Note on quotient structures of CKL-algebras

In this paper, we consider properties of quotient CKL-algebras by three types of ideals such as positive implicative, commutative and implicative. We show that (1) the class of all positive implicative ideals is identical with the class of all commutative and implicative ideals for any (not necessary bounded) CKL-algebra; (2) For every bounded CKL-algebra X, if I is commutative then the quotient algebra X/I is an involutive Rl-monoid, hence MV-algebra; (3) In addition, if I is implicative then X/I is a Boolean algebra. All of these results are proved through properties of quotient algebras X/I.