In this article three classes of ideals are discussed: associative ideals, commutative ideals, implicative ideals and positive implicative ideals, and their elementary properties. Some of their properties and the relationships between them have not been proven yet, and will be completed in the following article.
In this article the notion of the power of an element of BCI-algebra and its period in the book , sections 1.4 to 1.5 are firstly given. Then the definition of BCI-homomorphism is defined and the fundamental theorem of homomorphism, the first isomorphism theorem and the second isomorphism theorem are proved following the book , section 1.6.
It is known that commutative BCK-algebras form a variety, but BCK-algebras do not . Therefore H. Yutani introduced the notion of quasicommutative BCK-algebras. In this article we first present the notion and general theory of quasi-commutative BCI-algebras. Then we discuss the reduction of the type of quasi-commutative BCK-algebras and some special classes of quasicommutative BCI-algebras.