Ideals of BCI-algebras and their Properties

For simplicity, we use the following convention: X denotes a BCI-algebra, X1 denotes a non empty subset of X, A, I denote ideals of X, x, y, z denote elements of X, and a denotes an element of A. The following four propositions are true: (1) For all elements x, y, z, u ofX such that x ≤ y holds u\(z\x) ≤ u\(z\y). (2) For all elements x, y, z, u of X holds x \ (y \ z) \ (x \ (y \ u)) ≤ z \ u. (3) For all elements x, y, z, u, v of X holds x\(y\(z\u))\(x\(y\(z\v))) ≤ v \ u. (4) For all elements x, y of X holds 0X \ (x \ y) \ (y \ x) = 0X . Let us consider X and let a be an element of X. The initial section of a is defined by: (Def. 1) The initial section of a = {x ∈ X: x ≤ a}.