MATEMATIQKI VESNIK

58 (2006), 39–46

UDK 512.552

originalni nauqni rad research paper

ON PSEUDO-BCI IDEALS OF PSEUDO-BCI ALGEBRAS

Young Bae Jun, Hee Sik Kim and J. Neggers

Abstract. The notions of pseudo-atoms, pseudo-BCI ideals and pseudo-BCI homomorphisms in pseudo-BCI algebras are introduced. Characterizations of a pseudo-BCI ideal are displayed, and conditions for a subset to be a pseudo-BCI ideal are given. The concept of a ⋄- medial pseudo-BCI algebra is also introduced, and its characterization is provided. We show that every pseudo-BCI homomorphic image and preimage of a pseudo-BCI ideal is also a pseudo-BCI ideal.

1. Introduction

G. Georgescu and A. Iorgulescu [1] introduced the notion of a pseudo-BCK algebra as an extended notion of BCK-algebras. In [2], Y. B. Jun, one of the present authors, gave a characterization of pseudo-BCK algebra, and provided conditions for a pseudo-BCK algebra to be ∧-semi-lattice ordered (resp. ∩-semi-lattice ordered). Y. B. Jun et al. [4] introduced the notion of (positive implicative) pseudo-ideals in a pseudo-BCK algebra, and then they investigated some of their properties. In [2], W. A. Dudek and Y. B. Jun introduced the notion of pseudo-BCI algebras as an extension of BCI-algebras, and investigated some properties. In this paper, we introduce the concepts of pseudo-atoms, pseudo-BCI ideals and pseudo-BCI homomorphisms in pseudo-BCI algebras. We display characterizations of a pseudo-BCI ideal, and provide conditions for a subset to be a pseudo-BCI ideal. We also introduced the notion of a ⋄-medial pseudo-BCI algebra, and give its characterization. We prove that every pseudo-BCI homomorphic image and preimage of a pseudo-BCI ideal is also a pseudo-BCI ideal.

2. Preliminaries

Recall that a BCI-algebra is an algebra (X, ∗, 0) of type (2,0) satisfying the following axioms: for every x, y, z ∈ X, • ((x ∗ y) ∗ (x ∗ z)) ∗ (z ∗ y) = 0,

AMS Subject Classification: 06 F 35, 03 G 25

Keywords and phrases: Pseudo-BCK/BCI-algebra, pseudo-atom, pseudo-BCI ideal, pseudo-BCI homomorphism.

The first author was supported by Korea Research Foundation Grant (KRF-2001-005-D00002).

• (x ∗ (x ∗ y)) ∗ y = 0,

• x ∗ x = 0,

• x ∗ y = 0 and y ∗ x = 0 imply x = y.

For any BCI-algebra X, the relation ≤ defined by x ≤ y if and only if x∗y = 0 is a partial order on X. A nonempty subset I of a BCI-algebra X is called a BCIideal of X if it satisfies

• 0 ∈ I,

• ∀x, y ∈ X, x ∗ y ∈ I, y ∈ I ⇒ x ∈ I.

3. Properties of Pseudo-BCI algebras

Definition 3.1. A pseudo-BCI algebra is a structure 𝔛 = (X, ⪯, ∗, ⋄, 0), where “⪯” is a binary relation on a set X, “∗” and “⋄” are binary operations on X and “0” is an element of X, verifying the axioms: for all x, y, z ∈ X, (a1) (x ∗ y) ⋄ (x ∗ z) ⪯ z ∗ y, (x ⋄ y) ∗ (x ⋄ z) ⪯ z ⋄ y, (a2) x ∗ (x ⋄ y) ⪯ y, x ⋄ (x ∗ y) ⪯ y, (a3) x ⪯ x, (a4) x ⪯ y, y ⪯ x ⟹ x = y, (a5) x ⪯ y ⟺ x ∗ y = 0 ⟺ x ⋄ y = 0.

Note that every pseudo-BCI algebra satisfying x ∗ y = x ⋄ y for all x, y ∈ X is a BCI-algebra. Every pseudo-BCK algebra is a pseudo-BCI algebra.

Proposition 3.2. [2] In a pseudo-BCI algebra 𝔛 the following holds: (p1) x ⪯ 0 ⇒ x = 0. (p2) x ⪯ y ⇒ z ∗ y ⪯ z ∗ x, z ⋄ y ⪯ z ⋄ x. (p3) x ⪯ y, y ⪯ z ⇒ x ⪯ z. (p4) (x ∗ y) ⋄ z = (x ⋄ z) ∗ y. (p5) x ∗ y ⪯ z ⇔ x ⋄ z ⪯ y. (p6) (x ∗ y) ∗ (z ∗ y) ⪯ x ∗ z, (x ⋄ y) ⋄ (z ⋄ y) ⪯ x ⋄ z. (p7) x ⪯ y ⇒ x ∗ z ⪯ y ∗ z, x ⋄ z ⪯ y ⋄ z. (p8) x ∗ 0 = x = x ⋄ 0. (p9) x ∗ (x ⋄ (x ∗ y)) = x ∗ y and x ⋄ (x ∗ (x ⋄ y)) = x ⋄ y.

Example 3.3. Let X = [0, ∞] and let ≤ be the usual order on X. Define binary operations “∗” and “⋄” on X by

x ∗ y : = \{ \begin{matrix}0 & if x ≤ y, \\ \frac{2x}{π} arctan(ln(\frac{x}{y})) & if y < x, \end{matrix}
x ⋄ y : = \{ \begin{matrix}0 & if x ≤ y, \\ xe^{− tan( \frac{πy}{2x})} & if y < x, \end{matrix}

On pseudo-BCI ideals of pseudo-BCI algebras 41

for all x, y ∈ X. Then 𝔛 := (X, ≤, ∗, ⋄, 0) is a pseudo-BCK algebra, and so a pseudo-BCI algebra.

Proposition 3.4 In a pseudo-BCI algebra 𝔛, the following holds for all x, y ∈ X:

(i) 0 ∗ (x ⋄ y) ⪯ y ⋄ x. (ii) 0 ⋄ (x ∗ y) ⪯ y ∗ x. (iii) 0 ∗ (x ∗ y) = (0 ⋄ x) ⋄ (0 ∗ y). (iv) 0 ⋄ (x ⋄ y) = (0 ∗ x) ∗ (0 ⋄ y).

Proof. (i) and (ii). We have 0 ∗ (x ⋄ y) = (x ⋄ x) ∗ (x ⋄ y) ⪯ y ⋄ x and 0 ⋄ (x ∗ y) = (x ∗ x) ⋄ (x ∗ y) ⪯ y ∗ x by (a1) and (a3).

(iii) and (iv). Using (a3) and (p4), we obtain

(0 ⋄ x) ⋄ (0 ∗ y) = (((x ∗ y) ∗ (x ∗ y)) ⋄ x) ⋄ (0 ∗ y)
= (((x ∗ y) ⋄ x) ∗ (x ∗ y)) ⋄ (0 ∗ y)
= (((x ⋄ x) ∗ y) ∗ (x ∗ y)) ⋄ (0 ∗ y)
= ((0 ∗ y) ∗ (x ∗ y)) ⋄ (0 ∗ y)
= ((0 ∗ y) ⋄ (0 ∗ y)) ∗ (x ∗ y)
= 0 ∗ (x ∗ y)

and

(0 ∗ x) ∗ (0 ⋄ y) = (((x ⋄ y) ⋄ (x ⋄ y)) ∗ x) ∗ (0 ⋄ y)
= (((x ⋄ y) ∗ x) ⋄ (x ⋄ y)) ∗ (0 ⋄ y)
= (((x ∗ x) ⋄ y) ⋄ (x ⋄ y)) ∗ (0 ⋄ y)
= ((0 ⋄ y) ⋄ (x ⋄ y)) ∗ (0 ⋄ y)
= ((0 ⋄ y) ∗ (0 ⋄ y)) ⋄ (x ⋄ y)
= 0 ⋄ (x ⋄ y).

Definition 3.5. An element w of a pseudo-BCI algebra 𝔛 is called a pseudoatom if for every x ∈ X, x ⪯ w implies x = w.

Obviously, 0 is a pseudo-atom of 𝔛.

Proposiiton 3.6. Let 𝔛 be a pseudo-BCI algebra. If an element w of 𝔛 satisfies the identity y ∗ (y ⋄ (w ∗ x)) = w ∗ x for all x, y ∈ X, then w is a pseudoatom of 𝔛.

Proof. Let y ∈ X be such that y ⪯ w. Then

w = w ∗ 0 = y ∗ (y ⋄ (w ∗ 0)) = y ∗ (y ⋄ w) = y ∗ 0 = y.

Hence w is a pseudo-atom of 𝔛.

Proposition 3.7. Let 𝔛 be a pseudo-BCI algebra and let w be a pseudo-atom of 𝔛. Then the following are true.

(i) w = x ⋄ (x ∗ w), ∀x ∈ X. (ii) (x ∗ y) ⋄ (x ∗ w) = w ∗ y, ∀x, y ∈ X. (iii) w ∗ (x ⋄ y) ⪯ y ⋄ (x ∗ w), ∀x, y ∈ X. (iv) (w ⋄ x) ∗ (y ⋄ z) ⪯ (z ⋄ (y ∗ w)) ⋄ x, ∀x, y, z ∈ X.

(v) 0 ⋄ (y ∗ w) = w ∗ y, ∀y ∈ X. Proof. (i) Since x ⋄ (x ∗ w) ⪯ w by (a2), it follows that w = x ⋄ (x ∗ w). (ii) For every x, y ∈ X, we have

(x ∗ y) ⋄ (x ∗ w) = (x ⋄ (x ∗ w)) ∗ y = w ∗ y

by (p4) and (i).

(iii) Using (i), (a2), (p4) and (p7), we have

w ∗ (x ⋄ y) = (x ⋄ (x ∗ w)) ∗ (x ⋄ y) = (x ∗ (x ⋄ y)) ⋄ (x ∗ w) ⪯ y ⋄ (x ∗ w).

(iv) Using (p4), (p7) and (iii), we get (w ⋄ x) ∗ (y ⋄ z) = (w ∗ (y ⋄ z)) ⋄ x ⪯ (z ⋄ (y ∗ w)) ⋄ x.

(v) For every y ∈ X, we obtain

w ∗ y = (w ⋄ 0) ∗ (y ⋄ 0)  by (p8)
⪯ (0 ⋄ (y ∗ w)) ⋄ 0  by (iv)
= 0 ⋄ (y ∗ w)  by (p8)
⪯ w ∗ y,  by Proposition 3.4(ii)

and so 0 ⋄ (y ∗ w) = w ∗ y.

Definition 3.8. A pseudo-BCI algebra 𝔛 is said to be ⋄-medial if it satisfies the following identity: (M1) (x ∗ y) ⋄ (z ∗ u) = (x ∗ z) ⋄ (y ∗ u), ∀x, y, z, u ∈ X.

Proposition 3.9. A pseudo-BCI algebra 𝔛 is ⋄-medial if and only if it satisfies: (M2) x ⋄ (y ∗ z) = (x ∗ y) ⋄ (0 ∗ z), ∀x, y, z ∈ X.

Proof. Assume that 𝔛 is ⋄-medial. Putting z = 0 and u = z in (M1) and using (p8), we have

(x ∗ y) ⋄ (0 ∗ z) = (x ∗ 0) ⋄ (y ∗ z) = x ⋄ (y ∗ z).

Suppose that 𝔛 satisfies the condition (M2). Then

(x ∗ y) ⋄ (z ∗ u) = (x ⋄ (z ∗ u)) ∗ y  by (p4)
= ((x ∗ z) ⋄ (0 ∗ u)) ∗ y  by (M2)
= ((x ∗ z) ∗ y) ⋄ (0 ∗ u)  by (p4)
= (x ∗ z) ⋄ (y ∗ u).  by (M2)

Therefore 𝔛 is ⋄-medial.

Proposition 3.10. Every ⋄-medial pseudo-BCI algebra 𝔛 satisfies the following identities.

(i) x ∗ y = 0 ⋄ (y ∗ x). (ii) 0 ⋄ (0 ∗ x) = x. (iii) x ⋄ (x ∗ y) = y.

Proof. (i) For any x, y ∈ X, we have

x ∗ y = (x ∗ y) ⋄ 0 = (x ∗ y) ⋄ (x ∗ x)
= (x ∗ x) ⋄ (y ∗ x) = 0 ⋄ (y ∗ x).

(ii) If we put y = 0 in (i), then we have (ii).

(iii) Using (ii), (a3) and (p8), we get

x ⋄ (x ∗ y) = (x ∗ 0) ⋄ (x ∗ y) = (x ∗ x) ⋄ (0 ∗ y) = 0 ⋄ (0 ∗ y) = y.

4. Pseudo-BCI ideals

Let 𝔛 be a pseudo-BCI -algebra. For any nonempty subset J of X and any element y of X, we denote

∗(y, J) := {x ∈ X | x ∗ y ∈ J} and ⋄ (y, J) := {x ∈ X | x ⋄ y ∈ J}.

Note that ∗(y, J) ∩ ⋄(y, J) = {x ∈ X | x ∗ y ∈ J, x ⋄ y ∈ J}.

Definition 4.1. A nonempty subset J of a pseudo-BCI algebra 𝔛 is called a pseudo-BCI ideal of 𝔛 if it satisfies (I1) 0 ∈ J, (I2) ∀y ∈ J, ∗(y, J) ⊆ J and ⋄ (y, J) ⊆ J.

Note that if 𝔛 is a pseudo-BCI algebra satisfying x ∗ y = x ⋄ y for all x, y ∈ X, then the notion of a pseudo-BCI ideal and a BCI-ideal coincide.

Proposition 4.2. Let J be a pseudo-BCI ideal of a pseudo-BCI algebra 𝔛. If x ∈ J and y ⪯ x, then y ∈ J.

Proof is straightforward.

Theorem 4.3. For any element a of a pseudo-BCI algebra 𝔛, the initial section ↓ a := {x ∈ X | x ⪯ a} is a pseudo-BCI ideal of 𝔛 if and only if the following implications hold:

(i) ∀x, y, z ∈ X, x ∗ y ⪯ z, y ⪯ z ⇒ x ⪯ z, (ii) ∀x, y, z ∈ X, x ⋄ y ⪯ z, y ⪯ z ⇒ x ⪯ z.

Proof. Assume that for each a ∈ X, ↓ a is a pseudo-BCI ideal of 𝔛. Let x, y, z ∈ X be such that x ∗ y ⪯ z, x ⋄ y ⪯ z, and y ⪯ z. Then x ∗ y ∈ ↓z, x ⋄ y ∈ ↓z, and y ∈ ↓z, that is, y ∈ ↓z, x ∈ ∗(y, ↓z) and x ∈ ⋄(y, ↓z). Since ↓z is a pseudo-BCI ideal of 𝔛, it follows from (I2) that x ∈ ↓z, i.e., x ⪯ z. Conversely, consider ↓z for any z ∈ X. Obviously 0 ∈ ↓z. For every y ∈ ↓z, let a ∈ ∗(y, ↓z) and b ∈ ⋄(y, ↓z).

Then a ∗ y ∈ ↓z and b ⋄ y ∈ ↓z, i.e., a ∗ y ⪯ z and b ⋄ y ⪯ z. Since y ∈ ↓z, it follows from the hypothesis that a ⪯ z and b ⪯ z, i.e., a ∈ ↓z and b ∈ ↓z. This shows that ∗(y, ↓z) ⊆ ↓z and ⋄(y, ↓z) ⊆ ↓z. Hence ↓z is a pseudo-BCI ideal of 𝔛 for every z ∈ X.

Theorem 4.4. If J is a pseudo-BCI ideal of a pseudo-BCI algebra 𝔛, then (i) ∀x, y, z ∈ X, x, y ∈ J, z ∗ y ⪯ x ⇒ z ∈ J, (ii) ∀a, b, c ∈ X, a, b ∈ J, c ⋄ b ⪯ a ⇒ c ∈ J.

Proof. Suppose that J is a pseudo-ideal of 𝔛 and let x, y, z ∈ X be such that x, y ∈ J and z ∗ y ⪯ x. Then (z ∗ y) ⋄ x = 0 ∈ J, and so z ∗ y ∈ ⋄(x, J) ⊆ J. It follows that z ∈ ∗(y, J) ⊆ J so that (i) is valid. Now let a, b, c ∈ X be such that a, b ∈ J and c ⋄ b ⪯ a. Then (c ⋄ b) ∗ a = 0 ∈ J, and thus c ⋄ b ∈ ∗(a, J) ⊆ J. Hence c ∈ ⋄(b, J) ⊆ J, which shows (ii).

A pseudo-BCI subalgebra of a pseudo-BCI algebra 𝔛 is a subset S of 𝔛 which satisfies x ∗ y ∈ S and x ⋄ y ∈ S for all x, y ∈ S. We provide conditions for a pseudo-BCI subalgebra to be a pseudo-BCI ideal.

Theorem 4.5. Let J be a pseudo-BCI subalgebra of a pseudo-BCI algebra 𝔛. Then J is a pseudo-BCI ideal of 𝔛 if and only if

∀x, y ∈ X, x ∈ J, y ∈ X − J ⇒ y ∗ x ∈ X − J and y ⋄ x ∈ X − J.

Proof. Assume that J is a pseudo-BCI ideal of 𝔛 and let x, y ∈ X be such that x ∈ J and y ∈ X − J. If y ∗ x ∉ X − J, then y ∗ x ∈ J, i.e., y ∈ ∗(x, J) ⊆ J which is a contradiction. Hence y ∗ x ∈ X − J. Now if y ⋄ x ∉ X − J, then y ⋄ x ∈ J and so y ∈ ⋄(x, J) ⊆ J. This is a contradiction, and therefore y ⋄ x ∈ X − J. Conversely, assume that

∀x, y ∈ X, x ∈ J, y ∈ X − J ⇒ y ∗ x ∈ X − J and y ⋄ x ∈ X − J.

Since J is a pseudo-BCI subalgebra, therefore 0 ∈ J. For every x ∈ J, let y ∈ ∗(x, J). Then y ∗ x ∈ J. If y ∉ J, then y ∗ x ∈ X − J by assumption. This is a contradiction, and so y ∈ J which shows that ∗(x, J) ⊆ J. Now let z ∈ ⋄(x, J). Then z ⋄ x ∈ J. It follows from the hypothesis that z ∈ J so that ⋄(x, J) ⊆ J. Consequently, J is a pseudo-BCI ideal of 𝔛.

Using [2, Theorem 3.5], we know that every pseudo-BCI algebra 𝔛 contains a maximal pseudo-BCK algebra K(𝔛) := {x ∈ X | 0 ⪯ x}.

Proposition 4.6. Let 𝔛 be a pseudo-BCI algebra. If x ∈ K(𝔛) and y ∈ X − K(𝔛), then x ∗ y ∈ X − K(𝔛) and x ⋄ y ∈ X − K(𝔛).

Proof. If x ∗ y ∈ K(𝔛), then x ⋄ (x ∗ y) ∈ K(𝔛) because K(𝔛) is a pseudo-BCI subalgebra of 𝔛. Hence 0 ⪯ x⋄(x∗y) ⪯ y, and so y ∈ K(𝔛). This is a contradiction. Now if x ⋄ y ∈ K(𝔛), then x ∗ (x ⋄ y) ∈ K(𝔛) and so 0 ⪯ x ∗ (x ⋄ y) ⪯ y by (a2). Therefore y ∈ K(𝔛), a contradiction.

Theorem 4.7. Let 𝔛 be a pseudo-BCI algebra. Then the maximal pseudo-BCK algebra K(𝔛) is a pseudo-BCI ideal of 𝔛.

Proof. Let x, y ∈ X be such that x ∈ K(𝔛) and y ∈ X − K(𝔛). Using (a1) and (p8), we have

(y ∗ x) ⋄ y = (y ∗ x) ⋄ (y ∗ 0) ⪯ 0 ∗ x = 0

and

(y ⋄ x) ∗ y = (y ⋄ x) ∗ (y ⋄ 0) ⪯ 0 ⋄ x = 0

since x ∈ K(𝔛). It follows from (p1) that (y ∗ x) ⋄ y = 0 and (y ⋄ x) ∗ y = 0 so that y ∗ x ⪯ y and y ⋄ x ⪯ y. If y ∗ x ∈ K(𝔛), then 0 ⪯ y ∗ x ⪯ y, and so y ∈ K(𝔛) which is a contradiction. Now if y ⋄ x ∈ K(𝔛), then 0 ⪯ y ⋄ x ⪯ y which implies that y ∈ K(𝔛), a contradiction. Hence y ∗ x ∈ X − K(𝔛) and y ⋄ x ∈ X − K(𝔛). By means of Theorem 4.5, we know that K(𝔛) is a pseudo-BCI ideal of 𝔛.

Theorem 4.8. Let J be a pseudo-BCI ideal of a pseudo-BCI algebra 𝔛. Then the following are equivalent.

(i) J contains the maximal pseudo-BCK algebra K(𝔛). (ii) ∀x, y ∈ X, x ⪯ y, x ∈ J ⇒ y ∈ J.

Proof. The sufficiency is straightforward. Assume that K(𝔛) ⊂ J. Let x, y ∈ X be such that x ⪯ y and x ∈ J. Then x ∗ y = 0, and so

0 = 0 ⋄ 0 = 0 ⋄ (x ∗ y) = (x ∗ x) ⋄ (x ∗ y) ⪯ y ∗ x.

Thus y ∗ x ∈ K(𝔛) ⊂ J, which implies that y ∈ ∗(x, J) ⊆ J.

Definition 4.9. Let 𝔛 and 𝔜 be pseudo-BCI algebras. A mapping 𝔣 : 𝔛 → 𝔜 is called a pseudo-BCI homomorphism if 𝔣(x∗y) = 𝔣(x)∗𝔣(y) and 𝔣(x⋄y) = 𝔣(x)⋄𝔣(y) for all x, y ∈ X.

Note that if 𝔣 : 𝔛 → 𝔜 is a pseudo-BCI homomorphism, then 𝔣(0_{𝔛}) = 0_{𝔜} where 0_{𝔛} and 0_{𝔜} are zero elements of 𝔛 and 𝔜, respectively.

Theorem 4.10. Let 𝔣 : 𝔛 → 𝔜 be a pseudo-BCI homomorphism of pseudo-BCI algebras 𝔛 and 𝔜. (i) If J is a pseudo-BCI ideal of 𝔜, then 𝔣^{−1}(J) is a pseudo-BCI ideal of 𝔛. (ii) If 𝔣 is surjective and I is a pseudo-BCI ideal of 𝔛, then 𝔣(I) is a pseudo-BCI ideal of 𝔜.

Proof. (i) Assume that J is a pseudo-BCI ideal of 𝔜. Obviously 0_{𝔛} ∈ 𝔣^{−1}(J). For every y ∈ 𝔣^{−1}(J), let

a ∈ ∗(y, 𝔣^{−1}(J)) and b ∈ ⋄(y, 𝔣^{−1}(J)).

Then a ∗ y ∈ 𝔣^{−1}(J) and b ⋄ y ∈ 𝔣^{−1}(J). It follows that 𝔣(a) ∗ 𝔣(y) = 𝔣(a ∗ y) ∈ J and 𝔣(b) ⋄ 𝔣(y) = 𝔣(b ⋄ y) ∈ J so that 𝔣(a) ∈ ∗(𝔣(y), J) ⊆ J and 𝔣(b) ∈ ⋄(𝔣(y), J) ⊆ J because J is a pseudo-BCI ideal of 𝔛 and 𝔣(y) ∈ J. Hence a ∈ 𝔣^{−1}(J) and b ∈ 𝔣^{−1}(J), which shows that ∗(y, 𝔣^{−1}(J)) ⊆ 𝔣^{−1}(J) and ⋄(y, 𝔣^{−1}(J)) ⊆ 𝔣^{−1}(J). Hence 𝔣^{−1}(J) is a pseudo-BCI ideal of 𝔛.

(ii) Assume that 𝔣 is surjective and let I be a pseudo-BCI ideal of 𝔛. Obviously, 0_{𝔜} ∈ 𝔣(I). For every y ∈ 𝔣(I), let a, b ∈ Y be such that a ∈ ∗(y, 𝔣(I)) and b ∈ ⋄(y, 𝔣(I)). Then a ∗ y ∈ 𝔣(I) and b ⋄ y ∈ 𝔣(I). It follows that there exist x_{∗}, x_{⋄} ∈ I such that 𝔣(x_{∗}) = a ∗ y and 𝔣(x_{⋄}) = b ⋄ y. Since y ∈ 𝔣(I), there exists x_{y} ∈ I such that 𝔣(x_{y}) = y. Also since 𝔣 is surjective, there exist x_{a}, x_{b} ∈ X such that 𝔣(x_{a}) = a and 𝔣(x_{b}) = b. Hence

𝔣(x_{a} ∗ x_{y}) = 𝔣(x_{a}) ∗ 𝔣(x_{y}) = a ∗ y ∈ 𝔣(I)

and

𝔣(x_{b} ⋄ x_{y}) = 𝔣(x_{b}) ⋄ 𝔣(x_{y}) = b ⋄ y ∈ 𝔣(I),

which imply that x_{a} ∗ x_{y} ∈ I and x_{b} ⋄ x_{y} ∈ I. Since I is a pseudo-BCI ideal of 𝔛, we get x_{a} ∈ ∗(x_{y}, I) ⊆ I and x_{b} ∈ ⋄(x_{y}, I) ⊆ I, and thus a = 𝔣(x_{a}) ∈ 𝔣(I) and b = 𝔣(x_{b}) ∈ 𝔣(I). This shows that ∗(y, 𝔣(I)) ⊆ 𝔣(I) and ⋄(y, 𝔣(I)) ⊆ 𝔣(I). Therefore 𝔣(I) is a pseudo-BCI ideal of 𝔜.

Corollary 4.11. Let 𝔣 : 𝔛 → 𝔜 be a pseudo-BCI homomorphism of pseudo-BCI algebras 𝔛 and 𝔜. Then the kernel

Ker(𝔣) := {x ∈ X | 𝔣(x) = 0_{𝔜}}

of 𝔣 is a pseudo-BCI ideal of 𝔛.

Proof is straightforward.

REFERENCES

[1] G. Georgescu and A. Iorgulescu, Pseudo-BCK algebras: an extension of BCK algebras, Combinatorics, computability and logic, 97–114, Springer Ser. Discrete Math. Theor. Comput. Sci., Springer, London, 2001.

[2] W. A. Dudek and Y. B. Jun, On pseudo-BCI algebras, (submitted).

[3] Y. B. Jun, Characterizations of pseudo-BCK algebras, Scientiae Mathematicae Japonicae Online 7 (2002), 225–230.

[4] Y. B. Jun, M. Kondo and K. H. Kim, Pseudo-ideals of pseudo-BCK algebras, Scientiae Mathematicae Japonicae 58 (2003), 93–97.

(received 28.08.2004)

Y. B. Jun, Department of Mathematics Education, Gyeongsang National University, Chinju (Jinju) 660-701, Korea

E-mail: ybjun@nongae.gsnu.ac.kr

H. S. Kim, Department of Mathematics, Hanyang University, Seoul 133-791, Korea

E-mail: heekim@hanyang.ac.kr

J. Neggers, Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487-0350, USA

E-mail: jneggers@gp.as.ua.edu