Abstract
The paper contains an investigation of the notion of BCI-algebras and BCK-algebras. The concept of quantale was created in 1984 to develop a framework for non-commutative spaces and quantum mechanics with a view toward non-commutative logic. Implicational al- gebras like pseudo-BCK algebras are quantum B-algebras, and every quantum B-algebra can be recovered from its spectrum which is a quantale. [25]. A brief introduction to quantum mechanics is given. A new generalization of BCI=BCK- algebra and some there properties have been given. The BCI- quantum algebra and BCk- quantum algebra have been studied. The BCI- Lie algebra and BCK- Lie algebra are intro- duced. Various examples have been given.