Pseudo-BCH-algebras

On branchwise commutative pseudo-BCH algebras

“…Then (X ∪ Y ; * , , 0) is a pseudo-BCH algebra (see [15]). From [14] it follows that in any pseudo-BCH algebra X, for all x, y ∈ X, we have:…”

“…Remark. By Theorem 3.4 of [14], a pseudo-BCH algebra is a pseudo-BCI algebra if and only if it satisfies the following implication:…”

Strong ideals and horizontal ideals in pseudo-BCH-algebras

“…Following the terminology of [20], the set {a ∈ X : a = 0 * (0 * a)} will be called the centre of X. W shall denote it by Cen X. By Proposition 4.1 of [20], Cen X is [18] Andrzej Walendziak the set of all minimal elements of X, that is, Cen X = {a ∈ X :…”

“…[20]) Let I be an ideal of X. For any x, y ∈ X, if y ∈ I and x y, then x ∈ I.Proposition 2.10 ([20])Let X be a pseudo-BCH-algebra and I be a subset of X satisfying (I1).…”

“…[20]) An ideal I of X is closed if and only if I is a subalgebra of X.Proposition 3.2 ([20])Every ideal of a finite pseudo-BCH-algebra is closed.Example 3.3 Let M be the set of all matrices of the form A = x y 0 1 , where x and y are rational numbers such that x > 0. Evidently, (M ; •, E), where • is the usual multiplication of matrices and E = 1 0 0 1 , is a group.…”