Atoms and ideals of pseudo-BCI-algebras

Grzegorz Dymek

Atoms and ideals of pseudo-BCI-algebras

Abstract. The main subject of the paper are atoms and ideals of a pseudo-BCI- algebra. Many different characterizations of them are given. Some connections between ideals and subalgebras are presented. Conditions for the set At(X) of atoms of a pseudo-BCI-algebra X to be an ideal of X are established.

2010 Mathematics Subject Classification: 03G25, 06F35.

Key words and phrases: pseudo-BCI-algebra, atom, (closed) ideal.

1. Introduction. The notion of BCI-algebras has been introduced by K. Is´eki in 1966 (see [9]). BCI-algebras are algebraic formulation of the BCI-system in combi- natory logic which has application in the language of functional programming. The name of BCI-algebras originates from the combinatories B, C, I in combinatory logic.

The notion of pseudo-BCI-algebras has been introduced by W. A. Dudek and Y. B. Jun in [1] as an extension of BCI-algebras and it was investigated by several authors in [2], [11] and [13]. These algebras have connections with pseudo-BCK- algebras, pseudo-BL-algebras and pseudo-MV-algebras introduced by G. Georgescu and A. Iorgulescu in [3], [4] and [5], respectively. More about those algebras the reader can find in [8].

Atoms and ideals of algebras are important algebraic notions and for BCI- algebras they have been extensively investigated by many authors (for example [6], [7], [12] and [14]). In the paper we study atoms and ideals of pseudo-BCI- algebras. In Section 3 we present many characterizations of the notion of an atom of a pseudo-BCI-algebra X and investigate the set At(X) of atoms of X . In this section we study also the notion of branches of a pseudo-BCI-algebra. Ideals of a pseudo-BCI-algebra are the main subject of Section 4. We characterize them and give some connections between ideals and subalgebras of a pseudo-BCI-algebra.

The fact that every subalgebra of a p-semisimple pseudo-BCI-algebra is an ideal

is shown. We prove also theorem saying that if the set At(X) is finite, then every

ideal of a pseudo-BCI-algebra X is a subalgebra. Finally, we establish conditions

for the set At(X) to be an ideal of a pseudo-BCI-algebra X . For the convenience of the reader, in Section 2 we give the necessary material needed in the sequel, thus making our exposition self-contained.

2. Preliminaries. A pseudo-BCI-algebra is a structure X = (X, ¬, ∗, ◦, 0), where ¬ is 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 and 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. Notice also that every pseudo-BCI-algebra satisfying 0 ¬ x for all x ∈ X is a pseudo-BCK-algebra.

Proposition 2.1 ([1], [11], [13]) Let X be a pseudo-BCI-algebra. The following holds for all x, y, z ∈ X:

(b1) x ¬ 0 ⇒ x = 0,

(b2) x ¬ y ⇒ z ∗ y ¬ z ∗ x, z ◦ y ¬ z ◦ x, (b3) x ¬ y, y ¬ z ⇒ x ¬ z,

(b4) (x ∗ y) ◦ z = (x ◦ z) ∗ y, (b5) x ∗ y ¬ z ⇔ x ◦ z ¬ y,

(b6) (x ∗ y) ∗ (z ∗ y) ¬ x ∗ z, (x ◦ y) ◦ (z ◦ y) ¬ x ◦ z, (b7) x ¬ y ⇒ x ∗ z ¬ y ∗ z, x ◦ z ¬ y ◦ z,

(b8) x ∗ 0 = x = x ◦ 0,

(b9) x ∗ (x ◦ (x ∗ y)) = x ∗ y, x ◦ (x ∗ (x ◦ y)) = x ◦ y, (b10) 0 ∗ (x ◦ y) ¬ y ◦ x,

(b11) 0 ◦ (x ∗ y) ¬ y ∗ x,

(b12) 0 ∗ (x ∗ y) = (0 ◦ x) ◦ (0 ∗ y), (b13) 0 ◦ (x ◦ y) = (0 ∗ x) ∗ (0 ◦ y), (b14) 0 ∗ x = 0 ◦ x.

Proposition 2.2 ([1]) A structure X = (X, ¬, ∗, ◦, 0) is a pseudo-BCI-algebra if

and only if it satisfies (a1), (a4), (a5) and (b8).

Example 2.3 ([11]) Let X 1 = [0, ∞) and let ¬ be the usual order on X ¹ . Define binary operations ∗ and ◦ on X ¹ by

x ∗ y =

 0 if x ¬ y,

2x

π arc tg(ln( ^x _y )) if y < x, x ◦ y =

 0 if x ¬ y,

xe ^{− tg(}

⁾ if y < x,

for all x, y ∈ X 1 . Then X 1 = (X ₁ , ¬, ∗, ◦, 0) is a pseudo-BCK-algebra, and hence it is a pseudo-BCI-algebra.

Example 2.4 ([2]) Let X 2 = R ² and define binary operations ∗ and ◦ and a binary relation ¬ on X ² by

(x 1 , y 1 ) ∗ (x ² , y 2 ) = (x 1 − x ² , (y 1 − y ² )e ^−x

), (x ₁ , y 1 ) ◦ (x 2 , y 2 ) = (x ₁ − x 2 , y 1 − y 2 e ^x

^−x

),

(x 1 , y 1 ) ¬ (x ² , y 2 ) ⇔ (x ¹ , y 1 ) ∗ (x ² , y 2 ) = (0, 0) = (x 1 , y 1 ) ◦ (x ² , y 2 )

for all (x 1 , y 1 ), (x 2 , y 2 ) ∈ X ² . Then X ² = (X 2 , ¬, ∗, ◦, (0, 0)) is a pseudo-BCI-algebra.

Notice that X ² is not a pseudo-BCK-algebra because there exists (x, y) = (1, 1) ∈ X ² such that (0, 0)  (x, y).

Example 2.5 Let X be a direct product of pseudo-BCI-algebras X ¹ and X ² from Examples 2.3 and 2.4, respectively. Then X is a pseudo-BCI-algebra, where X = [0, ∞) × R ² and binary operations ∗ and ◦ and binary relation ¬ are defined on X by

(x 1 , y 1 , z 1 ) ∗ (x ² , y 2 , z 2 ) =

 (0, y 1 − y ² , (z 1 − z ² )e ^−y

) if x 1 ¬ x ² , ( ^2x _π

arc tg(ln( ^x _x

)), y 1 − y ² , (z 1 − z ² )e ^−y

) if x 2 < x 1 , (x ₁ , y 1 , z 1 ) ◦ (x 2 , y 2 , z 2 ) =

 (0,y 1 − y ² , z 1 − z ² e ^y

^−y

) if x 1 ¬ x ² , (x ₁ e ^{− tg(}

⁾ , y 1 − y 2 , z 1 − z 2 e ^y

^−y

) if x ₂ < x 1 , (x ₁ , y 1 , z 1 ) ¬ (x 2 , y 2 , z 2 ) ⇔ x 1 ¬ x ² and y ₁ = y ₂ and z ₁ = z ₂ .

Notice that X is not a pseudo-BCK-algebra because there exists (x, y, z) = (0, 1, 1) ∈ X such that (0, 0, 0)  (x, y, z).

By a subalgebra of a pseudo-BCI-algebra X , we mean a non-empty subset S of X which satisfies

x ∗ y ∈ S and x ◦ y ∈ S

for all x, y ∈ S.

Proposition 2.6 ([1]) For any pseudo-BCI-algebra X the set K(X) = {x ∈ X : 0 ¬ x}

is a subalgebra of X , and so it is a pseudo-BCK-algebra.

Remark 2.7 Note that if X is a pseudo-BCK-algebra, then X = K(X).

Example 2.8 Let X 1 be the pseudo-BCI-algebra from Example 2.3. Then, as we easily see, K(X ₁ ) = X ₁ .

Example 2.9 Let X ² be the pseudo-BCI-algebra from Example 2.4. Then it is easily seen that K(X 2 ) = {(0, 0)}.

Example 2.10 Let X be the pseudo-BCI-algebra from Example 2.5. Then, by sim- ple calculation, we get K(X) = {(x, 0, 0) : x ­ 0}.

A pseudo-BCI-algebra X is said to be p-semisimple if it satisfies for all x ∈ X, 0 ¬ x ⇒ x = 0.

Note that if X is a p-semisimple pseudo-BCI-algebra, then K(X) = {0}.

Remark 2.11 It is not difficult to see that the pseudo-BCI-algebras X ¹ and X from Examples 2.3 and 2.5, respectively, are not p-semisimple, and the pseudo- BCI-algebra X ² from Example 2.4 is the p-semisimple algebra.

Proposition 2.12 ([2]) Let X be a pseudo-BCI-algebra. Then the following are equivalent: for all a, b, x, y ∈ X,

(i) X is p-semisimple, (ii) x ¬ y ⇒ x = y,

(iii) x ∗ (x ◦ y) = y = x ◦ (x ∗ y), (iv) 0 ∗ (0 ◦ x) = x = 0 ◦ (0 ∗ x),

(v) x ∗ (0 ◦ y) = y ◦ (0 ∗ x), (vi) x ∗ y = 0 ◦ (y ∗ x), (vii) x ◦ y = 0 ∗ (y ◦ x), (viii) x ∗ a = x ∗ b ⇒ a = b,

(ix) a ∗ x = b ∗ x ⇒ a = b.

Remark 2.13 By Proposition 2.12 we have that a p-semisimple pseudo-BCI-algebra is a pseudo-BCI-algebra where ¬ can be replaced by =.

Next proposition is obvious and its proof is omitted.

Proposition 2.14 Every subalgebra of a p-semisimple pseudo-BCI-algebra is p- semisimple.

Next proposition says that p-semisimple pseudo-BCI-algebras and groups are categorically isomorphic.

Proposition 2.15 ([2]) A pseudo-BCI-algebra X = (X, ¬, ∗, ◦, 0) is p-semisimple if and only if (X, +, −, e) is a group, where, for any x, y ∈ X, x + y = x ∗ (0 ◦ y) = y ◦ (0 ∗ x), −x = 0 ∗ x = 0 ◦ x and e = 0. In this case, x ∗ y = x − y, x ◦ y = −y + x and x ¬ y iff x ∗ y = 0 = x ◦ y.

Example 2.16 Let X 2 be the pseudo-BCI-algebra from Example 2.4. Then (X 2 , +, −, (0, 0)) is a group under the following addition:

(x 1 , y 1 ) + (x 2 , y 2 ) = (x 1 + x 2 , y 1 e ^x

+ y 2 )

for all (x 1 , y 1 ), (x 2 , y 2 ) ∈ X ² , with the neutral element (0, 0) and with the inverse

−(x, y) = (−x, −ye ^−x ) of an element (x, y) ∈ X ² .

3. Atoms of pseudo-BCI-algebras. An element a of a pseudo-BCI-algebra X is called an atom of X if for every x ∈ X the following holds

x ¬ a ⇒ x = a.

We will denote by At(X) the set of all atoms of X . Obviously, 0 ∈ At(X). Notice that At(X) ∩ K(X) = {0}. Indeed, if a ∈ At(X) ∩ K(X), then 0 ¬ a and, by above implication, a = 0.

Moreover, observe that 0 is the only atom of a pseudo-BCK-algebra. Therefore, for a pseudo-BCK-algebra X , At(X) = {0}.

Proposition 3.1 ([2]) Let X be a pseudo-BCI-algebra. Then At(X) is a p-semisimple pseudo-BCI-subalgebra of X .

Proposition 3.2 ([2]) Let X be a pseudo-BCI-algebra. Then X is p-semisimple if and only if X = At(X).

Now, we give characterizations of atoms of a pseudo-BCI-algebra.

Theorem 3.3 Let X be a pseudo-BCI-algebra. Then for all a, x ∈ X, the following

are equivalent:

(i) a ∈ At(X),

(ii) x ∗ (x ◦ a) = a = x ◦ (x ∗ a).

Proof (i) ⇒ (ii): Let a ∈ At(X) and x ∈ X. By (a2), we have that x ∗ (x ◦ a) ¬ a and x ◦ (x ∗ a) ¬ a. Thus, by definition of an atom, x ∗ (x ◦ a) = a = x ◦ (x ∗ a).

(ii) ⇒ (i): Let a, x ∈ X be such that x ¬ a. Then x∗a = 0. We show that a ¬ x.

Indeed, by (ii), (b4) and (a3), we get a∗x = (x◦(x∗a))∗x = (x∗x)◦(x∗a) = 0◦0 = 0.

Now, by (a4), x = a. Therefore, a ∈ At(X). 

Theorem 3.4 Let X be a pseudo-BCI-algebra. Then for all a, x, y, z ∈ X, the following are equivalent:

(i) a ∈ At(X),

(ii) x ∗ (x ◦ a) = a = x ◦ (x ∗ a), (iii) a ∗ (x ◦ y) ¬ y ◦ (x ∗ a),

(iv) (a ◦ x) ∗ (y ◦ z) ¬ (z ◦ (y ∗ a)) ◦ x, (v) a ∗ x = 0 ◦ (x ∗ a).

Proof (i) ⇒ (ii): Follows from Theorem 3.3.

(ii) ⇒ (iii): We have by (ii), (b4), (a2) and (b7), a∗(x◦y) = (x◦(x∗a))∗(x◦y) = (x ∗ (x ◦ y)) ◦ (x ∗ a) ¬ y ◦ (x ∗ a).

(iii) ⇒ (iv): By (b4), (iii) and (b7) we obtain (a ◦ x) ∗ (y ◦ z) = (a ∗ (y ◦ z)) ◦ x ¬ (z ◦ (y ∗ a)) ◦ x.

(iv) ⇒ (v): From (b8) and (iv) it follows that a ∗ x = (a ◦ 0) ∗ (x ◦ 0) ¬ (0 ◦ (x ∗ a)) ◦ 0 = 0 ◦ (x ∗ a). On the other hand, by (b11), 0 ◦ (x ∗ a) ¬ a ∗ x. Now, by (a4), a ∗ x = 0 ◦ (x ∗ a).

(v) ⇒ (i): Let a, x ∈ X. Assume that x ¬ a. Then x ∗ a = 0. Hence, we have by (v), a ∗ x = 0 ◦ (x ∗ a) = 0 ◦ 0 = 0. So, a ¬ x. Finally, by (a4) we get x = a.

Therefore, a ∈ At(X). 

Next theorem is analogous to Theorem 3.4.

Theorem 3.5 Let X be a pseudo-BCI-algebra. Then for all a, x, y, z ∈ X, the following are equivalent:

(i) a ∈ At(X),

(ii) x ∗ (x ◦ a) = a = x ◦ (x ∗ a), (iii) a ◦ (x ∗ y) ¬ y ∗ (x ◦ a),

(iv) (a ∗ x) ◦ (y ∗ z) ¬ (z ∗ (y ◦ a)) ∗ x, (v) a ◦ x = 0 ∗ (x ◦ a).

We have another characterizations of atoms of a pseudo-BCI-algebra.

Theorem 3.6 Let X be a pseudo-BCI-algebra. Then for all a, x, y ∈ X, the follo-

wing are equivalent:

(i) a ∈ At(X),

(ii) x ∗ (x ◦ a) = a = x ◦ (x ∗ a), (iii) a ∗ x = (y ∗ x) ◦ (y ∗ a), (iv) a ◦ x = y ◦ (y ∗ (a ◦ x)).

Proof (i) ⇒ (ii): Follows from Theorem 3.3.

(ii) ⇒ (iii): We have by (b4) and (ii), (y ∗ x) ◦ (y ∗ a) = (y ◦ (y ∗ a)) ∗ x = a ∗ x.

(iii) ⇒ (iv): From (b8) and (iii) we get y ◦ (y ∗ (a ◦ x)) = (y ∗ 0) ◦ (y ∗ (a ◦ x)) = (a ◦ x) ∗ 0 = a ◦ x.

(iv) ⇒ (i): Let a, x ∈ X and let x ¬ a. Then, x ∗ a = 0. Hence, by (b8) and (iv) it follows a = a ◦ 0 = x ◦ (x ∗ (a ◦ 0)) = x ◦ (x ∗ a) = x ◦ 0 = x. Thus, a ∈ At(X). 

Next theorem is analogous to Theorem 3.6.

Theorem 3.7 Let X be a pseudo-BCI-algebra. Then for all a, x, y ∈ X, the follo- wing are equivalent:

(i) a ∈ At(X),

(ii) x ∗ (x ◦ a) = a = x ◦ (x ∗ a), (iii) a ◦ x = (y ◦ x) ∗ (y ◦ a), (iv) a ∗ x = y ∗ (y ◦ (a ∗ x)).

Theorem 3.8 Let X be a pseudo-BCI-algebra. Then for all a, x ∈ X, the following are equivalent:

(i) a ∈ At(X),

(ii) a ∗ x = (0 ∗ x) ◦ (0 ∗ a), (iii) a ◦ x = (0 ◦ x) ∗ (0 ◦ a).

Proof (i) ⇒ (ii): Let x ∈ X and a ∈ At(X). By Theorem 3.6(iii) we have a ∗ x = (y ∗ x) ◦ (y ∗ a) for any y ∈ X. Now, putting y = 0, we obtain (ii).

(ii) ⇒ (i): We have by (b12) and (b14) that a ∗ x = (0 ∗ x) ◦ (0 ∗ a) = 0 ◦ (x ∗ a) for any a, x ∈ X. Now, by Theorem 3.4(v), we get (i).

(i) ⇔ (iii): Analogous. 

Theorem 3.9 Let X be a pseudo-BCI-algebra. Then for all a ∈ X, the following are equivalent:

(i) a ∈ At(X),

(ii) 0 ∗ (0 ◦ a) = a = 0 ◦ (0 ∗ a).

Proof (i) ⇒ (ii): Follows from Theorem 3.3.

(ii) ⇒ (i): Let a, x ∈ X be such that x ¬ a. Then, x ∗ a = 0. By (ii), (b4), (b12) and (b14), we have a ∗ x = (0 ◦ (0 ∗ a)) ∗ x = (0 ∗ x) ◦ (0 ∗ a) = 0 ◦ (x ∗ a) = 0 ◦ 0 = 0.

Hence, a ¬ x. Finallly, by (a4), a = x. Thus, a ∈ At(X). 

Note that by (b14) we have 0 ∗ (0 ◦ x) = 0 ◦ (0 ◦ x) = 0 ◦ (0 ∗ x) = 0 ∗ (0 ∗ x) for any x ∈ X. Hence, by Theorem 3.9, we can write

At(X) = {x ∈ X : x = 0 ∗ (0 ∗ x)}.

Remark 3.10 Notice that, by (b9) and (b14), the following holds 0 ∗ x = 0 ◦ x = 0 ∗ (0 ∗ (0 ∗ x)) for any x ∈ X. Hence, it follows that 0 ∗ x = 0 ◦ x ∈ At(X) for any x ∈ X.

Example 3.11 Let X 1 be the pseudo-BCI-algebra from Example 2.3. Then it is easy to see that At(X 1 ) = {0}.

Example 3.12 Let X 2 be the pseudo-BCI-algebra from Example 2.4. Then it is seen that At(X 2 ) = X 2 .

Example 3.13 Let X be the pseudo-BCI-algebra from Example 2.5. Then, by sim- ple calculation, we see that At(X) = {(0, y, z) : y, z ∈ R}.

Theorem 3.14 Let X be a pseudo-BCI-algebra and let a ∈ At(X). Then, for any x, y ∈ X, the following hold:

(i) (a ∗ x) ◦ (a ∗ y) = 0 ◦ (x ∗ y), (ii) (a ◦ x) ∗ (a ◦ y) = 0 ∗ (x ◦ y).

Proof (i) Let X = (X, ¬, ∗, ◦, 0) be a pseudo-BCI-algebra. Since, by Proposition 3.1, At(X) is a p-semisimple pseudo-BCI-subalgebra of X , we obtain, by Proposition 2.15, that (At(X), +, −, 0) is a group, where a + b = a ∗ (0 ◦ b) = b ◦ (0 ∗ a) and

−a = 0∗a = 0◦a for any a, b ∈ At(X); and then a∗b := a−b and a◦b := −b+a for any a, b ∈ At(X). Moreover, we know that 0 ∗ x ∈ At(X) for any x ∈ X. Therefore, by Theorem 3.8, for any a ∈ At(X) and x ∈ X we have:

a ∗ x = (0 ∗ x) ◦ (0 ∗ a) = −(0 ∗ a) + (0 ∗ x).

Thus, by (b12) and (b14),

(a ∗ x) ◦ (a ∗ y) = −(a ∗ y) + (a ∗ x)

= −(−(0 ∗ a) + (0 ∗ y)) + (−(0 ∗ a) + (0 ∗ x))

= (−(0 ∗ y) + (0 ∗ a)) + (−(0 ∗ a) + (0 ∗ x))

= (−(0 ∗ y) + (0 ∗ a) − (0 ∗ a)) + (0 ∗ x)

= −(0 ∗ y) + (0 ∗ x)

= (0 ∗ x) ◦ (0 ∗ y)

= 0 ◦ (x ∗ y).

(ii) Analogous. _

Let X be a pseudo-BCI-algebra. For any a ∈ X we define a subset V (a) of X as follows

V (a) = {x ∈ X : a ¬ x}.

Note that V (a) is non-empty, because a ¬ a gives a ∈ V (a). If a ∈ At(X), then the set V (a) is called a branch of X . Notice also that V (0) = K(X) and it is a pseudo-BCK-part of X .

Observe that, if X is a pseudo-BCK-algebra, then X = V (0) and V (b) ⊆ V (a) for any a, b ∈ X such that a ¬ b.

Proposition 3.15 ([2]) Let X be a pseudo-BCI-algebra. Then V (a) ∩ V (b) = ∅

for any a, b ∈ At(X).

Proposition 3.16 ([2]) Let X be a pseudo-BCI-algebra. Then for each x ∈ X there exists a unique a ∈ At(X) such that x ∈ V (a).

Proposition 3.17 ([2]) Let X be a pseudo-BCI-algebra. Then

X = [

a ∈At(X)

V (a).

Example 3.18 Let X ¹ be the pseudo-BCI-algebra from Example 2.3. Then At(X 1 ) = {0} and V (a) = [a, ∞) for all a ∈ X ¹ .

Example 3.19 Let X ² be the pseudo-BCI-algebra from Example 2.4. Then At(X 2 ) = X 2 and V ((x, y)) = {(x, y)} for all (x, y) ∈ X ² .

Example 3.20 Let X be the pseudo-BCI-algebra from Example 2.5. Then At(X) = {(0, y, z) : y, z ∈ R}. It easy to see that, for (0, a ¹ , a 2 ) ∈ At(X), the sets

V ((0, a 1 , a 2 )) = {(x, a ¹ , a 2 ) ∈ X : x ­ 0}

are branches of X such that

V ((0, a 1 , a 2 )) ∩ V ((0, b ¹ , b 2 )) = ∅ for (0, a 1 , a 2 ), (0, b 1 , b 2 ) ∈ At(X) and

X = [

(0,a

,a

)∈At(X)

V ((0, a 1 , a 2 )).

Proposition 3.21 ([2]) Let X be a pseudo-BCI-algebra and let a ∈ At(X). If x, y ∈ V (a), then 0 ∗ x = 0 ◦ x = 0 ∗ y = 0 ◦ y.

Proposition 3.22 Let X be a pseudo-BCI-algebra and let x ∈ X and a, b ∈ At(X).

If x ∈ V (b), then a ∗ x = a ∗ b and a ◦ x = a ◦ b.

Proof If x ∈ V (b), then b ¬ x. Hence, by (b2), a ∗ x ¬ a ∗ b and a ◦ x ¬ a ◦ b. Since a, b ∈ At(X) and At(X) is a subalgebra of X , a ∗ b, a ◦ b ∈ At(X). So, a ∗ x ¬ a ∗ b implies a ∗ x = a ∗ b and a ◦ x ¬ a ◦ b implies a ◦ x = a ◦ b. 

Corollary 3.23 Let X be a pseudo-BCI-algebra. If a ∈ At(X), then for all x ∈ X, a ∗ x ∈ At(X) and a ◦ x ∈ At(X).

Proposition 3.24 Let X be a pseudo-BCI-algebra and let x, y ∈ X and a, b ∈ At(X) . If x ∈ V (a) and y ∈ V (b), then x ∗ y ∈ V (a ∗ b) and x ◦ y ∈ V (a ◦ b).

Proof Since x ∈ V (a), we have a ¬ x, i.e., by (b7), a ∗ y ¬ x ∗ y and a ◦ y ¬ x ◦ y.

Since y ∈ V (b), by Proposition 3.22, a ∗ y = a ∗ b and a ◦ y = a ◦ b. Thus, we get a ∗ y = a ∗ b ¬ x ∗ y and a ◦ y = a ◦ b ¬ x ◦ y, so x ∗ y ∈ V (a ∗ b) and x ◦ y ∈ V (a ◦ b). 

Theorem 3.25 Let X be a pseudo-BCI-algebra and let x, y ∈ X. Then, the follo- wing are equivalent:

(i) x and y belong to the same branch of X , (ii) x ∗ y ∈ K(X),

(iii) x ◦ y ∈ K(X).

Proof (i) ⇒ (ii): Assume that x ∈ V (a) and y ∈ V (a) for some a ∈ At(X). Then a ¬ x and a ¬ y. Hence, by (b7), a ∗ y ¬ x ∗ y. By Proposition 3.22 and (a3) we have a ∗ y = a ∗ a = 0. So 0 ¬ x ∗ y implies x ∗ y ∈ K(X).

(ii) ⇒ (i): Now, assume that x ∗ y ∈ K(X). Let x ∈ V (a) and y ∈ V (b) for some a, b ∈ At(X). We show that a = b. From Proposition 3.24 we infer x ∗ y ∈ V (a ∗ b).

Since x ∗ y ∈ K(X) = V (0) and x ∗ y ∈ V (a ∗ b), from Proposition 3.15 it follows a ∗ b = 0, i.e., a ¬ b. Thus, since b is an atom of X , a = b.

(ii) ⇔ (iii): Analogous. 

4. Ideals of pseudo-BCI-algebras. Let X be a pseudo-BCI-algebra. A subset I of X is called an ideal of X if it satisfies for all x, y ∈ X:

(I1) 0 ∈ I,

(I2) if x ∗ y ∈ I and y ∈ I, then x ∈ I.

Under this definition, {0} and X are the simplest examples of ideals.

Proposition 4.1 Let I be an ideal of a pseudo-BCI-algebra X . Then, for any x, y ∈ X, if y ∈ I and x ¬ y, then x ∈ I.

Proof Straightforward. _

Proposition 4.2 Let I be a subset of a pseudo-BCI-algebra X . Then I is an ideal of X if and only if it satisfies conditions (I1) and

(I2’) for all x, y ∈ X, if x ◦ y ∈ I and y ∈ I, then x ∈ I.

Proof It suffices to prove that if (I2) is satisfied, then (I2’) is also satisfied. The proof of the converse of this implication is analogous. Suppose that x ◦ y ∈ I and y ∈ I. From (a2) we know that x∗(x◦y) ¬ y. Then, by Proposition 4.1, x∗(x◦y) ∈ I.

Hence, since x ◦ y ∈ I, by (I2) we obtain x ∈ I and the proof is complete. 

Theorem 4.3 Let I be a subalgebra of a pseudo-BCI-algebra X . Then I is an ideal of X if and only if for any x, y ∈ X, if x ∈ I and y ∈ X\I, then y ∗ x ∈ X\I.

Proof Assume that I is an ideal of X . Let x, y ∈ X be such that x ∈ I and y ∈ X\I. If y ∗ x / ∈ X\I, then y ∗ x ∈ I. Hence, since x ∈ I, we have, by (I2), y ∈ I.

A contradiction. So, y ∗ x ∈ X\I

Conversely, since I is a subalgebra of X , 0 ∈ I and (I1) is satisfied. Now, let x ∗ y ∈ I and y ∈ I. If x / ∈ I, then by assumption, x ∗ y ∈ X\I. A contradiction. So x ∈ I and (I2) is also satisfied. Thus I is an ideal of X . 

Now, we have theorem analogous to Theorem 4.3.

Theorem 4.4 Let I be a subalgebra of a pseudo-BCI-algebra X . Then I is an ideal of X if and only if for any x, y ∈ X, if x ∈ I and y ∈ X\I, then y ◦ x ∈ X\I.

Remark 4.5 Theorems 4.3 and 4.4 generalize Theorem 4.5 of [11].

An ideal I of a pseudo-BCI-algebra X is called closed if I is a subalgebra of X . Proposition 4.6 An ideal I of a pseudo-BCI-algebra X is closed if and only if for any x ∈ I, 0 ∗ x = 0 ◦ x ∈ I.

Proof First note that by (b14), 0 ∗ x = 0 ◦ x for any x ∈ X. If I is closed, then x ∈ I and 0 ∈ I imply 0 ∗ x ∈ I.

Conversely, let x, y ∈ I. We show that x ∗ y ∈ I. We have by (b6) and (b8), (x∗y)∗(0∗y) ¬ x∗0 = x. Since x ∈ I, we get, by Proposition 4.1, (x∗y)∗(0∗y) ∈ I.

Thus, we obtain x∗y ∈ I because I is an ideal of X and 0∗y ∈ I. Similarly, x◦y ∈ I

for any x, y ∈ I. Therefore, I is a subalgebra of X , i.e., a closed ideal. 

Proposition 4.7 ([11]) Let X be a pseudo-BCI-algebra. Then K(X) is an ideal of X .

Corollary 4.8 Let X be a pseudo-BCI-algebra. Then K(X) is a closed ideal of X .

Example 4.9 Let X be the pseudo-BCI-algebra from Example 2.5 (see also Exam- ple 3.13). Then we can see that At(X) = {(0, y, z) : y, z ∈ R} is not an ideal of X (although it is a subalgebra of X , see Proposition 3.1) and K(X) = {(x, 0, 0) : x ­ 0}

is a closed ideal of X . We also have that I ^α = {(x, y, αy) : x ­ 0, y ∈ R} ⊆ X for any α ∈ R and I = {(x, 0, z) : x ­ 0, z ∈ R} ⊆ X are ideals of X , but only I ⁰ and I are closed ideals of X . By simple calculation we can get also that J ^β = {(0, y, βy) : y ∈ R} ⊆ At(X) for any β ∈ R and J = {(0, 0, z) : z ∈ R} ⊆ At(X) are ideals of X and only J 0 and J are closed ideals of X .

Theorem 4.10 Let I be a subset of a p-semisimple pseudo-BCI-algebra X . Then I is a closed ideal of X if and only if it is a subalgebra of X .

Proof Let X be a p-semisimple pseudo-BCI-algebra. If I is a closed ideal of X , then obviously, it is a subalgebra of X . Assume that I is a subalgebra of X . It suffices to prove that I is an ideal of X . Let x, y ∈ X be such that x ∗ y ∈ I and y ∈ I. Then, by Proposition 2.12(iii,vi), x = y ◦ (y ∗ x) = y ◦ (0 ◦ (x ∗ y)) ∈ I, because 0, y, x ∗ y ∈ I and I is a subalgebra of X . So I is a closed ideal of X . 

Corollary 4.11 Every subalgebra of a p-semisimple pseudo-BCI-algebra is an ideal.

Now, we wonder when an ideal of a pseudo-BCI-algebra is closed. First, we focus on p-semisimple pseudo-BCI-algebras.

Proposition 4.12 Let X be a finite p-semisimple pseudo-BCI-algebra and let I be an ideal of X such that I 6= X. Then |I| ¬ |X\I|.

Proof Suppose |I| > |X\I|. Let a ∈ I and x ∈ X\I. Note that x∗a ∈ X\I. Indeed, if x ∗ a ∈ I, then, since a ∈ I, x ∈ I and we get a contradiction. Since |I| > |X\I|, so for some x ∈ X and a ¹ , a 2 ∈ I such that a ¹ 6= a ² , we have x ∗a ¹ = x ∗a ² . Since X is p-semisimple, by Proposition 2.12(viii), a 1 = a 2 and we obtain a contradiction. _

Theorem 4.13 Every ideal of a finite p-semisimple pseudo-BCI-algebra is closed.

Proof Let I be an ideal of a finite p-semisimple pseudo-BCI-algebra X . If I = X, then, obviously, it is closed. Assume that I 6= X. By Proposition 4.6 it suffices to prove that 0 ∗ x ∈ I for any x ∈ I. Suppose that there exists a ∈ I such that 0 ∗ a ∈ X\I. Notice that y ∗ a ∈ X\I for any y ∈ X\I. Indeed, if y ∗ a ∈ I and a ∈ I, then, since I is an ideal, y ∈ I and we get a contradiction. Take |X| = n,

|I| = k and |X\I| = l. By Proposition 4.12, k ¬ l. Let X\I = {y 1 , y 2 , . . . , y l }. Now, y 1 ∗a, y 2 ∗a, . . . , y ^l ∗a are distinct l elements of X\I, because otherwise y ⁱ ∗a = y ^j ∗a for i 6= j implies y ⁱ = y j , by Proposition 2.12(ix). Hence, since 0 ∗ a ∈ X\I, we have 0 ∗ a = y ^m ∗ a for some m = 1, 2, . . . , l. Again by Proposition 2.12(ix), we get y m = 0. This is a contradiction, because 0 ∈ I. Therefore, I is a closed ideal of X . 

Let X be a pseudo-BCI-algebra and A a subset of X. Note that if B = A∩At(X), then

B = {x ∈ A : x = 0 ∗ (0 ∗ x)}.

Now we have the following.

Proposition 4.14 Let X be a pseudo-BCI-algebra. If I is an ideal of X , then J = I ∩ At(X) is an ideal of At(X).

Proof Let I be an ideal of a pseudo-BCI-algebra X such that I ∩ At(X) = J. We have

J = {x ∈ I : x = 0 ∗ (0 ∗ x)}.

Obviously, 0 ∈ J. Let x, y ∈ At(X) be such that x ∗ y ∈ J and y ∈ J. Then x ∗ y, y ∈ I and x ∗ y, y ∈ At(X). Hence x ∈ I. So, since x ∈ At(X) it follows that

x ∈ J. Thus J is an ideal of At(X). 

Theorem 4.15 Let X be a pseudo-BCI-algebra. If I is a subalgebra of X , then J = I ∩ At(X) is a closed ideal of At(X).

Proof First, we prove that J is a subalgebra of At(X). Let x, y ∈ J. We have, by (b12), (b13) and (b14),

x ∗ y = (0 ◦ (0 ∗ x)) ∗ (0 ◦ (0 ∗ y)) = 0 ∗ ((0 ∗ x) ◦ (0 ∗ y)) = 0 ∗ (0 ∗ (x ∗ y)) and

x ◦ y = (0 ∗ (0 ◦ x)) ◦ (0 ∗ (0 ◦ y)) = 0 ∗ ((0 ◦ x) ∗ (0 ◦ y)) = 0 ∗ (0 ∗ (x ◦ y)) Hence x ∗ y ∈ J and x ◦ y ∈ J and J is a subalgebra of At(X).

Now, we prove that J is an ideal of At(X). By Theorem 4.3 it suffices to show that for x, y ∈ At(X), if x ∈ J and y ∈ At(X)\J, then y ∗ x ∈ At(X)\J. Suppose that y ∗ x ∈ J. Then, since x ∈ J, it follows y ∈ J and we get a contradiction. So y ∗ x ∈ At(X)\J and J is an ideal of At(X).

Thus J is a closed ideal of At(X). _

Theorem 4.16 Let X be a pseudo-BCI-algebra and let At(X) be finite. Then every ideal of X is closed.

Proof Let I be an ideal of X and let J = I ∩ At(X). By Proposition 4.6 it suffices to prove that 0 ∗ x ∈ I for any x ∈ I. Let x ∈ I. By Proposition 3.16 there exists a unique a ∈ At(X) such that x ∈ V (a). Notice, that a ∈ I. Indeed, x ∈ V (a) implies a ∗ x = 0 ∈ I and since also x ∈ I, we get a ∈ I. Hence,

a ∈ I ∩ At(X) = J.

By Proposition 3.21, 0 ∗ a = 0 ∗ x for a, x ∈ V (a). By Remark 3.10, 0 ∗ a = 0 ∗ x ∈ At(X). Now, suppose that 0 ∗ x /∈ I. Then 0 ∗ x ∈ X\I, i.e., 0 ∗ a ∈ X\I. Hence,

0 ∗ a ∈ (X\I) ∩ At(X) = At(X)\J.

By Proposition 4.14, J is an ideal of At(X). Since At(X) is finite, we have, by The- orem 4.13, J is closed. Thus, 0, a ∈ J imply 0∗a ∈ J and we obtain a contradiction.

So 0 ∗ x ∈ I and therefore, I is closed. 

Let X be a pseudo-BCI-algebra. As we already know, usually At(X) is not an ideal of X . Therefore, now we want to establish conditions for it to be an ideal.

Firstly, we do this by defining the maps p _∗ and p _◦ . Define the maps p _∗ , p _◦ : X → X as follow

p _∗ (x) = x ∗ (0 ∗ (0 ∗ x)) for all x ∈ X, p _◦ (x) = x ◦ (0 ∗ (0 ∗ x)) for all x ∈ X.

Proposition 4.17 Let X be a pseudo-BCI-algebra. Then, for any x ∈ X, the following are equivalent:

(i) x ∈ At(X), (ii) p _∗ (x) = 0, (iii) p _◦ (x) = 0.

Proof (i) ⇒ (ii): Let x ∈ At(X). Then 0 ∗ (0 ∗ x) = x. Hence, by (a3), p _∗ (x) = x ∗ (0 ∗ (0 ∗ x)) = x ∗ x = 0.

(ii) ⇒ (iii): Let p ∗ (x) = x ∗ (0 ∗ (0 ∗ x)) = 0. Then x ¬ 0 ∗ (0 ∗ x). Hence, also x ◦ (0 ∗ (0 ∗ x)) = 0. So, p ◦ (x) = 0.

(iii) ⇒ (i): Assume that p ◦ (x) = 0, i.e., x ◦ (0 ∗ (0 ∗ x)) = 0. Then x ¬ 0 ∗ (0 ∗ x).

Since, by (a2) and (b14), 0 ∗ (0 ∗ x) = 0 ◦ (0 ∗ x) ¬ x, we get x = 0 ∗ (0 ∗ x). Thus,

x ∈ At(X). 

Theorem 4.18 Let X be a pseudo-BCI-algebra. If p ∗ is a homomorphism of X ,

then At(X) is an ideal of X .

Proof Suppose p _∗ is a homomorphism of X . Let x ∗ y ∈ At(X) and y ∈ At(X).

Then, by (b8) and Proposition 4.17, we get

p _∗ (x) = p _∗ (x) ∗ 0 = p ∗ (x) ∗ p ∗ (y) = p _∗ (x ∗ y) = 0.

Hence, again by Proposition 4.17, x ∈ At(X). So, At(X) is an ideal of X . 

Similar fact holds for the map p _◦ as well.

Theorem 4.19 Let X be a pseudo-BCI-algebra. If p ◦ is a homomorphism of X , then At(X) is an ideal of X .

Now, we define the notion of horizontal ideal of a pseudo-BCI-algebra X . We use it to give conditions for At(X) to be an ideal of X . An ideal I of a pseudo-BCI- algebra X is called a horizontal ideal of X if I ∩ K(X) = {0}. Obviously, {0} is a horizontal ideal of X .

Example 4.20 Let X be the pseudo-BCI-algebra from Example 2.5 (see also Exam- ple 4.9). Then we can see that I α = {(x, y, αy) : x ­ 0, y ∈ R} ⊆ X for any α ∈ R and I = {(x, 0, z) : x ­ 0, z ∈ R} ⊆ X are not horizontal ideals of X and J ^β = {(0, y, βy) : y ∈ R} ⊆ At(X) for any β ∈ R and J = {(0, 0, z) : z ∈ R} ⊆ At(X) are horizontal ideals of X .

Proposition 4.21 Let X be a pseudo-BCI-algebra and let I ⊆ X. If I is a closed horizontal ideal of X , then I ⊆ At(X).

Proof Let I be a closed horizontal ideal of X . For any x ∈ I we have, by (a2) and (b14), 0 ∗ (0 ∗ x) = 0 ∗ (0 ◦ x) ¬ x. Hence, by Proposition 4.1, 0 ∗ (0 ∗ x) ∈ I. Since I is a closed ideal, we get x ∗ (0 ∗ (0 ∗ x)) ∈ I. On the other hand, 0 ∗ (0 ∗ x) ¬ x implies x, 0 ∗ (0 ∗ x) ∈ V (0 ∗ (0 ∗ x)). By Theorem 3.25, x ∗ (0 ∗ (0 ∗ x)) ∈ K(X).

Thus, x ∗ (0 ∗ (0 ∗ x)) = 0, because I ∩ K(X) = {0}. Hence, x ¬ 0 ∗ (0 ∗ x) and by (a4), x = 0 ∗ (0 ∗ x). Therefore, x ∈ At(X), i.e., I ⊆ At(X). 

The converse of Proposition 4.21 does not hold which is shown in the following example.

Example 4.22 Let X be the pseudo-BCI-algebra from Example 2.5 (see also Exam- ples 4.9 and 4.20). Then for β 6= 0, J ^β is the horizontal ideal of X such that J β ⊆ At(X). But it is not closed ideal of X .

Now we establish conditions for a subalgebra I ⊆ At(X) of X to be a horizontal ideal of X .

Theorem 4.23 Let X be a pseudo-BCI-algebra and let I ⊆ At(X) be a subalgebra of X . Then the following are equivalent:

(i) I is a horizontal ideal of X ,

(ii) for all x ∈ X and a, b ∈ I, x ∗ b = a ∗ b implies x = a, (iii) for all x ∈ X and a ∈ I, x ∗ a = 0 ∗ a implies x = 0, (iv) for all x, y ∈ X and a ∈ I, x ∗ a = y ∗ a implies x = y,

(v) for all x, y ∈ K(X) and a, b ∈ I, x ∗ a = y ∗ b implies x = y and a = b, (vi) for all x, y ∈ K(X) and a ∈ I, x ∗ a = y ∗ a implies x = y,

(vii) for all x ∈ K(X) and a ∈ I, x ∗ a = 0 ∗ a implies x = 0.

Proof (i) ⇒ (ii): Suppose I is a horizontal ideal of X . Let x ∗ b = a ∗ b, where x ∈ X and a, b ∈ I. Since I is a subalgebra of X , we have a ∗ b ∈ I, and so x ∗ b ∈ I.

It follows that x ∈ I, because I is an ideal. Hence, x ∈ At(X). Thus, by Theorem 3.4(ii,v), we have x = b ◦ (b ∗ x) = b ◦ (0 ◦ (x ∗ b)) = b ◦ (0 ◦ (a ∗ b)) = b ◦ (b ∗ a) = a and so (ii) holds.

(ii) ⇒ (iii): Obvious.

(iii) ⇒ (iv): Let x ∗ a = y ∗ a, where x, y ∈ X and a ∈ I. Then, by (b4) and (a3), (x ◦ y) ∗ a = (x ∗ a) ◦ y = (y ∗ a) ◦ y = (y ◦ y) ∗ a = 0 ∗ a. By (iii), x ◦ y = 0, i.e., x ¬ y. Similarly, (y ◦ x) ∗ a = 0 ∗ a and again by (iii), y ◦ x = 0, i.e., y ¬ x. Finally, by (a4), x = y and (iv) holds.

(iv) ⇒ (v): Let x, y ∈ K(X) and a, b ∈ I. Then 0 ¬ x and 0 ¬ y, i.e., 0 ◦ x = 0 ◦ y = 0. If x ∗ a = y ∗ b, then, by (b12) and the fact that a, b ∈ At(X), we obtain a = 0 ◦(0∗a) = (0◦x)◦(0∗a) = 0∗(x∗a) = 0∗(y ∗b) = (0◦y)◦(0∗b) = 0◦(0∗b) = b and hence x ∗ a = y ∗ a. By (iv), x = y. Thus, (v) holds.

(v) ⇒ (vi): Obvious.

(vi) ⇒ (vii): Obvious.

(vii) ⇒ (i): Since I is a subalgebra of p-semisimple algebra At(X), it is an ideal of At(X) (see Corollary 4.11). So it is an ideal of X . Hence, it is sufficient to prove that I ∩ K(X) = {0}. Let x ∈ I ∩ K(X). Then x ∈ I and x ∈ K(X). Thus we have x ∗ x = 0 = 0 ∗ x. Since x ∈ I, by (vii), we get that x = 0. So I ∩ K(X) = {0} and

I is a horizontal ideal of X . 

We also have theorem analogous to Theorem 4.23.

Theorem 4.24 Let X be a pseudo-BCI-algebra and let I ⊆ At(X) be a subalgebra of X . Then the following are equivalent:

(i) I is a horizontal ideal of X ,

(ii) for all x ∈ X and a, b ∈ I, x ◦ b = a ◦ b implies x = a, (iii) for all x ∈ X and a ∈ I, x ◦ a = 0 ◦ a implies x = 0, (iv) for all x, y ∈ X and a ∈ I, x ◦ a = y ◦ a implies x = y,

(v) for all x, y ∈ K(X) and a, b ∈ I, x ◦ a = y ◦ b implies x = y and a = b, (vi) for all x, y ∈ K(X) and a ∈ I, x ◦ a = y ◦ a implies x = y,

(vii) for all x ∈ K(X) and a ∈ I, x ◦ a = 0 ◦ a implies x = 0.

We know that At(X) is a subalgebra of a pseudo-BCI-algebra X and At(X) ∩

K(X) = {0}. Therefore, if we put I = At(X) in Theorems 4.23 and 4.24, then we

get the following theorem giving conditions for At(X) to be an ideal of X .

Theorem 4.25 Let X be a pseudo-BCI-algebra. Then the following are equivalent:

(i) At(X) is an ideal of X ,

(ii) for all x ∈ X and a, b ∈ At(X), x ∗ b = a ∗ b implies x = a, (ii’) for all x ∈ X and a, b ∈ At(X), x ◦ b = a ◦ b implies x = a, (iii) for all x ∈ X and a ∈ At(X), x ∗ a = 0 ∗ a implies x = 0, (iii’) for all x ∈ X and a ∈ At(X), x ◦ a = 0 ◦ a implies x = 0,

(iv) for all x, y ∈ X and a ∈ At(X), x ∗ a = y ∗ a implies x = y, (iv’) for all x, y ∈ X and a ∈ At(X), x ◦ a = y ◦ a implies x = y,

(v) for all x, y ∈ K(X) and a, b ∈ At(X), x ∗ a = y ∗ b implies x = y and a = b, (v’) for all x, y ∈ K(X) and a, b ∈ At(X), x ◦ a = y ◦ b implies x = y and a = b, (vi) for all x, y ∈ K(X) and a ∈ At(X), x ∗ a = y ∗ a implies x = y,

(vi’) for all x, y ∈ K(X) and a ∈ At(X), x ◦ a = y ◦ a implies x = y, (vii) for all x ∈ K(X) and a ∈ At(X), x ∗ a = 0 ∗ a implies x = 0, (vii’) for all x ∈ K(X) and a ∈ At(X), x ◦ a = 0 ◦ a implies x = 0.

References

[1] W.A. Dudek and Y.B. Jun, Pseudo-BCI algebras, East Asian Math. J. 24 (2008), 187–190.

[2] G. Dymek, p-semisimple pseudo-BCI algebras, J. Mult.-Valued Logic Soft Comput., to ap- pear.

[3] G. Georgescu and A. Iorgulescu, Pseudo-BCK algebras: an extension of BCK-algebras, Pro- ceedings of DMTCS’01: Combinatorics, Computability and Logic, Springer, London, 2001, 97–114.

[4] G. Georgescu and A. Iorgulescu, Pseudo-BL algebras: a noncommutative extension of BL- algebras, Abstracts of The Fifth International Conference FSTA 2000, Slovakia, February 2000, 90–92.

[5] G. Georgescu and A. Iorgulescu, Pseudo-MV algebras: a noncommutative extension of MV- algebras, The Proceedings The Fourth International Symposium on Economic Informatics, INFOREC Printing House, Bucharest, Romania, May (1999), 961–968.

[6] C.S. Hoo, Filters and ideals in BCI-algebras, Math. Japon. 36 (1991), 987–997.

[7] W. Huang and Y.B. Jun, Ideals and subalgebras in BCI-algebras, Southeast Asian Bull. Math.

26 (2002), 567–573.

[8] A. Iorgulescu, Algebras of logic as BCK algebras, Editura ASE, Bucharest, 2008.

[9] K. Is´eki, An algebra related with a propositional calculus, Proc. Japan Acad. 42 (1966), 26–29.

[10] K. Is´eki, On BCI-algebras, Math. Seminar Notes 8 (1980), 125–130.

[11] Y.B. Jun, H.S. Kim and J. Neggers, On pseudo-BCI ideals of pseudo BCI-algebras, Mat.

Vesnik 58 (2006), 39–46.

[12] Y.B. Jun, X.L. Xin and E.H. Roh, The role of atoms in BCI-algebras, Soochow J. Math. 30 (2004), 491–506.

[13] K.J. Lee and C.H. Park, Some ideals of pseudo-BCI algebras, J. Appl. Math. Informatics 27

(2009), 217–231.

[14] J. Meng and X.L. Xin, Characterizations of atoms in BCI-algebras, Math. Japon. 37 (1992), 359–361.

Grzegorz Dymek

Faculty of Mathematics and Natural Sciences, The John Paul II Catholic University of Lublin Konstantynów 1H, 20-708 Lublin, Poland

E-mail: gdymek@o2.pl

(Received: 2.02.2012)