- Email: [email protected]
BCI-implicative ideals of BCI-algebras
Information Sciences 177 (2007) 4987–4996 www.elsevier.com/locate/ins
BCI-implicative ideals of BCI-algebras Yong Lin Liu
a,*
, Yang Xu b, Jie Meng
c
a
b
Department of Mathematics, Wuyi University, Wuyishan, Fujian 354300, PR China Department of Applied Mathematics, Southwest Jiaotong University, Chengdu 610031, Sichuan, PR China c Department of Mathematics, Northwest University, Xi’an, Shaanxi 710069, PR China Received 12 May 2005; received in revised form 29 June 2007; accepted 2 July 2007
Abstract This paper introduces the notion of BCI-implicative ideals and characterizes BCI-implicative ideals and closed BCI-implicative ideals. Using these characterizations, the connections between BCI-implicative ideals and other ideals in BCI/BCK-algebras are investigated. Additionally, the extension property of BCI-implicative ideals is established. Finally, the implicative BCI-algebras are completely described using BCI-implicative ideals. The above work generalizes the corresponding results in BCK-algebras. Published by Elsevier Inc. Keywords: BCI-implicative ideal; Closed BCI-implicative ideal; Implicative BCI-algebra; Implicative BCK-algebra
1. Introduction One important task of artificial intelligence is to make the computers simulate human beings in dealing with certainty and uncertainty in information. It is well known that certain information processing, especially inferences based on certain information, is based on classical two-valued logic. Due to strict and complete logical foundation (classical logic), making inferences about certain information can be done with high confidence levels. Thus, it is natural and necessary to attempt to establish some rational logic system as the logical foundation for uncertain information processing. It is evident that this kind of logic cannot be two-valued logic itself but might form a certain extension of two-valued logic. Various kinds of non-classical logic systems have therefore been extensively researched in order to construct natural and efficient inference systems to deal with uncertainty. In recent years, motivated by both theory and application, the study of t-norm-based logic systems and the corresponding pseudo-logic systems has been become a greater focus in the field of logic. Here, t-norm-based logical investigations were first to the corresponding algebraic investigations, and in the case of pseudo-logic systems, algebraic development was first to the corresponding logical development.
*
Corresponding author. E-mail address: [email protected] (Y.L. Liu).
0020-0255/$ - see front matter Published by Elsevier Inc. doi:10.1016/j.ins.2007.07.003
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BCK and BCI-algebras are two classes of logic algebras. They were introduced by Imai and Iseki [8,11] and have been extensively investigated by many researchers (cf. [2,5,12–26]). BCI-algebras are generalizations of BCK-algebras. Iorgulescu [9,10] showed that pocrims and BCK-algebras with condition (S) are categorically isomorphic, and residuated lattices and bounded BCK lattices with condition (S) are categorically isomorphic. Hence, most of the algebras related to the t-norm-based logics, such as MTL [3], BL [4], hoop, MV [1], NM [3] and Boolean algebras etc., are extensions of BCK-algebra (i.e. they are subclasses of BCK-algebra). This shows that BCK/BCI-algebras are considerably general structures. Indeed, any results on BCK/BCI-algebras also hold for the aforementioned logic algebras. Using ideals to completely describe BCK/BCI-algebras was initiated by Iseki and Tanaka [12]. There are three important classes of BCI-algebras: commutative BCI-algebras, implicative BCI-algebras and positive implicative BCI-algebras. Meng [21] introduced BCI-commutative ideals and used these to completely describe commutative BCI-algebras. Liu and Zhang [20], and Wei and Jun [27] independently introduced BCI-positive implicative ideals and used these to completely describe positive implicative BCI-algebras. Hence, it is an interesting topic for researchers to investigate how to introduce so-called BCI-implicative ideals to completely describe implicative BCI-algebras. This paper introduces the notion of BCI-implicative ideals. It generalizes the notion of BCK-implicative ideals (i.e. implicative ideals). Furthermore, a characterization of BCI-implicative ideals is developed. We also prove that a nonempty subset of a BCI-algebra is a BCI-implicative ideal if and only if it is both a BCI-commutative ideal and a BCI-positive implicative ideal. This generalizes the well-known result: a nonempty subset of a BCK-algebra is an implicative ideal if and only if it is both a commutative ideal and a positive implicative ideal. A simpler characterization of a closed BCI-implicative ideal is obtained. Using this characterization, the extension property of BCI-implicative ideals is obtained and implicative BCI-algebras are completely described by BCI-implicative ideals. 2. Preliminaries An algebra (X; * , 0) of type (2, 0) is said to be a BCI-algebra if it satisfies the following conditions: for all x, y, z 2 X, (I) (II) (III) (IV)
((x * y) * (x * z)) * (z * y) = 0, (x * (x * y)) * y = 0, x * x = 0 and x * y = 0 and y * x = 0 imply x = y. If a BCI-algebra X satisfies the condition, (V) 0 * x = 0 for all x 2 X,
then X is called a BCK-algebra. Hence, a BCK-algebra is a BCI-algebra. In a BCI/BCK-algebra, we can define a binary relation 6 on X by x 6 y if and only if x * y = 0. Then (X; 6) is a partial ordering set. In a BCI-algebra X, the following hold: (1) (2) (3) (4) (5) (6)
(x * y) * z = (x * z) * y, x * (x * (x * y)) = x * y, ((x * z) * (y * z)) * (x * y) = 0, x * 0 = x, 0 * (x * y) = (0 * x) * (0 * y) and x 6 y implies x * z 6 y * z and z * y 6 z * x.
Throughout this paper, X always means a BCI-algebra without any specification. A nonempty subset I of a BCI-algebra X is called an ideal of X if it satisfies (I1) 0 2 I and (I2) x * y 2 I and y 2 I imply x 2 I.
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Definition 2.1 [20]. A nonempty subset I of a BCI-algebra X is called a BCI-positive implicative ideal of X if it satisfies (I1) and (I3) ((x * z) * z) * (y * z) 2 I and y 2 I imply x * z 2 I. Proposition 2.2 [20]. Let I be an ideal of BCI-algebra X. Then the following conditions are equivalent: (i) I is BCI-positive implicative. (ii) ((x * z) * z) * (y * z) 2 I implies (x * y) * z 2 I. (iii) ((x * y) * y) * (0 * y) 2 I implies x * y 2 I. Definition 2.3 [21]. A nonempty subset I of a BCI-algebra X is called a BCI-commutative ideal if it satisfies (I1) and (I4) (x * y) * z 2 I and z 2 I imply x * ((y * (y * x)) * (0 * (0 * (x * y)))) 2 I. Proposition 2.4 [21]. An ideal I of a BCI-algebra X is BCI-commutative if and only if x * y 2 I implies x * ((y * (y * x)) * (0 * (0 * (x * y)))) 2 I. 3. BCI-implicative ideals of BCI-algebras Definition 3.1. A nonempty subset I of a BCI-algebra X is called a BCI-implicative ideal if it satisfies (I1) and (I5) (((x * y) * y) * (0 * y)) * z 2 I and z 2 I imply x * ((y * (y * x)) * (0 * (0 * (x * y)))) 2 I, for all x,y,z 2 X. The following examples show that BCI-implicative ideals exist. Example 3.2. Any ideal of a p-semisimple BCI-algebra [15] is BCI-implicative. Example 3.3. Let X = {0, 1, 2, 3, 4, 5} with Cayley table as follows:
* 0 1 2 3 4 5
0
1
2
3
4
5
0 1 2 3 4 5
0 0 2 3 3 3
0 1 0 3 4 5
3 3 3 0 1 1
3 3 3 0 0 1
3 3 3 0 0 0
Then X is a proper BCI-algebra. By routine calculations, we have I = {0, 1, 2} is a BCI-implicative ideal of X. A characterization of the BCI-implicative ideal of X is obtained by the following: Theorem 3.4. Let I be an ideal of X. Then I is BCI-implicative if and only if (a) ððx yÞ yÞ ð0 yÞ 2 I implies x ððy ðy xÞÞ ð0 ð0 ðx yÞÞÞÞ 2 I: Proof. Suppose that I is a BCI-implicative ideal of X. Let x, y 2 X; if ((x * y) * y) * (0 * y) 2 I, then (((x * y) * y) * (0 * y)) * 0 2 I and 0 2 I. By (I5), x * ((y * (y * x)) * (0 * (0 * (x * y)))) 2 I. Hence, (a) holds. Conversely, suppose that the ideal I satisfies (a). For x,y,z 2 X, if (((x * y) * y) * (0 * y)) * z 2 I and z 2 I, by the definition of ideals we obtain ((x * y) * y) * (0 * y) 2 I. It follows from (a) that x * ((y * (y * x)) * (0 * (0 * (x * y)))) 2 I. This means that I is a BCI-implicative ideal. This completes the proof. h
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The connections between BCI-implicative ideals and ideals of X are obtained by the following theorem. Theorem 3.5. A BCI-implicative ideal is an ideal, but the converse is not true. Proof. Suppose that I is a BCI-implicative ideal, and let y = 0 in (I5). We have x * z 2 I, z 2 I imply x 2 I. This means that I is an ideal. The last part is shown by the following example. h Example 3.6. Let X = {0, 1, 2, 3, 4} be a proper BCI-algebra with Cayley table as follows: * 0 1 2 3 4
0
1
2
3
4
0 1 2 3 4
0 0 2 3 4
0 0 0 2 4
0 0 0 0 4
4 4 4 4 0
I = {0,1} is a non-trivial ideal of X; but not (((3 * 2) * 2) * (0 * 2)) * 0 = (2 * 2) * (0 * 2) = 0 2 {0, 1} and (3 * 2)))) = 3 * ((2 * 0) * 0) = 2; 62 {0, 1}.
a BCI-implicative ideal of X because 0 2 {0, 1}, but 3 * ((2 * (2 * 3)) * (0 * (0 *
The following three theorems demonstrate the close relations among BCI-positive implicative ideals, BCIcommutative ideals and BCI-implicative ideals in BCI-algebras. Theorem 3.7. A BCI-implicative ideal is a BCI-positive implicative ideal, but the converse is not true. Proof. Assume that I is a BCI-implicative ideal of X. It follows from Theorem 3.5 that I is an ideal. In order to prove that I is BCI-positive implicative, from Proposition 2.2 (iii), it suffices to show that, if ((x * y) * y) * (0 * y) 2 I, then x * y 2 I. Now, if ((x * y) * y) * (0 * y) 2 I, by Theorem 3.4 we have x * ((y * (y * x)) * (0 * (0 * (x * y)))) 2 I. Since ðx yÞ ½x ððy ðy xÞÞ ð0 ð0 ðx yÞÞÞÞ 6 ½ðy ðy xÞÞ ð0 ð0 ðx yÞÞÞ y ¼ ð0 ðy xÞÞ ð0 ð0 ðx yÞÞÞ ¼ ½0 ð0 ð0 ðx yÞÞÞ ðy xÞ ¼ ð0 ðx yÞÞ ðy xÞ ¼ ðð0 xÞ ð0 yÞÞ ðy xÞ ¼ ðð0 ðy xÞÞ xÞ ð0 yÞ ¼ ððð0 yÞ ð0 xÞÞ xÞ ð0 yÞ ¼ ð0 ð0 xÞÞ x ¼ ð0 xÞ ð0 xÞ ¼ 0 2 I; we have x * y 2 I. Thus, we prove that I is a BCI-positive implicative ideal. The final statement of the theorem is shown by the following example. Example 3.8. Let X = {0, 1, 2, 3, 4} be a proper BCI-algebra with Cayley table given by
* 0 1 2 3 4
0
1
2
3
4
0 1 2 3 4
0 0 2 3 4
0 1 0 3 4
0 0 0 0 4
4 4 4 4 0
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By routine calculations, {0, 2} is a non-trivial BCI-positive implicative ideal of X, but not a BCI-implicative ideal of X as follows: (((1 * 3) * 3) * (0 * 3)) * 0 = 0 2 {0, 2} and 0 2 {0, 2}, but 1 * ((3 * (3 * 1)) * (0 * (0 * (1 * 3)))) = 1 * (0 * 0) = 1 62 {0, 2}. The proof is complete. h Theorem 3.9. A BCI-implicative ideal is a BCI-commutative ideal, but the converse is not true. Proof. Suppose that I is a BCI-implicative ideal of X. By Theorem 3.5, I is an ideal. To prove that I is a BCIcommutative ideal, from Proposition 2.4, it suffices to show that, if x * y 2 I, then x * ((y * (y * x)) * (0 * (0 * (x * y)))) 2 I. By Theorem 3.4, for any x,y 2 X, if ((x * y) * y) * (0 * y) 2 I,then x * ((y * (y * x)) * (0 * (0 * (x * y)))) 2 I. As (((x * y) * y) * (0 * y)) * (x * y) = (0 * y) * (0 * y) = 0 2 I, hence, if x * y 2 I, then ((x * y) * y) * (0 * y) 2 I, and so x * ((y * (y * x)) * (0 * (0 * (x * y)))) 2 I. Therefore I is a BCI-commutative ideal. To show the last half, we consider Example 3.6. We have demonstrated that {0,1} is not a BCI-implicative ideal of X. By routine calculations, {0,1} is a BCI-commutative ideal. This completes the proof. h Theorem 3.10. Let I be a nonempty subset of a BCI-algebra X. Then I is a BCI-implicative ideal if and only if it is both a BCI-positive implicative ideal and a BCI-commutative ideal. Proof. Necessity: Theorems 3.7 and 3.9. Sufficiency: Assume that I is both a BCI-positive implicative ideal and a BCI-commutative ideal. Then I is an ideal [20]. Let ((x * y) * y) * (0 * y) 2 I. By Proposition 2.2 (iii), we have x * y 2 I. Then by Proposition 2.4, we have x * ((y * (y * x)) * (0 * (0 * (x * y)))) 2 I. It follows from Theorem 3.4 that I is a BCI-implicative ideal. This proof is complete. h The connections between BCI-implicative ideals and p-ideals of X are obtained by the following: Definition 3.11 [28]. A nonempty subset I of a BCI-algebra X is called a p-ideal of X if it satisfies (I1) and (I6) (x * z) * (y * z) 2 I and y 2 I imply x 2 I. Proposition 3.12 [28]. An ideal I of a BCI-algebra X is a p-ideal of X if and only if 0 * (0 * x) 2 I implies x 2 I. Theorem 3.13. A p-ideal is a BCI-implicative ideal, but the converse is not true. Proof. Suppose that I is a p-ideal. Then I is an ideal [28]. In order to prove that I is a BCI-implicative ideal, from Theorem 3.4, it suffices to show that, if ((x * y) * y) * (0 * y) 2 I, then x * ((y * (y * x)) * (0 * (0 * (x * y)))) 2 I. Now, if ((x * y) * y) * (0 * y) 2 I, since 0 ½0 ðx ððy ðy xÞÞ ð0 ð0 ðx yÞÞÞÞÞ ¼ 0 ½ð0 xÞ ððð0 yÞ ð0 ðy xÞÞÞ ð0 ð0 ð0 ðx yÞÞÞÞÞ ¼ 0 ½ð0 xÞ ððð0 yÞ ðð0 yÞ ð0 xÞÞÞ ð0 ðx yÞÞÞ ¼ 0 ½ð0 xÞ ðððð0 ð0 yÞÞ ð0 ð0 xÞÞÞ yÞ ð0 ðx yÞÞÞ ¼ 0 ½ð0 xÞ ðð0 ð0 ð0 xÞÞÞ ð0 ðx yÞÞÞ ¼ 0 ½ð0 xÞ ðð0 xÞ ðð0 xÞ ð0 yÞÞÞ ¼ 0 ½ð0 xÞ ððð0 ð0 xÞÞ ð0 ð0 yÞÞÞ xÞ ¼ 0 ½ð0 xÞ ð0 ð0 ð0 yÞÞÞ ¼ 0 ðð0 xÞ ð0 yÞÞ ¼ 0 ð0 ðx yÞÞ;
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and ½0 ð0 ðx yÞÞ ½ððx yÞ yÞ ð0 yÞ ¼ ½0 ðððx yÞ yÞ ð0 yÞÞ ð0 ðx yÞÞ ¼ ½ðð0 ðx yÞÞ ð0 yÞÞ ð0 ð0 yÞÞ ð0 ðx yÞÞ ¼ ð0 ð0 yÞÞ ð0 ð0 yÞÞ ¼ 0 2 I; we have 0 * [0 * (x * ((y * (y * x)) * (0 * (0 * (x * y)))))] 2 I. By Proposition 3.12, x * ((y * (y * x)) * (0 * (0 * (x * y)))) 2 I. This means that I is a BCI-implicative ideal. The final statement of Theorem 3.13 is shown by the following: Example 3.14 Let X = {0,1,2,3} be a proper BCI-algebra with Cayley table as follows: * 0 1 2 3
0
1
2
3
0 1 2 3
0 0 2 3
0 1 0 3
3 3 3 0
Routine calculations show that I = {0,1} is a non-trivial BCI-implicative ideal of X, but it is not a p-ideal as follows: 0 * (0 * 2) = 0 * 0 = 0 2 {0,1}, but 2 62 {0,1}. The proof is complete. h From [21], the concept of BCI-commutative ideals is a generalization of the concept of commutative ideals. From [20], the concept of BCI-positive implicative ideals is a generalization of the concept of positive implicative ideals. That is, if X is a BCK-algebra, the concept of BCI-commutative ideals corresponds to the concept of commutative ideals, and the concept of BCI-positive implicative ideals corresponds to the concept of positive implicative ideals. Next, we show that the concept of BCI-implicative ideals is exactly a generalization of the concept of implicative ideals. Definition 3.15 [23]. A nonempty subset I of a BCK-algebra X is called an implicative ideal of X if it satisfies (I1) and (I7) (x * (y * x)) * z 2 I and z 2 I imply x 2 I. Proposition 3.16 [23]. Let I be a nonempty subset of a BCK-algebra X. Then I is an implicative ideal of X if and only if it is both a commutative ideal and a positive implicative ideal of X. Theorem 3.17. In a BCK-algebra X, a nonempty subset I is a BCI-implicative ideal of X if and only if it is an implicative ideal of X. Proof. By Proposition 3.16, I is an implicative ideal of X; if and only if it is both a commutative ideal, a positive implicative ideal of X; if and only if it is both a BCI-commutative ideal, a BCI-positive implicative ideal of X; if and only if it is a BCI-implicative ideal of X by Theorem 3.10. The proof is complete. h 4. The ideal characterizations of implicative BCI-algebras In this section, some further properties of BCI-implicative ideals are investigated, and the implicative BCIalgebras are completely described via BCI-implicative ideals. An ideal I is called closed if 0 * x 2 I whenever x 2 I, for all x 2 X [6]. The following shows that the characterization of BCI-implicative ideals I in Theorem 3.4, has a simpler form if I is a closed ideal.
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Theorem 4.1. Let I be a closed ideal of a BCI-algebra X. Then I is BCI-implicative if and only if it satisfies(b) ððx yÞ yÞ ð0 yÞ 2 I implies x ðy ðy xÞÞ 2 I: Proof. Let I be a BCI-implicative ideal and ((x * y) * y) * (0 * y) 2 I. By Theorem 3.4 we have x * ((y * (y * x)) * (0 * (0 * (x * y)))) 2 I. As ðx ðy ðy xÞÞÞ ½x ððy ðy xÞÞ ð0 ð0 ðx yÞÞÞÞ 6 ½ðy ðy xÞÞ ð0 ð0 ðx yÞÞÞ ðy ðy xÞÞ ¼ 0 ð0 ð0 ðx yÞÞÞ ¼ 0 ðx yÞ; and 0 ðððx yÞ yÞ ð0 yÞÞ ¼ ðð0 ðx yÞÞ ð0 yÞÞ ð0 ð0 yÞÞ ¼ ððð0 xÞ ð0 yÞÞ ð0 yÞÞ ð0 ð0 yÞÞ ¼ ½ðð0 ð0 ð0 yÞÞÞ xÞ ð0 yÞ ð0 yÞ ¼ ððð0 yÞ xÞ ð0 yÞÞ ð0 yÞ ¼ ð0 xÞ ð0 yÞ ¼ 0 ðx yÞ; ðÞ hence (x * (y * (y * x))) * [x * ((y * (y * x)) * (0 * (0 * (x * y))))] 6 0 * (((x * y) * y) * (0 * y)). Since I is a closed ideal, 0 * (((x * y) * y) * (0 * y)) 2 I. Combining x * ((y * (y * x)) * (0 * (0 * (x * y)))) 2 I, we obtain x * (y * (y * x)) 2 I. That is, I satisfies (b). Conversely, if I satisfies (b) and ((x * y) * y) * (0 * y) 2 I, then x * (y * (y * x)) 2 I. Since ½x ððy ðy xÞÞ ð0 ð0 ðx yÞÞÞÞ ðx ðy ðy xÞÞÞ 6 ðy ðy xÞÞ ½ðy ðy xÞÞ ð0 ð0 ðx yÞÞÞ 6 0 ð0 ðx yÞÞ ¼ 0 ð0 ðððx yÞ yÞ ð0 yÞÞÞðbyðÞÞ 2 I; we have x * ((y * (y * x)) * (0 * (0 * (x * y)))) 2 I. Hence, I is BCI-implicative.
h
The extension property of BCI-implicative ideals is obtained by the following: Theorem 4.2. If I is a BCI-implicative ideal of a BCI-algebra X, then every closed ideal A of X containing I is BCI-implicative. Proof. Let ((x * y) * y) * (0 * y) 2 A. Putting u = ((x * y) * y) * (0 * y), then 0 * u 2 A, as A is closed. Since (((x * u) * y) * y) * (0 * y) = (((x * y) * y) * (0 * y)) * u = 0 2 I, by the implicativity of I and Theorem 3.4, we have (x * u) * ((y * (y * (x * u))) * (0 * (0 * ((x * u) * y)))) 2 I A. As 0 ð0 ððx uÞ yÞÞ ¼ 0 ð0 ððx yÞ uÞÞ ¼ 0 ðð0 ðx yÞÞ ð0 uÞÞ ¼ 0ðbyðÞÞ; we have (x * u) * (y * (y * (x * u))) 2 A, i.e., (x * (y * (y * (x * u)))) * u 2 x * (y * (y * (x * u))) 2 A. As
A. Combining u 2 A implies
ðx ðy ðy xÞÞÞ ðx ðy ðy ðx uÞÞÞÞ 6 ðy ðy ðx uÞÞÞ ðy ðy xÞÞ 6 ðy xÞ ðy ðx uÞÞ 6 ðx uÞ x ¼ 0 u 2 A; we obtain x * (y * (y * x)) 2 A. Thus we have proved x * (y * (y * x)) 2 A. By Theorem 4.1, A is BCI-implicative. h
that
((x * y) * y) * (0 * y) 2 A
implies
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Corollary 4.3. If zero ideal {0} of X is BCI-implicative, then all closed ideals of X are BCI-implicative. The following theorems show that implicative BCI-algebras are completely described by BCI-implicative ideals. Definition 4.4 [25]. A BCI-algebra is said to be implicative if it satisfies (x * (x * y)) * (y * x) = y * (y * x). Proposition 4.5 [24]. A BCI-algebra X is commutative if and only if it satisfies x * (x * y) = y * (y * (x * (x * y))). Proposition 4.6 [26]. A BCI-algebra is implicative if and only if it is both positive implicative and commutative. Proposition 4.7 [7]. A BCI-algebra X is positive implicative if and only if it satisfies x * y = ((x * y) * y) * (0 * y). Theorem 4.8. For any BCI-algebra X, the following are equivalent: (i) X is an implicative BCI-algebra. (ii) Every closed ideal of X is BCI-implicative. (iii) The zero ideal {0} of X is BCI-implicative. Proof. (i) ) (ii). Assume that X is an implicative BCI-algebra and I a closed ideal of X. By Proposition 4.6, X is positive implicative and commutative. To prove that I is BCI-implicative, from Theorem 4.1, it suffices to show that if ((x * y) * y) * (0 * y) 2 I, then x * (y * (y * x)) 2 I. Now, if ((x * y) * y) * (0 * y) 2 I, then 0 * (((x * y) * y) * (0 * y)) 2 I, as I is closed. Since ðx ðy ðy xÞÞÞ ðððx yÞ yÞ ð0 yÞÞ ¼ ðx ðy ðy xÞÞÞ ðx yÞðby Proposition 4:7Þ ¼ ðx ðx yÞÞ ðy ðy xÞÞ ¼ ðy ðy ðx ðx yÞÞÞÞ ðy ðy xÞÞðby Proposition 4:5Þ ¼ ðy ðy ðy xÞÞÞ ðy ðx ðx yÞÞÞ ¼ ðy xÞ ðy ðx ðx yÞÞÞ 6 ðx ðx yÞÞ x ¼ 0 ðx yÞ ¼ 0 ðððx yÞ yÞ ð0 yÞÞðbyðÞÞ 2 I; we have x * (y * (y * x)) 2 I. Hence I is BCI-implicative. (ii) ) (iii). It is clear as {0} is a closed ideal. (iii) ) (i). If zero ideal {0} is BCI-implicative, by Theorem 3.10, it is both BCI-commutative and BCI-positive implicative. Now we prove X is both a commutative BCI-algebra and a positive implicative BCIalgebra. For any x,y 2 X, since (x * (x * y)) * y = 0, by Proposition 2.4, we have (x * (x * y)) * ((y * (y * (x * (x * y)))) * (0 * (0 * ((x * (x * y)) * y)))) = 0, i.e., (x * (x * y)) * (y * (y * (x * (x * y)))) = 0 as 0 * (0 * ((x * (x * y)) * y)) = 0. On the other hand, (y * (y * (x * (x * y)))) * (x * (x * y)) 6 (x * (x * y)) * (x * (x * y)) = 0. Hence, x * (x * y) = y * (y * (x * (x * y))), and X is commutative by Proposition 4.5. For any x,y 2 X, if we let s = ((x * y) * y) * (0 * y), then (((x * s) * y) * y) * (0 * y) = (((x * y) * y) * (0 * y)) * s = 0. Applying Proposition 2.2 (iii), we have (x * y) * s = (x * s) * y = 0, i.e., (x * y) * (((x * y) * y) * (0 * y)) = 0. On the other hand, (((x * y) * y) * (0 * y)) * (x * y) = (0 * y) * (0 * y) = 0. Hence, ((x * y) * y) * (0 * y) = x * y, and X is positive implicative by Proposition 4.7. Thus, by Proposition 4.6, we determine that X is implicative. The proof is complete. h
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Theorem 4.9. Let I be a closed ideal of a BCI-algebra X. Then quotient algebra (X/I; * ,C0) is an implicative BCI-algebra if and only if I is a BCI-implicative ideal. Proof. Suppose that I is a closed BCI-implicative ideal of X. If ((Cx * Cy) * Cy) * (C0 * Cy) = C0, i.e., C((x * y) * y) * (0 * y) = C0 2 {C0}, then ((x * y) * y) * (0 * y) 2 I. By Theorem 4.1, we have x * (y * (y * x)) 2 I. Hence, Cx * (Cy * (Cy * Cx)) = Cx * (y * (y * x)) = I = C0 2 {C0}. Thus we have shown that zero ideal {C0} is BCI-implicative. By Theorem 4.8, (X/I; * ,C0) is an implicative BCI-algebra. Conversely, if X/I is implicative, by Theorem 4.8, zero ideal {C0} is BCI-implicative. If ((x * y) * y) * (0 * y) 2 I, then ((Cx * Cy) * Cy) * (C0 * Cy) = C((x * y) * y) * (0 * y) = I = C0 2 {C0}. By Theorem 4.1, Cx * (y * (y * x)) = Cx * (Cy * (Cy * Cx)) 2 {C0}. This means that x * (y * (y * x)) 2 I. By Theorem 4.1 again, I is a BCI-implicative ideal of X. We complete the proof. h Now, we see Example 3.8. By routine calculations, {0,1,2} is a closed BCI-implicative ideal of X. By Theorem 4.9, X/I = {{0,1,2},{3},{4}} is an implicative BCI-algebra. Note that X is not an implicative BCI-algebra by Theorem 4.8. For a BCI-algebra X, the subset B(X) = {x 2 X—0 6 x} of X is called the p-radical of X [15]. Corollary 4.10. For any BCI-algebra X, B(X) is a BCI-implicative ideal of X. Proof. Because X/B(X) is a p-semisimple BCI-algebra [15] and a p-semisimple BCI-algebra is an implicative BCI-algebra [25], by Theorem 4.9, B(X) is a BCI-implicative ideal of X. The proof is complete. h 5. Conclusions To investigate the structure of an algebraic system, it is clear that ideals with special properties play an important role. The present paper introduced and studied BCI-implicative ideals. A characterization of BCI-implicative ideals was obtained. The connections between BCI-implicative ideals and other ideals were established. Specifically we proved that a nonempty subset of a BCI-algebra is a BCI-implicative ideal if and only if it is both a BCI-commutative ideal and a BCI-positive implicative ideal. If a BCI-implicative ideal is closed, it was provided a simpler characterization form of BCI-implicative ideals. Using this form, the extension property of BCI-implicative ideals was established, and the implicative BCI-algebras were described completely. The above work generalizes the corresponding results in BCK-algebras. It is our hope that this work would serve as a foundation for further study of the theory of BCK/BCI-algebras. Additional research remains to be conducted for Theorems 4.2, 4.8, 4.9, where they need a condition under which the ideal is closed. Research can be undertaken to discover if the condition ‘‘closed’’ can be deleted. Acknowledgements The authors express their sincere thanks to the referees for their valuable suggestions and comments. This work is supported by the National Natural Science Foundation of P.R. China (Grant No. 60474022), the Natural Science Foundation of Fujian (Grant No. S0650032) and the Science and Technology Foundation of Fujian Education Department (Grant No. JA06065). References [1] R. Cignoli, I.M.L. D’Ottaviano, D. Mundici, Algebraic Foundations of Many-Valued Reasoning, Kluwer Academic Publishers, Dordrecht, 2000. [2] R. Cignoli, A. Torrens, Glivenko like theorems in natural expansions of BCK-logic, Math. Log. Quart. 50 (2) (2004) 111–125. [3] F. Esteva, L. Godo, Monoidal t-norm based logic: towards a logic for left-continuous t-norms, Fuzzy Sets Syst 124 (2001) 271–288. [4] P. Hajek, Metamathematics of Fuzzy Logic, Kluwer Academic Publishers, Dordrecht, 1998. [5] P. Hajek, Fleas and fuzzy logic, J. Mult.-Valued Logic Soft Comput 11 (1–2) (2005) 137–152. [6] C.S. Hoo, Closed ideals and p-semisimple BCI-algebras, Math. Jpn. 35 (1990) 1103–1112. [7] Y.S. Huang, Characterizations of implicative BCI-algebras, Soochow J. Math. 25 (1999) 375–386. [8] Y. Imai, K. Iseki, On axiom system of propositional calculus, Proc. Jpn. Acad. 42 (1966) 19–22.
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A. Iorgulescu, Some direct ascendents of Wajsberg and MV algebras, Sci. Math. Jpn. 57 (2003) 583–647. A. Iorgulescu, Pseudo-Iseki algebras. Connection with pseudo-BL algebras, Multi-Valued Logic Soft Comput. 11 (2005) 263–308. K. Iseki, An algebra related with a propositional calculus, Proc. Jpn. Acad. 42 (1966) 26–29. K. Iseki, S. Tanaka, Ideal theory of BCK-algebras, Math. Jpn. 21 (1976) 351–366. Y.B. Jun, W.H. Shim, Fuzzy strong implicative hyper BCK-ideals of hyper BCK-algebras, Inform. Sci. 170 (2005) 351–361. Y.B. Jun, Y. Xu, J. Ma, Redefined fuzzy implicative filters, Inform. Sci. 177 (2007) 1333–1530. T.D. Lei, C.C. Xi, p-radical in BCI-algebras, Math. Jpn. 30 (1985) 511–517. Y.L. Liu, S.Y. Liu, J. Meng, FSI-ideals and FSC-ideals of BCI-algebras, Bull. Kor. Math. Soc. 41 (2004) 167–179. Y.L. Liu, S.Y. Liu, Y. Xu, An answer to the Jun-Shim-Lele’s open problem on the fuzzy filters, J. Appl. Math. Comput. 21 (2006) 325–329. Y.L. Liu, S.Y. Liu, Y. Xu, Pseudo-BCK algebras and PD-posets, Soft Comput. 11 (1) (2007) 91–101. Y.L. Liu, X.H. Zhang, Pseudo NM-algebras and their properties (in Chinese), Chin. J. Eng. Math. 23 (4) (2006) 753–756. Y.L. Liu, X.H. Zhang, Characterization of weakly positive implicative BCI-algebras, J. Hanzhong Teachers College (Natural) (1) (1994) 4–8. J. Meng, An ideal characterization of commutative BCI-algebras, Pusan Kyongnam Math. J. 9 (1) (1993) 1–6. J. Meng, X. Guo, On fuzzy ideals in BCK/BCI-algebras, Fuzzy Sets Syst 149 (2005) 509–525. J. Meng, Y.B. Jun, BCK-algebras, Kyung Moon Sa Co., Seoul, Korea, 1994. J. Meng, X.L. Xin, Commutative BCI-algebras, Math. Jpn. 37 (1992) 569–572. J. Meng, X.L. Xin, Implicative BCI-algebras, Pure Appl. Math. 8 (2) (1992) 99–103. J. Meng, X.L. Xin, Positive implicative BCI-algebras, Pure Appl. Math. 9 (1) (1993) 19–22. S.M. Wei, Y.B. Jun, Weakly positive implicative BCI-algebras, Comm. Korean Math. Soc. 10 (1995) 815–821. X.H. Zhang, H. Jiang, S.A. Bhatti, On p-ideals of BCI-algebra, Punjab Univ. J. Math. 27 (1994) 121–128.
BCI-implicative ideals of BCI-algebras Yong Lin Liu
a,*
, Yang Xu b, Jie Meng
c
a
b
Department of Mathematics, Wuyi University, Wuyishan, Fujian 354300, PR China Department of Applied Mathematics, Southwest Jiaotong University, Chengdu 610031, Sichuan, PR China c Department of Mathematics, Northwest University, Xi’an, Shaanxi 710069, PR China Received 12 May 2005; received in revised form 29 June 2007; accepted 2 July 2007
Abstract This paper introduces the notion of BCI-implicative ideals and characterizes BCI-implicative ideals and closed BCI-implicative ideals. Using these characterizations, the connections between BCI-implicative ideals and other ideals in BCI/BCK-algebras are investigated. Additionally, the extension property of BCI-implicative ideals is established. Finally, the implicative BCI-algebras are completely described using BCI-implicative ideals. The above work generalizes the corresponding results in BCK-algebras. Published by Elsevier Inc. Keywords: BCI-implicative ideal; Closed BCI-implicative ideal; Implicative BCI-algebra; Implicative BCK-algebra
1. Introduction One important task of artificial intelligence is to make the computers simulate human beings in dealing with certainty and uncertainty in information. It is well known that certain information processing, especially inferences based on certain information, is based on classical two-valued logic. Due to strict and complete logical foundation (classical logic), making inferences about certain information can be done with high confidence levels. Thus, it is natural and necessary to attempt to establish some rational logic system as the logical foundation for uncertain information processing. It is evident that this kind of logic cannot be two-valued logic itself but might form a certain extension of two-valued logic. Various kinds of non-classical logic systems have therefore been extensively researched in order to construct natural and efficient inference systems to deal with uncertainty. In recent years, motivated by both theory and application, the study of t-norm-based logic systems and the corresponding pseudo-logic systems has been become a greater focus in the field of logic. Here, t-norm-based logical investigations were first to the corresponding algebraic investigations, and in the case of pseudo-logic systems, algebraic development was first to the corresponding logical development.
*
Corresponding author. E-mail address: [email protected] (Y.L. Liu).
0020-0255/$ - see front matter Published by Elsevier Inc. doi:10.1016/j.ins.2007.07.003
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BCK and BCI-algebras are two classes of logic algebras. They were introduced by Imai and Iseki [8,11] and have been extensively investigated by many researchers (cf. [2,5,12–26]). BCI-algebras are generalizations of BCK-algebras. Iorgulescu [9,10] showed that pocrims and BCK-algebras with condition (S) are categorically isomorphic, and residuated lattices and bounded BCK lattices with condition (S) are categorically isomorphic. Hence, most of the algebras related to the t-norm-based logics, such as MTL [3], BL [4], hoop, MV [1], NM [3] and Boolean algebras etc., are extensions of BCK-algebra (i.e. they are subclasses of BCK-algebra). This shows that BCK/BCI-algebras are considerably general structures. Indeed, any results on BCK/BCI-algebras also hold for the aforementioned logic algebras. Using ideals to completely describe BCK/BCI-algebras was initiated by Iseki and Tanaka [12]. There are three important classes of BCI-algebras: commutative BCI-algebras, implicative BCI-algebras and positive implicative BCI-algebras. Meng [21] introduced BCI-commutative ideals and used these to completely describe commutative BCI-algebras. Liu and Zhang [20], and Wei and Jun [27] independently introduced BCI-positive implicative ideals and used these to completely describe positive implicative BCI-algebras. Hence, it is an interesting topic for researchers to investigate how to introduce so-called BCI-implicative ideals to completely describe implicative BCI-algebras. This paper introduces the notion of BCI-implicative ideals. It generalizes the notion of BCK-implicative ideals (i.e. implicative ideals). Furthermore, a characterization of BCI-implicative ideals is developed. We also prove that a nonempty subset of a BCI-algebra is a BCI-implicative ideal if and only if it is both a BCI-commutative ideal and a BCI-positive implicative ideal. This generalizes the well-known result: a nonempty subset of a BCK-algebra is an implicative ideal if and only if it is both a commutative ideal and a positive implicative ideal. A simpler characterization of a closed BCI-implicative ideal is obtained. Using this characterization, the extension property of BCI-implicative ideals is obtained and implicative BCI-algebras are completely described by BCI-implicative ideals. 2. Preliminaries An algebra (X; * , 0) of type (2, 0) is said to be a BCI-algebra if it satisfies the following conditions: for all x, y, z 2 X, (I) (II) (III) (IV)
((x * y) * (x * z)) * (z * y) = 0, (x * (x * y)) * y = 0, x * x = 0 and x * y = 0 and y * x = 0 imply x = y. If a BCI-algebra X satisfies the condition, (V) 0 * x = 0 for all x 2 X,
then X is called a BCK-algebra. Hence, a BCK-algebra is a BCI-algebra. In a BCI/BCK-algebra, we can define a binary relation 6 on X by x 6 y if and only if x * y = 0. Then (X; 6) is a partial ordering set. In a BCI-algebra X, the following hold: (1) (2) (3) (4) (5) (6)
(x * y) * z = (x * z) * y, x * (x * (x * y)) = x * y, ((x * z) * (y * z)) * (x * y) = 0, x * 0 = x, 0 * (x * y) = (0 * x) * (0 * y) and x 6 y implies x * z 6 y * z and z * y 6 z * x.
Throughout this paper, X always means a BCI-algebra without any specification. A nonempty subset I of a BCI-algebra X is called an ideal of X if it satisfies (I1) 0 2 I and (I2) x * y 2 I and y 2 I imply x 2 I.
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Definition 2.1 [20]. A nonempty subset I of a BCI-algebra X is called a BCI-positive implicative ideal of X if it satisfies (I1) and (I3) ((x * z) * z) * (y * z) 2 I and y 2 I imply x * z 2 I. Proposition 2.2 [20]. Let I be an ideal of BCI-algebra X. Then the following conditions are equivalent: (i) I is BCI-positive implicative. (ii) ((x * z) * z) * (y * z) 2 I implies (x * y) * z 2 I. (iii) ((x * y) * y) * (0 * y) 2 I implies x * y 2 I. Definition 2.3 [21]. A nonempty subset I of a BCI-algebra X is called a BCI-commutative ideal if it satisfies (I1) and (I4) (x * y) * z 2 I and z 2 I imply x * ((y * (y * x)) * (0 * (0 * (x * y)))) 2 I. Proposition 2.4 [21]. An ideal I of a BCI-algebra X is BCI-commutative if and only if x * y 2 I implies x * ((y * (y * x)) * (0 * (0 * (x * y)))) 2 I. 3. BCI-implicative ideals of BCI-algebras Definition 3.1. A nonempty subset I of a BCI-algebra X is called a BCI-implicative ideal if it satisfies (I1) and (I5) (((x * y) * y) * (0 * y)) * z 2 I and z 2 I imply x * ((y * (y * x)) * (0 * (0 * (x * y)))) 2 I, for all x,y,z 2 X. The following examples show that BCI-implicative ideals exist. Example 3.2. Any ideal of a p-semisimple BCI-algebra [15] is BCI-implicative. Example 3.3. Let X = {0, 1, 2, 3, 4, 5} with Cayley table as follows:
* 0 1 2 3 4 5
0
1
2
3
4
5
0 1 2 3 4 5
0 0 2 3 3 3
0 1 0 3 4 5
3 3 3 0 1 1
3 3 3 0 0 1
3 3 3 0 0 0
Then X is a proper BCI-algebra. By routine calculations, we have I = {0, 1, 2} is a BCI-implicative ideal of X. A characterization of the BCI-implicative ideal of X is obtained by the following: Theorem 3.4. Let I be an ideal of X. Then I is BCI-implicative if and only if (a) ððx yÞ yÞ ð0 yÞ 2 I implies x ððy ðy xÞÞ ð0 ð0 ðx yÞÞÞÞ 2 I: Proof. Suppose that I is a BCI-implicative ideal of X. Let x, y 2 X; if ((x * y) * y) * (0 * y) 2 I, then (((x * y) * y) * (0 * y)) * 0 2 I and 0 2 I. By (I5), x * ((y * (y * x)) * (0 * (0 * (x * y)))) 2 I. Hence, (a) holds. Conversely, suppose that the ideal I satisfies (a). For x,y,z 2 X, if (((x * y) * y) * (0 * y)) * z 2 I and z 2 I, by the definition of ideals we obtain ((x * y) * y) * (0 * y) 2 I. It follows from (a) that x * ((y * (y * x)) * (0 * (0 * (x * y)))) 2 I. This means that I is a BCI-implicative ideal. This completes the proof. h
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The connections between BCI-implicative ideals and ideals of X are obtained by the following theorem. Theorem 3.5. A BCI-implicative ideal is an ideal, but the converse is not true. Proof. Suppose that I is a BCI-implicative ideal, and let y = 0 in (I5). We have x * z 2 I, z 2 I imply x 2 I. This means that I is an ideal. The last part is shown by the following example. h Example 3.6. Let X = {0, 1, 2, 3, 4} be a proper BCI-algebra with Cayley table as follows: * 0 1 2 3 4
0
1
2
3
4
0 1 2 3 4
0 0 2 3 4
0 0 0 2 4
0 0 0 0 4
4 4 4 4 0
I = {0,1} is a non-trivial ideal of X; but not (((3 * 2) * 2) * (0 * 2)) * 0 = (2 * 2) * (0 * 2) = 0 2 {0, 1} and (3 * 2)))) = 3 * ((2 * 0) * 0) = 2; 62 {0, 1}.
a BCI-implicative ideal of X because 0 2 {0, 1}, but 3 * ((2 * (2 * 3)) * (0 * (0 *
The following three theorems demonstrate the close relations among BCI-positive implicative ideals, BCIcommutative ideals and BCI-implicative ideals in BCI-algebras. Theorem 3.7. A BCI-implicative ideal is a BCI-positive implicative ideal, but the converse is not true. Proof. Assume that I is a BCI-implicative ideal of X. It follows from Theorem 3.5 that I is an ideal. In order to prove that I is BCI-positive implicative, from Proposition 2.2 (iii), it suffices to show that, if ((x * y) * y) * (0 * y) 2 I, then x * y 2 I. Now, if ((x * y) * y) * (0 * y) 2 I, by Theorem 3.4 we have x * ((y * (y * x)) * (0 * (0 * (x * y)))) 2 I. Since ðx yÞ ½x ððy ðy xÞÞ ð0 ð0 ðx yÞÞÞÞ 6 ½ðy ðy xÞÞ ð0 ð0 ðx yÞÞÞ y ¼ ð0 ðy xÞÞ ð0 ð0 ðx yÞÞÞ ¼ ½0 ð0 ð0 ðx yÞÞÞ ðy xÞ ¼ ð0 ðx yÞÞ ðy xÞ ¼ ðð0 xÞ ð0 yÞÞ ðy xÞ ¼ ðð0 ðy xÞÞ xÞ ð0 yÞ ¼ ððð0 yÞ ð0 xÞÞ xÞ ð0 yÞ ¼ ð0 ð0 xÞÞ x ¼ ð0 xÞ ð0 xÞ ¼ 0 2 I; we have x * y 2 I. Thus, we prove that I is a BCI-positive implicative ideal. The final statement of the theorem is shown by the following example. Example 3.8. Let X = {0, 1, 2, 3, 4} be a proper BCI-algebra with Cayley table given by
* 0 1 2 3 4
0
1
2
3
4
0 1 2 3 4
0 0 2 3 4
0 1 0 3 4
0 0 0 0 4
4 4 4 4 0
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By routine calculations, {0, 2} is a non-trivial BCI-positive implicative ideal of X, but not a BCI-implicative ideal of X as follows: (((1 * 3) * 3) * (0 * 3)) * 0 = 0 2 {0, 2} and 0 2 {0, 2}, but 1 * ((3 * (3 * 1)) * (0 * (0 * (1 * 3)))) = 1 * (0 * 0) = 1 62 {0, 2}. The proof is complete. h Theorem 3.9. A BCI-implicative ideal is a BCI-commutative ideal, but the converse is not true. Proof. Suppose that I is a BCI-implicative ideal of X. By Theorem 3.5, I is an ideal. To prove that I is a BCIcommutative ideal, from Proposition 2.4, it suffices to show that, if x * y 2 I, then x * ((y * (y * x)) * (0 * (0 * (x * y)))) 2 I. By Theorem 3.4, for any x,y 2 X, if ((x * y) * y) * (0 * y) 2 I,then x * ((y * (y * x)) * (0 * (0 * (x * y)))) 2 I. As (((x * y) * y) * (0 * y)) * (x * y) = (0 * y) * (0 * y) = 0 2 I, hence, if x * y 2 I, then ((x * y) * y) * (0 * y) 2 I, and so x * ((y * (y * x)) * (0 * (0 * (x * y)))) 2 I. Therefore I is a BCI-commutative ideal. To show the last half, we consider Example 3.6. We have demonstrated that {0,1} is not a BCI-implicative ideal of X. By routine calculations, {0,1} is a BCI-commutative ideal. This completes the proof. h Theorem 3.10. Let I be a nonempty subset of a BCI-algebra X. Then I is a BCI-implicative ideal if and only if it is both a BCI-positive implicative ideal and a BCI-commutative ideal. Proof. Necessity: Theorems 3.7 and 3.9. Sufficiency: Assume that I is both a BCI-positive implicative ideal and a BCI-commutative ideal. Then I is an ideal [20]. Let ((x * y) * y) * (0 * y) 2 I. By Proposition 2.2 (iii), we have x * y 2 I. Then by Proposition 2.4, we have x * ((y * (y * x)) * (0 * (0 * (x * y)))) 2 I. It follows from Theorem 3.4 that I is a BCI-implicative ideal. This proof is complete. h The connections between BCI-implicative ideals and p-ideals of X are obtained by the following: Definition 3.11 [28]. A nonempty subset I of a BCI-algebra X is called a p-ideal of X if it satisfies (I1) and (I6) (x * z) * (y * z) 2 I and y 2 I imply x 2 I. Proposition 3.12 [28]. An ideal I of a BCI-algebra X is a p-ideal of X if and only if 0 * (0 * x) 2 I implies x 2 I. Theorem 3.13. A p-ideal is a BCI-implicative ideal, but the converse is not true. Proof. Suppose that I is a p-ideal. Then I is an ideal [28]. In order to prove that I is a BCI-implicative ideal, from Theorem 3.4, it suffices to show that, if ((x * y) * y) * (0 * y) 2 I, then x * ((y * (y * x)) * (0 * (0 * (x * y)))) 2 I. Now, if ((x * y) * y) * (0 * y) 2 I, since 0 ½0 ðx ððy ðy xÞÞ ð0 ð0 ðx yÞÞÞÞÞ ¼ 0 ½ð0 xÞ ððð0 yÞ ð0 ðy xÞÞÞ ð0 ð0 ð0 ðx yÞÞÞÞÞ ¼ 0 ½ð0 xÞ ððð0 yÞ ðð0 yÞ ð0 xÞÞÞ ð0 ðx yÞÞÞ ¼ 0 ½ð0 xÞ ðððð0 ð0 yÞÞ ð0 ð0 xÞÞÞ yÞ ð0 ðx yÞÞÞ ¼ 0 ½ð0 xÞ ðð0 ð0 ð0 xÞÞÞ ð0 ðx yÞÞÞ ¼ 0 ½ð0 xÞ ðð0 xÞ ðð0 xÞ ð0 yÞÞÞ ¼ 0 ½ð0 xÞ ððð0 ð0 xÞÞ ð0 ð0 yÞÞÞ xÞ ¼ 0 ½ð0 xÞ ð0 ð0 ð0 yÞÞÞ ¼ 0 ðð0 xÞ ð0 yÞÞ ¼ 0 ð0 ðx yÞÞ;
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and ½0 ð0 ðx yÞÞ ½ððx yÞ yÞ ð0 yÞ ¼ ½0 ðððx yÞ yÞ ð0 yÞÞ ð0 ðx yÞÞ ¼ ½ðð0 ðx yÞÞ ð0 yÞÞ ð0 ð0 yÞÞ ð0 ðx yÞÞ ¼ ð0 ð0 yÞÞ ð0 ð0 yÞÞ ¼ 0 2 I; we have 0 * [0 * (x * ((y * (y * x)) * (0 * (0 * (x * y)))))] 2 I. By Proposition 3.12, x * ((y * (y * x)) * (0 * (0 * (x * y)))) 2 I. This means that I is a BCI-implicative ideal. The final statement of Theorem 3.13 is shown by the following: Example 3.14 Let X = {0,1,2,3} be a proper BCI-algebra with Cayley table as follows: * 0 1 2 3
0
1
2
3
0 1 2 3
0 0 2 3
0 1 0 3
3 3 3 0
Routine calculations show that I = {0,1} is a non-trivial BCI-implicative ideal of X, but it is not a p-ideal as follows: 0 * (0 * 2) = 0 * 0 = 0 2 {0,1}, but 2 62 {0,1}. The proof is complete. h From [21], the concept of BCI-commutative ideals is a generalization of the concept of commutative ideals. From [20], the concept of BCI-positive implicative ideals is a generalization of the concept of positive implicative ideals. That is, if X is a BCK-algebra, the concept of BCI-commutative ideals corresponds to the concept of commutative ideals, and the concept of BCI-positive implicative ideals corresponds to the concept of positive implicative ideals. Next, we show that the concept of BCI-implicative ideals is exactly a generalization of the concept of implicative ideals. Definition 3.15 [23]. A nonempty subset I of a BCK-algebra X is called an implicative ideal of X if it satisfies (I1) and (I7) (x * (y * x)) * z 2 I and z 2 I imply x 2 I. Proposition 3.16 [23]. Let I be a nonempty subset of a BCK-algebra X. Then I is an implicative ideal of X if and only if it is both a commutative ideal and a positive implicative ideal of X. Theorem 3.17. In a BCK-algebra X, a nonempty subset I is a BCI-implicative ideal of X if and only if it is an implicative ideal of X. Proof. By Proposition 3.16, I is an implicative ideal of X; if and only if it is both a commutative ideal, a positive implicative ideal of X; if and only if it is both a BCI-commutative ideal, a BCI-positive implicative ideal of X; if and only if it is a BCI-implicative ideal of X by Theorem 3.10. The proof is complete. h 4. The ideal characterizations of implicative BCI-algebras In this section, some further properties of BCI-implicative ideals are investigated, and the implicative BCIalgebras are completely described via BCI-implicative ideals. An ideal I is called closed if 0 * x 2 I whenever x 2 I, for all x 2 X [6]. The following shows that the characterization of BCI-implicative ideals I in Theorem 3.4, has a simpler form if I is a closed ideal.
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Theorem 4.1. Let I be a closed ideal of a BCI-algebra X. Then I is BCI-implicative if and only if it satisfies(b) ððx yÞ yÞ ð0 yÞ 2 I implies x ðy ðy xÞÞ 2 I: Proof. Let I be a BCI-implicative ideal and ((x * y) * y) * (0 * y) 2 I. By Theorem 3.4 we have x * ((y * (y * x)) * (0 * (0 * (x * y)))) 2 I. As ðx ðy ðy xÞÞÞ ½x ððy ðy xÞÞ ð0 ð0 ðx yÞÞÞÞ 6 ½ðy ðy xÞÞ ð0 ð0 ðx yÞÞÞ ðy ðy xÞÞ ¼ 0 ð0 ð0 ðx yÞÞÞ ¼ 0 ðx yÞ; and 0 ðððx yÞ yÞ ð0 yÞÞ ¼ ðð0 ðx yÞÞ ð0 yÞÞ ð0 ð0 yÞÞ ¼ ððð0 xÞ ð0 yÞÞ ð0 yÞÞ ð0 ð0 yÞÞ ¼ ½ðð0 ð0 ð0 yÞÞÞ xÞ ð0 yÞ ð0 yÞ ¼ ððð0 yÞ xÞ ð0 yÞÞ ð0 yÞ ¼ ð0 xÞ ð0 yÞ ¼ 0 ðx yÞ; ðÞ hence (x * (y * (y * x))) * [x * ((y * (y * x)) * (0 * (0 * (x * y))))] 6 0 * (((x * y) * y) * (0 * y)). Since I is a closed ideal, 0 * (((x * y) * y) * (0 * y)) 2 I. Combining x * ((y * (y * x)) * (0 * (0 * (x * y)))) 2 I, we obtain x * (y * (y * x)) 2 I. That is, I satisfies (b). Conversely, if I satisfies (b) and ((x * y) * y) * (0 * y) 2 I, then x * (y * (y * x)) 2 I. Since ½x ððy ðy xÞÞ ð0 ð0 ðx yÞÞÞÞ ðx ðy ðy xÞÞÞ 6 ðy ðy xÞÞ ½ðy ðy xÞÞ ð0 ð0 ðx yÞÞÞ 6 0 ð0 ðx yÞÞ ¼ 0 ð0 ðððx yÞ yÞ ð0 yÞÞÞðbyðÞÞ 2 I; we have x * ((y * (y * x)) * (0 * (0 * (x * y)))) 2 I. Hence, I is BCI-implicative.
h
The extension property of BCI-implicative ideals is obtained by the following: Theorem 4.2. If I is a BCI-implicative ideal of a BCI-algebra X, then every closed ideal A of X containing I is BCI-implicative. Proof. Let ((x * y) * y) * (0 * y) 2 A. Putting u = ((x * y) * y) * (0 * y), then 0 * u 2 A, as A is closed. Since (((x * u) * y) * y) * (0 * y) = (((x * y) * y) * (0 * y)) * u = 0 2 I, by the implicativity of I and Theorem 3.4, we have (x * u) * ((y * (y * (x * u))) * (0 * (0 * ((x * u) * y)))) 2 I A. As 0 ð0 ððx uÞ yÞÞ ¼ 0 ð0 ððx yÞ uÞÞ ¼ 0 ðð0 ðx yÞÞ ð0 uÞÞ ¼ 0ðbyðÞÞ; we have (x * u) * (y * (y * (x * u))) 2 A, i.e., (x * (y * (y * (x * u)))) * u 2 x * (y * (y * (x * u))) 2 A. As
A. Combining u 2 A implies
ðx ðy ðy xÞÞÞ ðx ðy ðy ðx uÞÞÞÞ 6 ðy ðy ðx uÞÞÞ ðy ðy xÞÞ 6 ðy xÞ ðy ðx uÞÞ 6 ðx uÞ x ¼ 0 u 2 A; we obtain x * (y * (y * x)) 2 A. Thus we have proved x * (y * (y * x)) 2 A. By Theorem 4.1, A is BCI-implicative. h
that
((x * y) * y) * (0 * y) 2 A
implies
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Corollary 4.3. If zero ideal {0} of X is BCI-implicative, then all closed ideals of X are BCI-implicative. The following theorems show that implicative BCI-algebras are completely described by BCI-implicative ideals. Definition 4.4 [25]. A BCI-algebra is said to be implicative if it satisfies (x * (x * y)) * (y * x) = y * (y * x). Proposition 4.5 [24]. A BCI-algebra X is commutative if and only if it satisfies x * (x * y) = y * (y * (x * (x * y))). Proposition 4.6 [26]. A BCI-algebra is implicative if and only if it is both positive implicative and commutative. Proposition 4.7 [7]. A BCI-algebra X is positive implicative if and only if it satisfies x * y = ((x * y) * y) * (0 * y). Theorem 4.8. For any BCI-algebra X, the following are equivalent: (i) X is an implicative BCI-algebra. (ii) Every closed ideal of X is BCI-implicative. (iii) The zero ideal {0} of X is BCI-implicative. Proof. (i) ) (ii). Assume that X is an implicative BCI-algebra and I a closed ideal of X. By Proposition 4.6, X is positive implicative and commutative. To prove that I is BCI-implicative, from Theorem 4.1, it suffices to show that if ((x * y) * y) * (0 * y) 2 I, then x * (y * (y * x)) 2 I. Now, if ((x * y) * y) * (0 * y) 2 I, then 0 * (((x * y) * y) * (0 * y)) 2 I, as I is closed. Since ðx ðy ðy xÞÞÞ ðððx yÞ yÞ ð0 yÞÞ ¼ ðx ðy ðy xÞÞÞ ðx yÞðby Proposition 4:7Þ ¼ ðx ðx yÞÞ ðy ðy xÞÞ ¼ ðy ðy ðx ðx yÞÞÞÞ ðy ðy xÞÞðby Proposition 4:5Þ ¼ ðy ðy ðy xÞÞÞ ðy ðx ðx yÞÞÞ ¼ ðy xÞ ðy ðx ðx yÞÞÞ 6 ðx ðx yÞÞ x ¼ 0 ðx yÞ ¼ 0 ðððx yÞ yÞ ð0 yÞÞðbyðÞÞ 2 I; we have x * (y * (y * x)) 2 I. Hence I is BCI-implicative. (ii) ) (iii). It is clear as {0} is a closed ideal. (iii) ) (i). If zero ideal {0} is BCI-implicative, by Theorem 3.10, it is both BCI-commutative and BCI-positive implicative. Now we prove X is both a commutative BCI-algebra and a positive implicative BCIalgebra. For any x,y 2 X, since (x * (x * y)) * y = 0, by Proposition 2.4, we have (x * (x * y)) * ((y * (y * (x * (x * y)))) * (0 * (0 * ((x * (x * y)) * y)))) = 0, i.e., (x * (x * y)) * (y * (y * (x * (x * y)))) = 0 as 0 * (0 * ((x * (x * y)) * y)) = 0. On the other hand, (y * (y * (x * (x * y)))) * (x * (x * y)) 6 (x * (x * y)) * (x * (x * y)) = 0. Hence, x * (x * y) = y * (y * (x * (x * y))), and X is commutative by Proposition 4.5. For any x,y 2 X, if we let s = ((x * y) * y) * (0 * y), then (((x * s) * y) * y) * (0 * y) = (((x * y) * y) * (0 * y)) * s = 0. Applying Proposition 2.2 (iii), we have (x * y) * s = (x * s) * y = 0, i.e., (x * y) * (((x * y) * y) * (0 * y)) = 0. On the other hand, (((x * y) * y) * (0 * y)) * (x * y) = (0 * y) * (0 * y) = 0. Hence, ((x * y) * y) * (0 * y) = x * y, and X is positive implicative by Proposition 4.7. Thus, by Proposition 4.6, we determine that X is implicative. The proof is complete. h
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Theorem 4.9. Let I be a closed ideal of a BCI-algebra X. Then quotient algebra (X/I; * ,C0) is an implicative BCI-algebra if and only if I is a BCI-implicative ideal. Proof. Suppose that I is a closed BCI-implicative ideal of X. If ((Cx * Cy) * Cy) * (C0 * Cy) = C0, i.e., C((x * y) * y) * (0 * y) = C0 2 {C0}, then ((x * y) * y) * (0 * y) 2 I. By Theorem 4.1, we have x * (y * (y * x)) 2 I. Hence, Cx * (Cy * (Cy * Cx)) = Cx * (y * (y * x)) = I = C0 2 {C0}. Thus we have shown that zero ideal {C0} is BCI-implicative. By Theorem 4.8, (X/I; * ,C0) is an implicative BCI-algebra. Conversely, if X/I is implicative, by Theorem 4.8, zero ideal {C0} is BCI-implicative. If ((x * y) * y) * (0 * y) 2 I, then ((Cx * Cy) * Cy) * (C0 * Cy) = C((x * y) * y) * (0 * y) = I = C0 2 {C0}. By Theorem 4.1, Cx * (y * (y * x)) = Cx * (Cy * (Cy * Cx)) 2 {C0}. This means that x * (y * (y * x)) 2 I. By Theorem 4.1 again, I is a BCI-implicative ideal of X. We complete the proof. h Now, we see Example 3.8. By routine calculations, {0,1,2} is a closed BCI-implicative ideal of X. By Theorem 4.9, X/I = {{0,1,2},{3},{4}} is an implicative BCI-algebra. Note that X is not an implicative BCI-algebra by Theorem 4.8. For a BCI-algebra X, the subset B(X) = {x 2 X—0 6 x} of X is called the p-radical of X [15]. Corollary 4.10. For any BCI-algebra X, B(X) is a BCI-implicative ideal of X. Proof. Because X/B(X) is a p-semisimple BCI-algebra [15] and a p-semisimple BCI-algebra is an implicative BCI-algebra [25], by Theorem 4.9, B(X) is a BCI-implicative ideal of X. The proof is complete. h 5. Conclusions To investigate the structure of an algebraic system, it is clear that ideals with special properties play an important role. The present paper introduced and studied BCI-implicative ideals. A characterization of BCI-implicative ideals was obtained. The connections between BCI-implicative ideals and other ideals were established. Specifically we proved that a nonempty subset of a BCI-algebra is a BCI-implicative ideal if and only if it is both a BCI-commutative ideal and a BCI-positive implicative ideal. If a BCI-implicative ideal is closed, it was provided a simpler characterization form of BCI-implicative ideals. Using this form, the extension property of BCI-implicative ideals was established, and the implicative BCI-algebras were described completely. The above work generalizes the corresponding results in BCK-algebras. It is our hope that this work would serve as a foundation for further study of the theory of BCK/BCI-algebras. Additional research remains to be conducted for Theorems 4.2, 4.8, 4.9, where they need a condition under which the ideal is closed. Research can be undertaken to discover if the condition ‘‘closed’’ can be deleted. Acknowledgements The authors express their sincere thanks to the referees for their valuable suggestions and comments. This work is supported by the National Natural Science Foundation of P.R. China (Grant No. 60474022), the Natural Science Foundation of Fujian (Grant No. S0650032) and the Science and Technology Foundation of Fujian Education Department (Grant No. JA06065). References [1] R. Cignoli, I.M.L. D’Ottaviano, D. Mundici, Algebraic Foundations of Many-Valued Reasoning, Kluwer Academic Publishers, Dordrecht, 2000. [2] R. Cignoli, A. Torrens, Glivenko like theorems in natural expansions of BCK-logic, Math. Log. Quart. 50 (2) (2004) 111–125. [3] F. Esteva, L. Godo, Monoidal t-norm based logic: towards a logic for left-continuous t-norms, Fuzzy Sets Syst 124 (2001) 271–288. [4] P. Hajek, Metamathematics of Fuzzy Logic, Kluwer Academic Publishers, Dordrecht, 1998. [5] P. Hajek, Fleas and fuzzy logic, J. Mult.-Valued Logic Soft Comput 11 (1–2) (2005) 137–152. [6] C.S. Hoo, Closed ideals and p-semisimple BCI-algebras, Math. Jpn. 35 (1990) 1103–1112. [7] Y.S. Huang, Characterizations of implicative BCI-algebras, Soochow J. Math. 25 (1999) 375–386. [8] Y. Imai, K. Iseki, On axiom system of propositional calculus, Proc. Jpn. Acad. 42 (1966) 19–22.
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