On ringoids

Document Type : Original Article


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

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

3 Department of Mathematics, Payame Noor University, 19395-4697, Tehran, Iran


In this paper, we introduce the notion of a ringoid, and we obtain left distributive ringoids over a field which are not rings. We introduce several different types of ringoids, and also we discuss on (r, s)-ringoids. Moreover, we discuss geometric observations of the parallelism of vectors in several ringoids.


[1] P.J. Allen, H.S. Kim, J. Neggers, On companion d-algebras, Mathematica Slovaca, 57 (2007),
[2] P.J. Allen, H.S. Kim, J. Neggers, Deformations of d/BCK-algebras, Bulletin of the Korean
Mathematical Society, 48 (2011), 315–324.
[3] O. Boru˙ vka, Foundations of the theory of groupoids and groups, John Wiley and Sons, New
York, 1976.
[4] R.H. Bruck, A survey of binary systems, Springer, New York, 1971.
[5] J.S. Han, H.S. Kim, J. Neggers, Fibonacci sequences in groupoids, Advances in Difference
Equations, 19 (2012), DOI:10.1186/1687-1847-2012-19.
[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, 7 (2019), 973–800.
[8] A. Iorgulescu, Algebras of logic as BCK-algebras, Editura ASE, Bucharest, 2008.
[9] H.S. Kim, J. Neggers, S.S. Ahn, On pre-commutative algebras, Mathematics, 7 (2019), 336–
[10] Y.L. Liu, H.S. Kim, J. Neggers, Some special elements and pseudo inverse functions in
groupoids, Mathematics, 7 (2019), 173–179.
[11] J. Meng, Y.B. Jun, BCK-algebras, Kyungmoon Sa, Seoul, 1994.
[12] J. Neggers, H.S. Kim, On d-algebras, Mathematica Slovaca, 49 (1999), 19–26.