The smallest class of subalgebras of a commutative BCK-algebra containing initial subsets

Document Type : Original Paper


1 Department of Mathematics, Shahid Chamran University of Ahvaz, Ahvaz, Iran

2 Department of Mathematics, Faculty of Sciences, Payame noor University, Tehran, Iran


In this paper, we assume that X is a BCK-algebra and y, t elements of X. We assign to these elements a set, denoted by F(y; t). We show that F(y; t) is a subalgebra of X. Then we prove that a BCK-algebra X is a Linear Commutative BCK-algebra if and only if every F(y; t) is an initial set of X. Moreover, we give a necessary and sufficient condition for F(y; t) to be an ideal. Finally, we show that the set consisting of all these sets forms a bounded distributive lattice.


Main Subjects

Volume 10, Issue 1
May 2020
Pages 158-171
  • Receive Date: 16 January 2019
  • Revise Date: 08 December 2019
  • Accept Date: 17 January 2020
  • First Publish Date: 21 May 2020