Shahid Bahonar University of Kerman Journal of Mahani Mathematical Research 2251-7952 11 3 2022 11 01 $\gamma$- BCK algebras 133 145 3360 10.22103/jmmr.2022.19322.1234 EN Arsham Borumand Saeid Department of pure Mathematics, Facultu of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, Iran. M Murali Krishna Rao Department of Mathematics, Sankethika Institute of Tech. and Management, Visakhapatnam, 530 041, India R Kumar Kona Department of Mathematics, GIS, GITAM (Deemed to be University), Visakhapatnam- 530 045, A.P., India Journal Article 2022 04 16 We know that $\Gamma-$ring, $\Gamma-$incline, $\Gamma-$semiring, $\Gamma-$semigroup are generalizations of<br />ring, incline, semiring and semigroup respectively. In this paper, we introduce the concept of $\Gamma-$BCK-algebras as a generalization of BCK-algebras and study $\Gamma-$BCK-algebras. We also introduce subalgebra, ideal, closed ideal, normal subalgebra, normal ideal and construct quotient of $\Gamma-$BCK-algebras. We prove that if $f: M\to L$ be a normal homomorphism of $\Gamma-$BCK-algebras $M$ and $N,$ then $\Gamma-$BCK-algebra $M/N$ is isomorphic to $Im (f)$ where $N =Ker (f).$ https://jmmrc.uk.ac.ir/article_3360_c7ccfa24ae9c04d2f4a786b35f1867e6.pdf