d-algebras, Scientiae Math-
ematicae Japonicae, 61 (2005), 447{449.
[2] P.J. Allen, H.S. Kim, J. Neggers, Bracket functions on groupoids, Communications of the
Korean Mathematical Society, 34 (2019), 375{381.
[3] O. Bor _ uvka, Foundations of the theory of groupoids and groups, John Wiley Sons, New York,
[4] R.H. Bruck, A survey of binary systems, Springer, New York, 1971.
[5] J.S. Han, H.S. Kim, J. Neggers, Strong and ordinary d-algebras, Journal of Multiple-Valued
Logic and Soft Computing, 16 (2010), 331{339.
[6] Y. Huang, BCI-algebras, Science Press, Beijing, 2006.
[7] I.H. Hwang, H.S. Kim, J. Neggers, Some implicativities for groupoids and BCK-algebras,
Mathematics Journal, 7 (2019), 973{800.
[8] A. Iorgulescu, Algebras of logic as BCK-algebras, Editura ASE, Bucharest, 2008.
[9] H.S. Kim, J. Neggers, The semigroups of binary systems and some perspectives, Bulletin of
the Korean Mathematical Society, 45 (2008), 651{661.
[10] H.S. Kim, J. Neggers, S.S. Ahn, On pre-commutative algebras, Mathematics Journal, 7 (2019),
[11] Y.L. Liu, H.S. Kim, J. Neggers, Some special elements and pseudo inverse functions in
groupoids, Mathematics Journal, 7 (2019), 173{179.
[12] J. Meng, Y.B. Jun, BCK-algebras, Kyungmoon Sa, Seoul, 1994.
[13] J. Neggers, Y.B. Jun, H.S. Kim, On d-ideals in d-algebras, Mathematica Slovaca, 49 (1999),
[14] J. Neggers, H.S. Kim, On d-algebras, Mathematica Slovaca, 49 (1999), 19{26.