BCI-algebras with Condition (S) and their Properties

The papers [5], [12], [3], [1], [6], [2], [10], [9], [4], [11], and [7] provide the notation and terminology for this paper. We introduce BCI stuctures with complements which are extensions of BCI structure with 0 and zero structure and are systems 〈 a carrier, an external complement, an internal complement, a zero 〉, where the carrier is a set, the external complement and the internal complement are binary operations on the carrier, and the zero is an element of the carrier. Let us mention that there exists a BCI structure with complements which is non empty and strict. Let A be a BCI structure with complements and let x, y be elements of A. The functor x · y yields an element of A and is defined as follows: (Def. 1) x · y = (the external complement of A)(x, y).