Bounded commutative BCK-algebras satisfying d.c.c. (English) Zbl 0638.03066

Hoo, C. S.

Math. Jap. 32, 217-225 (1987).

We prove that a bounded commutative BCK-algebra satisfying d.c.c. and containing an atom splits as the direct sum of a prime BCK ideal and a finite chain which is the interval between 0 and a minimal complemented element which is also a join prime. The complement of this element is the generator of the prime BCK ideal which is also a principal lattice ideal and the annihilator ideal of the atom.

MSC:

03G25	Other algebras related to logic
06B10	Lattice ideals, congruence relations

Keywords:

descending chain condition; bounded commutative BCK-algebra; prime BCK ideal