Neural Computing and Applications

, Volume 20, Issue 3, pp 319–328

On \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy ideals of BCI-algebras

Authors

    • Department of MathematicsHubei Institute for Nationalities
  • Young Bae Jun
    • Department of Mathematics EducationGyeongsang National University
Original Article

DOI: 10.1007/s00521-010-0376-6

Cite this article as:
Zhan, J. & Jun, Y.B. Neural Comput & Applic (2011) 20: 319. doi:10.1007/s00521-010-0376-6

Abstract

The concepts of \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy (p-, q- and a-) ideals of BCI-algebras are introduced and some related properties are investigated. In particular, we describe the relationships among ordinary fuzzy (p-, q- and a-) ideals, (∈, ∈ ∨ q)-fuzzy (p-, q- and a-) ideals and \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy (p-,q- and a-) ideals of BCI-algebras. Moreover, we prove that a fuzzy set μ of a BCI-algebra X is an \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy a-ideal of X if and only if it is both an \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy p-ideal and an \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy q-ideal. Finally, we give some characterizations of three particular cases of BCI-algebras by these generalized fuzzy ideals.

Keywords

(p-Semisimple; quasi-associative; associative) BCI-algebra(∈, ∈ ∨ q)-fuzzy (p-; q- and a-) ideal(p-; q- and a-) ideal\((\overline{\in};\overline{\in} \vee \overline{q})\)-fuzzy (p-; q- and a-) ideal

1 Introduction

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 inference 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.

As it is well known, BCK and BCI-algebras are two classes of algebras of logic. They were introduced by Imai and Iséki [7, 8] and have been extensively investigated by many researchers, see ([7, 8, 15, 16, 2022]). BCI-algebras are generalizations of BCK-algebras. Most of the algebras related to the t-norm based logic, such as MTL-algebras [5], BL-algebras [6], hoop, MV-algebras (i.e., lattice implication algebras [18]) and Boolean algebras et al., are extensions of BCK-algebras (i.e., they are subclasses of BCK-algebras). This shows that BCK/BCI-algebras are considerably general structures.

After the introduction of fuzzy sets by Zadeh [19], there have been a number of generalizations of this fundamental concept. A new type of fuzzy subgroup, that is, the (∈, ∈ ∨ q)-fuzzy subgroup, was introduced in an earlier paper of Bhakat and Das [2] by using the combined notions of “belongingness” and “quasicoincidence” of fuzzy points and fuzzy sets, which was introduced by Pu and Liu [17]. In fact, the (∈, ∈ ∨ q)-fuzzy subgroup is an important generalization of Rosenfeld’s fuzzy subgroup. It is now natural to investigate similar type of generalizations of the existing fuzzy subsystems with other algebraic structures. With this objective in view, Jun [9] introduced the concept of (α, β)-fuzzy ideals of a BCK/BCI-algebra and investigated related results. Davvaz [2] applied this theory to near-rings and obtained some useful results. Further, in [15], Ma et al. also discussed the properties of some kinds of (∈, ∈ ∨ q)-interval-valued fuzzy ideals of BCI-algebras.

In [20], we introduced the concepts of (∈, ∈ ∨ q)-fuzzy (p-,q- and a-) ideals of BCI-algebras and investigated some of their related properties. As a continuation of this paper, we further discuss this topic in the present paper. Section 3 is divided into four subsections. In Sect. 3.1, we describe the relationships among ordinary fuzzy ideals, (∈, ∈ ∨ q)-fuzzy ideals and \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy ideals of BCI-algebras. In Sect. 3.2, we describe the relationships among ordinary fuzzy p-ideals, (∈, ∈ ∨ q)-fuzzy p-ideals and \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy p-ideals of BCI-algebras. In Sect. 3.3, we describe the relationships among ordinary fuzzy q-ideals, (∈, ∈ ∨ q)-fuzzy q-ideals and \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy q-ideals of BCI-algebras. Further, in Sect. 3.4, the relationships among ordinary fuzzy a-ideals, (∈, ∈ ∨ q)-fuzzy a-ideals and \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy a-ideals of BCI-algebras are considered. Finally, we prove that a fuzzy set μ of a BCI-algebra X is an \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy a-ideal of X if and only if it is both an \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy p-ideal and an \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy q-ideal. In Sect. 4, we give some characterizations of three particular cases of BCI-algebras by these generalized fuzzy ideals.

2 Preliminaries

By a BCI-algebra, we mean an algebra (X,  ∗  , 0) of type (2,0) satisfying the axioms:
  1. (1)

    ((x  ∗  y) ∗ (x ∗ z)) ∗ (z ∗ y) = 0;

     
  2. (2)

    (x ∗ (x ∗ y)) ∗ y = 0;

     
  3. (3)

    x ∗ x = 0;

     
  4. (4)

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

     

We can define a partial ordering “≤” by xy if and only if x ∗ y = 0.

If a BCI-algebra X satisfies 0 ∗ x = 0 for all x ∈ X, then we say that X is a BCK-algebra. In what follows, let X denote a BCI-algebra unless otherwise specified.

Proposition 2.1

[3, 4, 7, 8]. In any BCI-algebra X, the following are true:
  1. (1)

    (x ∗ y) ∗ z = (x ∗ z) ∗ y,

     
  2. (2)

    (x ∗ z) ∗ (y ∗ z) ≤ x ∗ y,

     
  3. (3)

    (x ∗ y) ∗ (x ∗ z) ≤ z ∗ y,

     
  4. (4)

    x ∗ 0 = x,

     
  5. (5)

    0 ∗ (x ∗ y) = (0 ∗ x) ∗ (0 ∗ y),

     
  6. (6)

    x ∗ (x ∗ (x ∗ y)) = x ∗ y.

     

In [13, 22], we can see some types of ideals of BCI-algebras as follows:

A non-empty subset I of X is called an ideal of X if it satisfies (I1) 0 ∈ I; (I2) x ∗ y ∈ I and y ∈ I imply x ∈ I. A non-empty subset I of X is called a p-ideal if it satisfies (I1) and (I3) (x ∗ z)  ∗ (y ∗ z) ∈ I and y ∈ I imply x ∈ I. A non-empty subset I of X is called a q-ideal if it satisfies (I1) and (I4) x ∗ (y ∗ z) ∈ I and y ∈ I imply x ∗ z ∈ I. A non-empty subset I of X is called an a-ideal if it satisfies (I1) and (I5) (x ∗ z) ∗ (0 ∗ y) ∈ I and z ∈ I imply y ∗ x ∈ I.

Theorem 2.2 [13]

  1. (1)

    Every a- (resp., p-, q-) ideal of any BCI-algebra is an ideal, but the converse is not true;

     
  2. (2)

    A non-empty subset I of a BCI-algebra X is an a-ideal of X if and only if it is both a p-ideal and q-ideal.

     

We now review some fuzzy logic concepts. A fuzzy set of X is a function μ: X → [0, 1].

Definition 2.3 [10, 11, 14]

  1. (1)
    A fuzzy set μ of X is called a fuzzy ideal of X if it satisfies:
    • (F1) μ(0) ≥ μ(x), ∀x ∈ X

    • (F2) μ(x) ≥ min{μ(x ∗ y), μ(y)}, ∀xy ∈ X.

     
  2. (2)
    A fuzzy set μ of X is called a fuzzy p-ideal of X if it satisfies (F1) and
    • (F3) μ(x) ≥ min{μ((x ∗ z) ∗ (y ∗ z)), μ(y)}, for all xyz ∈ X.

     
  3. (3)
    A fuzzy set μ of X is called a fuzzy q-ideal of X if it satisfies (F1) and
    • (F4) μ(x ∗ z) ≥ min{μ(x ∗ (y ∗ z)), μ(y)}, for all xyz ∈ X.

     
  4. (4)
    A fuzzy set μ of X is called fuzzy a-ideal of X if it satisfies (F1) and
    • (F5) μ(y ∗ x) ≥ min{μ((x ∗ z) ∗ (0 ∗ y)), μ(z)}, for all xyz ∈ X.

     

For a fuzzy set μ of X and t ∈ (0, 1], the crisp set μt = {x ∈ X | μ(x) ≥ t} is called the level subset of μ.

Theorem 2.4 [10, 11, 14]

A fuzzy set μ of X is a fuzzy (p-, q- and a-) ideal of X if and only if each non-empty level subset μt is a (p-, q- and a-)ideal of X, respectively.

Theorem 2.5

[10, 11, 14]
  1. (1)

    Every fuzzy p-(resp., q-, a-) ideal of any BCI-algebra is a fuzzy ideal, but the converse are not true;

     
  2. (2)

    A fuzzy set μ of any BCI-algebra X is a fuzzy a-ideal of X if and only if it is both a fuzzy p-ideal and a fuzzy q-ideal.

     
A fuzzy set μ of X having the form
$$ \mu(y)=\left\{\begin{array}{ll} t &\hbox{ if } y=x, \\ 0 & \hbox{ otherwise,} \\ \end{array}\right. $$
is said to be a fuzzy point with supportxand valuet and is denoted by xt. A fuzzy point xt is said to belong to (resp., be quasi-coincident with) a fuzzy set μ, written as xt ∈ μ (resp., xtq μ) if μ(x) ≥ t (resp., μ(x) + t > 1). If xt ∈ μ or xtq μ, then we write xt ∈ ∨ q μ.

Definition 2.6

[9] A fuzzy set μ of X is called an (∈, ∈ ∨ q)-fuzzy ideal of X if for all tr ∈ (0, 1] and xy ∈ X,
  • (GF1) xt ∈ μ implies 0t ∈ ∨ q μ,

  • (GF2) (x ∗ y)t ∈ μ and yr ∈ μ imply xmin{t,r} ∈ ∨ q μ.

Lemma 2.7

[9] The conditions (GF1) and (GF2) in Definition 2.6, are equivalent to the following conditions, respectively:
  • (GF3) ∀x ∈ X,  μ(0) ≥ min{μ(x), 0.5},

  • (GF4) ∀xy ∈ X,  μ(x) ≥ min{μ(x ∗ y), μ(y), 0.5}.

Lemma 2.8

[9] Let μ be a fuzzy set of X. Then μt is an ideal of X for all 0.5 < t ≤ 1 if and only if it satisfies:
  • (GF5) ∀x ∈ X, max{μ(0), 0.5} ≥ μ(x),

  • (GF6) ∀xy ∈ X, max{μ(x), 0.5} ≥ min{μ(x ∗ y), μ(y)}.

Definition 2.9

[20]
  1. (1)
    An (∈, ∈ ∨ q)-fuzzy ideal μ of X is called an (∈, ∈ ∨ q)-fuzzy p-ideal of X if it satisfies:
    • (GF7) μ(x) ≥ min{μ((x ∗ z) ∗ (y ∗ z)), μ(y), 0.5}, for all xyz ∈ X.

     
  2. (2)
    An (∈, ∈ ∨ q)-fuzzy ideal of X is called an (∈, ∈ ∨ q)-fuzzy q-ideal of X if it satisfies:
    • (GF8) μ(x ∗ z) ≥ min{μ(x ∗ (y ∗ z)), μ(y), 0.5}, for all xyz ∈ X.

     
  3. (3)
    An (∈, ∈ ∨ q)-fuzzy ideal of X is called an (∈, ∈ ∨ q)-fuzzy a-ideal of X if it satisfies:
    • (GF9) μ(y ∗ x) ≥ min{μ((x ∗ z) ∗ (0 ∗ y)), μ(z), 0.5}, for all xyz ∈ X.

     

Theorem 2.10

[9, 20] A fuzzy set μ of X is an (∈, ∈ ∨ q)-fuzzy (p-, q- and a-) ideal of X if and only if each non-empty level subset μt is a (p-, q- and a-) ideal of X for all 0 < t ≤ 0.5, respectively.

Theorem 2.11

[20]
  1. (1)

    Every fuzzy (p-, q- and a-) ideal of any BCI-algebra is an (∈, ∈ ∨ q)-fuzzy (p-, q- and a-) ideal, but the converse are not true;

     
  2. (2)

    A fuzzy set μ of any BCI-algebra X is an (∈, ∈ ∨ q)-fuzzy a-ideal of X if and only if it is both an (∈, ∈ ∨ q)-fuzzy p-ideal and an (∈, ∈ ∨ q)-fuzzy q-ideal.

     

3 Some kinds of generalized fuzzy ideals

This section will be divided into four subsections. In Sect. 3.1, we describe the relationships among ordinary fuzzy filters, (∈, ∈ ∨ q)-fuzzy ideals and \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy ideals of BCI-algebras. In Sect. 3.2, we describe the relationships among ordinary fuzzy p-ideals, (∈, ∈ ∨ q)-fuzzy p-ideals and \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy p-ideals of BCI-algebras. In Sect. 3.3, we describe the relationships among ordinary fuzzy q-ideals, (∈, ∈ ∨ q)-fuzzy q-ideals and \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy q-ideals of BCI-algebras. Further, the relationships among ordinary fuzzy a-ideals, (∈, ∈ ∨ q)-fuzzy a-ideals and \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy a-ideals of BCI-algebras are considered in Sect. 3.4.

3.1 Generalized fuzzy ideals

Consider J = {t| t ∈ (0, 1] and μt is an empty set or an ideal of X}. In [9], Jun obtained the following results:
  1. (1)

    If J = (0, 1], then μ is an ordinary fuzzy ideal of X (Theorem 2.4);

     
  2. (2)

    If J = (0, 0.5], then μ is an (∈, ∈ ∨ q)-fuzzy ideal of X (Theorem 2.10).

     
Naturally, we consider the following questions:
  1. (1)

    If J = (0.5, 1], what kind of fuzzy ideals of X will be μ?

     
  2. (2)

    If J = (α, β], (α, β ∈ (0, 1]), whether μ will be a kind of fuzzy ideals of X or not?

     
  3. (3)

    Can we give a description for the relationship among the above generalized fuzzy ideals?

     

Definition 3.1.1

A fuzzy set μ of X is called an \((\overline{\in},\overline{\in} \vee \overline{q})\) -fuzzy ideal of X if for all tr ∈ (0, 1] and for all xy ∈ X,
  • (F1’) \( 0_t \overline{\in}\mu\) implies \(x_t \overline{\in} \vee \overline{q}\mu\),

  • (F2’) \( x_{\min\{t,r\}}\overline{\in}\mu\) implies \((x \ast y)_t \overline{\in} \vee \overline{q}\) or \(y_r \overline{\in} \vee \overline{q} \mu\) .

Example 3.1.2

Let X = {0, 1, 2, 3, 4, 5} be a proper BCI-algebra with Cayley table as follows:
https://static-content.springer.com/image/art%3A10.1007%2Fs00521-010-0376-6/MediaObjects/521_2010_376_Figa_HTML.gif

Define a fuzzy set μ of X by μ(0) = μ(1) = 0.6, μ(2) = 0.5, μ(3) = 0.4 and μ(4) = μ(5) = 0.2. It is now routine to verify that μ is an \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy ideal of X.

Theorem 3.1.3

A fuzzy set μ of X is an \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy ideal of X if and only if it satisfies (GF5) and (GF6)

Proof

(F1’)\(\Rightarrow\) (GF5) If there exists x ∈ X such that max{μ(0), 0.5} < μ(x) = t, then 0.5 < t ≤ 1,  μ(0) < t and x ∈ μt. Thus, \( 0_t \overline{\in}\ \mu\) . By (F1’), we have \(x_t\ \ \overline{\in} \vee \overline{q} \mu\), that is, μ(x) < t or μ(x) + t ≤ 1. Since μ(x) = t, then t ≤ 0.5, contradiction.

(GF5)\(\Rightarrow\) (F1’) Let \( 0_t\ \overline{\in}\ \mu\), then μ(0) < t.
  1. (a)

    If μ(0) ≥ μ(x), then μ(x) < t, and so \(x_t\ \ \overline{\in}\ \mu\). That is, \(x_t\ \ \overline{\in} \vee \overline{q}\ \mu\) .

     
  2. (b)

    If μ(0) < μ(x), by (GF5), 0.5 ≥ μ(x). (i) If μ(x) < t, then \(x_t\ \ \overline{\in}\ \mu\), and so \(x_t\ \ \overline{\in} \vee \overline{q} \mu\) . (ii) If μ(x) ≥ t, then t ≤ μ(x) ≤ 0.5, it follows that \(x_t\ \ \overline{q}\ \mu\), and thus \(x_{t}\ \ \overline{\in} \vee \overline{q} \mu\).

     

(F2’)\(\Rightarrow\) (GF6) If there exist xy ∈ X such that max{μ(x), 0.5} < t = min{μ(x ∗ y), μ(y)}, then 0.5 < t ≤ 1, \(x_t\ \overline{\in}\ \mu\) and (x ∗ y)t ∈ μ, yt  ∈ μ. By (F2’), we have \((x \ast y)_t\overline{q}\ \mu\) or \(y_t\ \ \overline{q}\ \mu\). Then (t ≤ μ(x ∗ y) and t + μ(x ∗ y) ≤ 1) or (t ≤ μ(y) and t + μ(y) ≤ 1). Thus, t ≤ 0.5, contradiction.

(GF6)\(\Rightarrow\) (F2’) Let \(x_{\min\{t,r\}}\overline{\in} \mu\), then μ(x) < min{tr}.
  1. (a)

    If μ(x) ≥ min{μ(x ∗ y), μ(y)}, then min{μ(x ∗ y), μ(y)} < min{tr}, and consequently, μ(x ∗ y) < t or μ(y) < r. It follows that \( (x \ast y)_t\overline{\in}\mu\) or \(y_r \overline{\in} \mu\). Thus, \( (x \ast y)_t\overline{\in}\vee \overline{q} \mu\) or \(y_r \overline{\in} \vee \overline{q} \mu\).

     
  2. (b)

    If μ(x) < min{μ(x ∗ y),  μ(y)}, then by (GF6), 0.5 ≥ min{μ(x ∗ y), μ(y)}. Putting (x ∗ y)t ∈ μ or yr  ∈ μ, then t ≤ μ(x ∗ y) ≤ 0.5 or r ≤ μ(y) ≤ 0.5. It follows that \((x \ast y)_t \overline{q} \mu\) or \( y_r \overline{q}\mu\), and thus \((x \ast y)_t\overline{\in} \vee \overline{q} \mu\) or \(y_r \overline{\in} \vee \overline{q} \mu\).

     

This complete the proof. \(\square\)

The following is a consequence of Theorem 3.1.3 and Lemma 2.8.

Theorem 3.1.4

A fuzzy set μ of X is an \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy ideal of X if and only if μt(≠ \(\emptyset\)) is an ideal of X for all 0.5 < t ≤ 1.

Corollary 3.1.5

Every fuzzy ideal of X is an \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy ideal, but the converse is not true.

Remark 3.1.6

Let μ be a fuzzy set of X and J = {t ∈ (0, 1] and μt is an empty set or an ideal of X}. In particular,
  1. (1)

    If J = (0, 1], then μ is an ordinary fuzzy ideal of X (Theorem 2.4);

     
  2. (2)

    If J = (0, 0.5], then μ is an (∈, ∈ ∨ q)-fuzzy ideal of X (Theorem 2.10);

     
  3. (3)

    If J = (0.5, 1], then μ is an \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy ideal of X (Theorem 3.1.4).

     

Definition 3.1.7

Let α, β ∈ [0, 1] and α <  β. Then a fuzzy set μ of X is called a fuzzy ideal with thresholds (α, β] of X if it satisfies,
  • (F1′′) ∀x ∈ X, max{μ(0), α} ≥ min{μ(x), β},

  • (F2′′) ∀xy ∈ X, max{μ(x), α} ≥ min{μ(x ∗ y), μ(y), β}.

We now characterize the fuzzy ideals with thresholds by using their level subsets.

Theorem 3.1.8

A fuzzy set μ of X is a fuzzy ideal with thresholds (α, β] of X if and only if μt(≠ \(\emptyset\)) is an ideal of X for all α < t ≤ β

Proof

It is similar to the proof of Theorem 3.1.4.□

Remark 3.1.9

  1. (1)
    By Definition 3.1.7, we have the following result: if μ is a fuzzy ideal with thresholds (α, β] of X, then we can conclude that
    1. (i)

      μ is an ordinary fuzzy ideal when α = 0, β = 1;

       
    2. (ii)

      μ is an (∈, ∈ ∨ q)-fuzzy ideal when α = 0, β = 0.5;

       
    3. (iii)

      μ is an \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy ideal when α = 0.5, β = 1.

       
     
  2. (2)

    By Definition 3.1.7, we can define other fuzzy ideals of X, same as the fuzzy ideal with thresholds (0.3, 0.9], with thresholds (0.4, 0.6] of X, etc.

     
  3. (3)

    However, the fuzzy ideal with thresholds of X may not be the usual fuzzy ideal, or may not be an (∈, ∈ ∨ q)-fuzzy ideal, or may not be an \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy ideal, respectively. These situations can be shown in the following example:

     

Example 3.1.10

Consider the BCI-algebra X as in Example 3.1.2. Define a fuzzy set μ of X by μ(0) = 0.6, μ(1) = μ(2) = 0.8, μ(3) = 0.4 and μ(4) = μ(5) = 0.2.

Then, we have
$$ \mu_t=\left\{ \begin{array}{ll} \{0,1,2,3,4,5\} & \hbox{ if } 0< t\le 0.2,\\ \{0,1,2,3\} & \hbox{ if } 0.2<t\le 0.4,\\ \{0,1,2\} & \hbox{ if } 0.4<t\le 0.6, \\ \{1,2\} & \hbox{ if } 0.6<t\le 0.8,\\ \emptyset & \hbox{ if } 0.8<t\le 1.\\ \end{array}\right. $$

Thus, μ is a fuzzy ideal with thresholds (0.4, 0.6] of X. But μ is not a fuzzy ideal, nor an (∈, ∈ ∨ q)-fuzzy ideal of X, nor is an \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy ideal of X.

For any fuzzy set μ of X and t ∈ (0, 1], we denote

Q(μ; t) = {x ∈ X|xtq μ} and [μ]t = {x ∈ X| xt ∈ ∨ q μ}. It is clear that [μ]t = μtQ(μ;t).

Theorem 3.1.11

A fuzzy set μ of X is an (∈, ∈ ∨ q)-fuzzy ideal of X if and only if [μ]t(≠ \(\emptyset\)) is an ideal of X for all t ∈ (0, 1].

Proof

Let μ be an (∈, ∈ ∨ q)-fuzzy ideal of X, then μ(0) ≥ min{μ(x), 0.5}, for all x ∈ [μ]t. Since x ∈ [μ]t, then xt ∈ ∨ q μ, that is, μ(x) ≥ t or μ(x) + t > 1.

Case 1: μ(x) ≥ t.
  1. (1)

    If t > 0.5, then μ(0) ≥ min{μ(x), 0.5} ≥ min{t, 0.5} = 0.5, and so μ(0) + t > 1, that is, 0tq μ.

     
  2. (2)

    If t ≤ 0.5, then μ(0) ≥ t, that is, 0t ∈ μ.

     
Case 2: μ(x) + t > 1.
  1. (1)

    If t > 0.5, then μ(0) ≥ min{μ(x), 0.5} > min{1 − t, 0.5} = 1 − t, that is, μ(0) + t > 1. Thus, 0tq μ.

     
  2. (2)

    If t ≤ 0.5, then μ(0) > min{1 − t, 0.5} = 0.5, and so 0t ∈ μ. Thus, in any case, 0 ∈ [μ]t.

     

Now, let x ∗ yy ∈ [μ]t for t ∈ (0, 1]. Then, (x ∗ y)t ∈ ∨ q μ or y ∈ ∨ q μ, that is, μ(x ∗ y) ≥ t or μ(x ∗ y) + t > 1, and μ(y) ≥ t or μ(y) + t > 1. Since μ is an (∈, ∈ ∨ q)-fuzzy ideal of X, we have μ(x) ≥ min{μ(x ∗ y), μ(y), 0.5}.

Case 1: μ(x ∗ y) ≥ t and μ(y) ≥ t.
  1. (1)

    If t > 0.5, then μ(x) ≥ min{μ(x ∗ y), μ(y), 0.5} ≥ min{t, 0.5} = 0.5, and thus, xtq μ.

     
  2. (2)

    If t ≤ 0.5, then μ(x) ≥ t, and thus, xt ∈ μ.

     
Case 2: μ(x ∗ y) ≥ t and μ(y) + t > 1.
  1. (1)

    If t > 0.5, then μ(x) ≥ min{μ(x ∗ y), μ(y), 0.5} ≥ min{t, μ(y), 0.5} = min{μ(y), 0.5} > min{1 − t, 0.5} = 1 − t, i.e., μ(x) + t > 1, and thus, xtq μ.

     
  2. (2)

    If t < 0.5, then μ(x) ≥ min{μ(x ∗ y), μ(y), 0.5} ≥ min{t, 1 − t, 0.5} = t, and so, xt ∈ μ.

     

Case 3: μ(x ∗ y) + t > 1 and μ(y) ≥ t.

The proof is similar to the case 2.

Case 4: μ(x ∗ y) + t > 1 and μ(y) + t > 1.
  1. (1)

    If t > 0.5, then μ(x) ≥ min{μ(x ∗ y), μ(y), 0.5} > min{1 − t, 1 − t, 0.5} = 1 − t, i.e., μ(x) + t > 1, and thus, xtq μ.

     
  2. (2)

    If t ≤ 0.5, then μ(x) ≥ min{μ(x ∗ y), μ(y), 0.5} > min{1 − t, 0.5} = 0.5 ≥ t, and so, xt ∈ μ.

     

Thus, in any case, we have xt ∈ ∨ q μ, and so, x ∈ [μ]t. Therefore, [μ]t is an ideal of X.

Conversely, let μ be a fuzzy set of X and t ∈ (0, 1] be such that [μ]t is an ideal of X. If μ(0) < t < min{μ(x), 0.5} for some t ∈ (0, 0.5), then μ(0) < t < 0.5. Since 0 ∈ [μ]t, then μ(0) ≥ t or μ(0) + t > 1, a contradiction. If μ(x) < t < min{μ(x ∗ y), μ(y), 0.5} for some t ∈ (0, 0.5), then μ(x ∗ y) ≥ t and μ(y) ≥ t, that is, \(x \ast y,y{\in \mu_t\subseteq}[\mu]_t\), which implies, x ∈ [μ]t. Hence, we have μ(x) ≥ t or μ(x) + t > 1, a contradiction. Therefore, μ is an (∈, ∈ ∨ q)-fuzzy ideal of X. \(\square\)

3.2 Generalized fuzzy p-ideals

Consider J = {t| t ∈ (0, 1] and μt is an empty set or a p-ideal of X}. In [20], we obtained the following results:
  1. (1)

    If J = (0, 1], then μ is an ordinary fuzzy p-ideal of X (Theorem 2.4);

     
  2. (2)

    If J = (0, 0.5], then μ is an (∈, ∈ ∨ q)-fuzzy p-ideal of X (Theorem 2.10).

     
Naturally, we consider the following questions:
  1. (i)

    If J = (0.5, 1], what kind of fuzzy p-ideals of X will be μ?

     
  2. (ii)

    If J = (α, β], (α, β ∈ (0, 1]), whether μ will be a kind of fuzzy p-ideals of X or not?

     
  3. (iii)

    Can we give a description for the relationship among the above generalized fuzzy p-ideals ?

     

Definition 3.2.1

An \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy ideal μ of X is called an \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy p-ideal of X if it satisfies:
  • (F3’) max{μ(x), 0.5} ≥ min{μ((x ∗ z) ∗ (y ∗ z)), μ(y)}, for all xyz ∈ X.

Example 3.2.2

Let X = {0, 1, 2, 3} be a proper BCI-algebra with Cayley table as follows:
https://static-content.springer.com/image/art%3A10.1007%2Fs00521-010-0376-6/MediaObjects/521_2010_376_Figb_HTML.gif

Define a fuzzy set μ of X by μ(0) = 0.8,  μ(1) = 0.6,  μ(2) = 0.2 and μ(3) = 0.3. It is now routine to verify that μ is an \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy p-ideal of X, but μ is neither a fuzzy p-ideal, nor an (∈, ∈ ∨ q)-fuzzy p-ideal of X.

Lemma 3.2.3 [20]

A fuzzy set μ of X is an \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy p-ideal of X if and only if it satisfies (GF5) and (F3’).

Theorem 3.2.4

A fuzzy set μ of X is an \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy p-ideal of X if and only if μt(≠ \(\emptyset\)) is a p-ideal of X for all 0.5 < t ≤ 1.

Proof

It is similar to the proof of Theorem 3.1.4. \(\square\)

Corollary 3.2.5

Every fuzzy p-ideal of X is an \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy p-ideal, but the converse is not true.

Remark 3.2.6

Let μ be a fuzzy set of X and J = {t ∈ (0, 1] and μt is an empty set or a p-ideal of X}. In particular,
  1. (1)

    If J = (0, 1], then μ is an ordinary fuzzy p-ideal of X (Theorem 2.4);

     
  2. (2)

    If J = (0, 0.5], then μ is an (∈, ∈ ∨ q)-fuzzy p-ideal of X (Theorem 2.10);

     
  3. (3)

    If J = (0.5, 1], then μ is an \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy p-ideal of X (Theorem 3.2.4).

     

Definition 3.2.7 [20]

Let α, β ∈ [0, 1] and α < β. Then a fuzzy set μ of X is called a fuzzy p-ideal with thresholds (α, β] of X if it satisfies (F1′′) and
  • (F3′′) \(\forall x,y,z\in X, \max\{\mu(x), \alpha \}\ge \min\{\mu((x \ast z) \ast (y \ast z)),\mu(y), \beta\}\).

We now characterize the fuzzy p-ideals with thresholds by using their level p-ideals.

Theorem 3.2.8 [20]

A fuzzy set μ of X is a fuzzy p-ideal with thresholds (α, β] of X if and only if μt(≠ \(\emptyset\)) is a p-ideal of X for all α < t ≤ β.

Remark 3.2.9

  1. (1)
    By Definition 3.2.7, we have the following result: if μ is a fuzzy p-ideal with thresholds (α, β] of X, then we can conclude that
    1. (i)

      μ is an ordinary fuzzy p-ideal when α = 0, β = 1;

       
    2. (ii)

      μ is an (∈, ∈ ∨ q)-fuzzy p-ideal when α = 0, β = 0.5;

       
    3. (iii)

      μ is an \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy p-ideal when α = 0.5, β = 1.

       
     
  2. (2)

    By Definition 3.2.7, we can define other fuzzy p-ideals of X, same as the fuzzy p-ideal with thresholds (0.3, 0.9], with thresholds (0.4, 0.6] of X, etc.

     
  3. (3)

    However, the fuzzy p-ideal with thresholds of X may not be the usual fuzzy p-ideal, or may not be an (∈, ∈ ∨ q)-fuzzy p-ideal, or may not be an \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy p-ideal, respectively. These situations can be shown in the following example:

     

Example 3.2.10

Consider the BCI-algebra X as in Example 3.2.2. Define a fuzzy set μ of X by μ(0) = 0.6,  μ(1) = 0.8,  μ(2) = 0.2 and μ(3) = 0.3.

Then, we have
$$ \mu_t=\left\{\begin{array}{ll} \{0,1,2,3\} & \hbox{ if } 0<t\le 0.2,\\ \{0,1,3\} & \hbox{ if } 0.2<t\le 0.3,\\ \{0,1\} & \hbox{ if } 0.3<t\le 0.6, \\ \{1\} & \hbox{ if } 0.6<t\le 0.8,\\ \emptyset & \hbox{ if } 0.8<t\le 1. \end{array}\right. $$

Thus, μ is a fuzzy p-ideal with thresholds (0.3, 0.6] of X. But μ is not a fuzzy p-ideal, not an (∈, ∈ ∨ q)-fuzzy p-ideal of X, nor is an \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy p-ideal of X.

Finally, we consider the characterization that [μ]t(≠ \(\emptyset\)) is a p-ideal of X for all t ∈ (0, 1].

Theorem 3.2.11

A fuzzy set μ of X is an (∈, ∈ ∨ q)-fuzzy p-ideal of X if and only if [μ]t(≠ \(\emptyset\)) is a p-ideal of X for all t ∈ (0, 1].

Proof

It is similar to the proof of Theorem 3.1.11. \(\square\)

3.3 Generalized fuzzy q-ideals

In this section, we introduce the concept of \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy q-ideals of BCI-algebras and investigate the relationships among these generalized fuzzy q-ideals.

Definition 3.3.1

An \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy ideal μ of X is called an \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy q-ideal of X if it satisfies:
  • (F4’) max{μ(x ∗ z), 0.5} ≥ min{μ(x ∗ (y ∗ z)), μ(y)}, for all xyz ∈ X.

Example 3.3.2

Let X = {0, 1, 2} be a proper BCI-algebra with Cayley table as follows:
https://static-content.springer.com/image/art%3A10.1007%2Fs00521-010-0376-6/MediaObjects/521_2010_376_Figc_HTML.gif

Define a fuzzy set μ of X by μ(0) = 0.8,  μ(1) = 0.2 and μ(2) = 0.5. It is now routine to verify that μ is an \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy q-ideal of X, but μ is neither a fuzzy q-ideal, nor an (∈, ∈ ∨ q)-fuzzy q-ideal of X.

Lemma 3.3.3

A fuzzy set μ of X is an \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy q-ideal of X if and only if it satisfies (GF5) and (F4’).

Proof

It is similar to the proof of Theorem 3.2.3. \(\square\)

Theorem 3.3.4

A fuzzy set μ of X is an \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy q-ideal of X if and only if μt(≠ \(\emptyset\)) is a q-ideal of X for all 0.5 < t ≤ 1.

Proof

It is similar to the proof of Theorem 3.1.4. \(\square\)

Corollary 3.3.5

Every fuzzy q-ideal of X is an \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy q-ideal, but the converse is not true.

Remark 3.3.6

Let μ be a fuzzy set of X and J = {t ∈ (0, 1] and μt is an empty set or a q-ideal of X}. In particular,
  1. (1)

    If J = (0, 1], then μ is an ordinary fuzzy q-ideal of X (Theorem 2.4);

     
  2. (2)

    If J = (0, 0.5], then μ is an (∈, ∈ ∨ q)-fuzzy q-ideal of X (Theorem 2.10);

     
  3. (3)

    If J = (0.5, 1], then μ is an \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy q-ideal of X (Theorem 3.3.4).

     

Definition 3.3.7

Let α, β ∈ [0, 1] and α < β. Then a fuzzy set μ of X is called a fuzzy q-ideal with thresholds (α, β] of X if it satisfies (F1′′) and
  • (F4′′) ∀xyz ∈ X, max{μ(x ∗ z), α} ≥ min{μ(x ∗ (y ∗ z)), μ(y), β}.

We now characterize the fuzzy q-ideals with thresholds by using their level q-ideals.

Theorem 3.3.8

A fuzzy set μ of X is a fuzzy q-ideal with thresholds (α, β] of X if and only if μt(≠ \(\emptyset\)) is a q-ideal of X for all α < t ≤ β.

Proof

It is similar to the proof of Theorem 3.2.8. \(\square\)

Remark 3.3.9

  1. (1)
    By Definition 3.3.7, we have the following result: if μ is a fuzzy q-ideal with thresholds (α, β] of X, then we can conclude that
    1. (1)

      μ is an ordinary fuzzy q-ideal when α = 0, β = 1;

       
    2. (2)

      μ is an (∈, ∈ ∨ q)-fuzzy q-ideal when α = 0, β = 0.5;

       
    3. (3)

      μ is an \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy q-ideal when α = 0.5, β = 1.

       
     
  2. (2)

    By Definition 3.3.7, we can define other fuzzy q-ideals of X, same as the fuzzy q-ideal with thresholds (0.3, 0.9], with thresholds (0.4, 0.6] of X, etc.

     
  3. (3)

    However, the fuzzy q-ideal with thresholds of X may not be the usual fuzzy q-ideal, or may not be an (∈, ∈ ∨ q)-fuzzy q-ideal, or may not be an \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy q-ideal, respectively. These situations can be shown in the following example:

     

Example 3.3.10

Let X = {0, 1, 2, 3, 4, 5} be a proper BCI-algebra with Cayley table as follows:
https://static-content.springer.com/image/art%3A10.1007%2Fs00521-010-0376-6/MediaObjects/521_2010_376_Figd_HTML.gif

Define a fuzzy set μ of X by μ(0) = 0.6,  μ(1) = μ(2) = μ(3) = 0.8,  μ(4) = 0.3 and μ(5) = 0.2.

Then, we have
$$ \mu_t=\left\{ \begin{array}{ll} \{0,1,2,3,4,5\} & \hbox{ if } 0<t\le 0.2,\cr \{0,1,2,3,4\} & \hbox{ if } 0.2<t\le 0.3,\cr \{0,1,2,3\} & \hbox{ if } 0.3<t\le 0.6, \cr \{1,2,3\} & \hbox{ if } 0.6<t\le 0.8,\cr \emptyset & \hbox{ if } 0.8<t\le 1. \end{array} \right. $$

Thus, μ is a fuzzy q-ideal with thresholds (0.3, 0.6] of X. But μ is not a fuzzy q-ideal, not an (∈, ∈ ∨ q)-fuzzy q-ideal of X, nor is an \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy q-ideal of X.

Finally, we consider the characterization that [μ]t(≠ \(\emptyset\)) is a q-ideal of X for all t ∈ (0, 1].

Theorem 3.3.11

A fuzzy set μ of X is an (∈, ∈ ∨ q)-fuzzy q-ideal of X if and only if [μ]t(≠ \(\emptyset\)) is a q-ideal of X for all t ∈ (0, 1].

Proof

It is similar to the proof of Theorem 3.1.11. \(\square\)

3.4 Generalized fuzzy a-ideals

In this section, we introduce the concept of \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy a-ideals of BCI-algebras and investigate some related properties.

Finally, we prove that a fuzzy set μ of a BCI-algebra X is an \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy a-ideal of X if and only if it is both an \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy p-ideal and an \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy q-ideal.

Definition 3.4.1

An \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy ideal μ of X is called an \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy a-ideal of X if it satisfies:
  • (F5’) max{μ(y ∗ x), 0.5} ≥ min{μ((x ∗ z) ∗ (0 ∗ y)), μ(z)}, for all xyz ∈ X.

Example 3.4.2

Consider the BCI-algebra X as in Example 3.2.2. Define a fuzzy set μ of X by μ(0) = μ(1) = 0.8,  μ(2) = 0.5 and μ(3) = 0.2. It is now routine to verify that μ is an \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy a-ideal of X, but μ is neither a fuzzy a-ideal, nor an (∈, ∈ ∨ q)-fuzzy a-ideal of X.

Lemma 3.4.3

A fuzzy set μ of X is an \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy a-ideal of X if and only if it satisfies (GF5) and (F5’).

Proof

It is similar to the proof of Theorem 3.2.3. \(\square\)

Theorem 3.4.4

A fuzzy set μ of X is an \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy a-ideal of X if and only if μt(≠ \(\emptyset\)) is an a-ideal of X for all 0.5 < t ≤ 1.

Proof

It is similar to the proof of Theorem 3.1.4. \(\square\)

Corollary 3.4.5

Every fuzzy a-ideal of X is an \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy a-ideal, but the converse is not true.

Remark 3.4.6

Let μ be a fuzzy set of X and J = {t ∈ (0, 1] and μt is an empty set or an a-ideal of X}. In particular,
  1. (1)

    If J = (0, 1], then μ is an ordinary fuzzy a-ideal of X (Theorem 2.4);

     
  2. (2)

    If J = (0, 0.5], then μ is an (∈, ∈ ∨ q)-fuzzy a-ideal of X (Theorem 2.10);

     
  3. (3)

    If J = (0.5, 1], then μ is an \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy a-ideal of X (Theorem 3.4.4).

     

Definition 3.4.7

Let α, β ∈ [0, 1] and α < β. Then a fuzzy set μ of X is called a fuzzy a-ideal with thresholds (α, β] of X if it satisfies (F1”) and
  • (F5′′) ∀xyz ∈ X, max{μ(y ∗ x), α} ≥ min{μ((x ∗ z) ∗ (0 ∗ y)), μ(z), β}.

We now characterize the fuzzy a-ideals with thresholds by using their level a-ideals.

Theorem 3.4.8

A fuzzy set μ of X is a fuzzy a-ideal with thresholds (α, β] of X if and only if μt(≠ \(\emptyset\)) is an a-ideal of X for all α < t ≤ β.

Proof

It is similar to the proof of Theorem 3.2.8. \(\square\)

Remark 3.4.9

  1. (1)
    By Definition 3.4.7, we have the following result: if μ is a fuzzy a-ideal with thresholds (α, β] of X, then we can conclude that
    1. (i)

      μ is an ordinary fuzzy a-ideal when α = 0, β = 1;

       
    2. (ii)

      μ is an (∈, ∈ ∨ q)-fuzzy a-ideal when α = 0, β = 0.5;

       
    3. (iii)

      μ is an \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy a-ideal when α = 0.5, β = 1.

       
     
  2. (2)

    By Definition 3.4.7, we can define other fuzzy a-ideals of X, same as the fuzzy a-ideal with thresholds (0.3, 0.9], with thresholds (0.4, 0.6] of X, etc.

     
  3. (3)

    However, the fuzzy a-ideal with thresholds of X may not be the usual fuzzy a-ideal, or may not be an (∈, ∈ ∨ q)-fuzzy a-ideal, or may not be an \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy a-ideal, respectively. These situations can be shown in the following example:

     

Example 3.4.10

Consider the BCI-algebra X as in Example 3.2.2. Define a fuzzy set μ of X by μ(0) = 0.6,  μ(1) = 0.8,  μ(2) = 0.2 and μ(3) = 0.3.

Then, we have
$$ \mu_t=\left\{ \begin{array}{ll} \{0,1,2,3\} & \hbox{ if } 0<t\le 0.2,\cr \{0,1,3\} & \hbox{ if } 0.2<t\le 0.3,\cr \{0,1\} & \hbox{ if } 0.3<t\le 0.6, \cr \{1\} & \hbox{ if } 0.6<t\le 0.8,\cr \emptyset & \hbox{ if } 0.8<t\le 1. \end{array}\right. $$

Thus, μ is a fuzzy a-ideal with thresholds (0.3, 0.6] of X. But μ is not a fuzzy a-ideal, not an (∈, ∈ ∨ q)-fuzzy a-ideal of X, nor is an \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy a-ideal of X.

Next, we consider the characterization that [μ]t(≠ \(\emptyset\)) is an a-ideal of X for all t ∈ (0, 1].

Theorem 3.4.11

A fuzzy set μ of X is an ( ∈ ,  ∈ ∨q)-fuzzy a-ideal of X if and only if [μ]t(≠ \(\emptyset\)) is an a-ideal of X for all t ∈ (0, 1].

Proof

It is similar to the proof of Theorem 3.1.11. \(\square\)

Finally, we prove that a fuzzy set μ of a BCI-algebra X is an \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy a-ideal of X if and only if it is both an \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy p-ideal and an \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy q-ideal.

The following is a consequence of Theorem 2.2.

Lemma 3.4.12

A non-empty subset μt of X is an a-ideal of X if and only if it is both a p-ideal and q-ideal for all t ∈ (0.5, 1].

Theorem 3.4.13

A fuzzy set μ of X is an \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy a-ideal of X if and only if it is both an \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy p-ideal and an \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy q-ideal.

Proof

Let μ be an \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy a-ideal of X. By Theorem 3.4.4, we know non-empty subset μt is an a-ideal of X for all t ∈ (0.5, 1]. By Lemma 3.4.12, μt is both a p-ideal and a q-ideal of X for all t ∈ (0.5, 1]. It follows from Theorem 3.2.4 and 3.3.4 that μ is both an \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy p-ideal and an \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy q-ideal of X.

Conversely, assume that μ is both an \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy p-ideal and an \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy q-ideal of X. By Theorem 3.2.4 and 3.3.4, we know non-empty subset μt is both a p-ideal and a q-ideal of X for all t ∈ (0.5, 1]. By Lemma 3.4.12, μt is an a-ideal of X for all t ∈ (0.5, 1]. It follows from Theorem 3.4.4 that μ is an \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy a-ideal of X. \(\square\)

4 Characterizations of three particular cases of BCI-algebras by these generalized fuzzy ideals

In this section, we give some characterizations of three particular cases of BCI-algebras by these generalized fuzzy ideals.

Definition 4.1

[13, 22]
  1. (1)

    A BCI-algebra X is called p-semisimple if it satisfies 0 ∗ (0 ∗ x) = x, for all x ∈ X;

     
  2. (2)

    A BCI-algebra X is called quasi-associative if it satisfies (x ∗ y) ∗ zx ∗ (y ∗ z), for all xyz ∈ X;

     
  3. (3)

    A BCI-algebra X is called associative if it satisfies (x ∗ y) ∗ z = x ∗ (y ∗ z), for all xyz ∈ X.

     

Theorem 4.2

[13, 22].
  1. (1)

    A BCI-algebra X is p-semisimple if and only if every ideal of X is a p-ideal.

     
  2. (2)

    A BCI-algebra X is quasi-associative if and only if every ideal of X is a q-ideal.

     
  3. (3)

    A BCI-algebra X is associative if and only if every ideal of X is an a-ideal.

     
The following theorem shows the connection between above three particular cases of BCI-algebras.

Theorem 4.3

[13]. A BCI-algebra is associative if and only if it is both quasi-associative and p-semisimple.

In [14], Liu et al. give some characterizations of three particular cases of BCI-algebras by these fuzzy p-(resp., q, a-)ideals.

Theorem 4.4

  1. (1)

    A BCI-algebra X is p-semisimple if and only if every fuzzy ideal of X is a fuzzy p-ideal.

     
  2. (2)

    A BCI-algebra X is quasi-associative if and only if every fuzzy ideal of X is a fuzzy q-ideal.

     
  3. (3)

    A BCI-algebra X is associative if and only if every fuzzy ideal of X is a fuzzy a-ideal.

     

Next, we give some characterizations of three particular cases of BCI-algebras by (∈, ∈ ∨ q)-fuzzy p- (resp., q-, a-) ideals.

Theorem 4.5

  1. (1)

    A BCI-algebra X is p-semisimple if and only if every fuzzy ideal of X is an (∈, ∈ ∨ q)-fuzzy p-ideal.

     
  2. (2)

    A BCI-algebra X is quasi-associative if and only if every fuzzy ideal of X is an (∈, ∈ ∨ q)-fuzzy q-ideal.

     
  3. (3)

    A BCI-algebra X is associative if and only if every fuzzy ideal of X is an (∈, ∈ ∨ q)-fuzzy a-ideal.

     

Proof

We only prove (i) and the proofs of (ii) and (iii) are similar. Let X be a p-semisimple BCI-algebra. Then, by Theorem 4.4(i) and 2.11(i), we know that every fuzzy ideal of X is an (∈, ∈ ∨ q)-fuzzy p-ideal.

Conversely, assume that every fuzzy ideal of X is an (∈, ∈ ∨ q)-fuzzy p-ideal. Let I be an ideal of X. Define a fuzzy set μ of X by
$$ \mu(x)=\left\{ \begin{array}{ll} 0.5 & \hbox{ if } x\in I,\\ 0 & \hbox{ otherwise.} \end{array}\right. $$

It is easy to check that μ is a fuzzy ideal of X. By hypothesis, μ is an (∈, ∈ ∨ q)-fuzzy p-ideal of X. By Theorem 2.10, I = μ0.5 is a p-ideal of X. This shows that every ideal of X is a p-ideal. It follows from Theorem 4.2(i) that X is p-semisimple. \(\square\)

Similarly, we can give some characterizations of three particular cases of BCI-algebras by \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy p- (resp., q-, a-) ideals.

Theorem 4.5

  1. (1)

    A BCI-algebra X is p-semisimple if and only if every fuzzy ideal of X is an \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy p-ideal.

     
  2. (2)

    A BCI-algebra X is quasi-associative if and only if every fuzzy ideal of X is an \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy q-ideal.

     
  3. (3)

    A BCI-algebra X is associative if and only if every fuzzy ideal of X is an \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy a-ideal.

     

Summarizing Theorem 2.11, 3.4.13, 4.2, 4.3. 4.4, 4.5 and 4.6, we have:

Corollary 4.7

For any BCI-algebra X, the following are equivalent:
  1. (1)

    X is p-semisimple;

     
  2. (2)

    Every ideal of X is a p-ideal;

     
  3. (3)

    Every fuzzy ideal of X is a fuzzy p-ideal;

     
  4. (4)

    Every fuzzy ideal of X is an (∈, ∈ ∨ q)-fuzzy p-ideal;

     
  5. (5)

    Every fuzzy ideal of X is an \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy p-ideal.

     

Corollary 4.8

For any BCI-algebra X, the following are equivalent:
  1. (1)

    X is quasi-associative;

     
  2. (2)

    Every ideal of X is a q-ideal;

     
  3. (3)

    Every fuzzy ideal of X is a fuzzy q-ideal;

     
  4. (4)

    Every fuzzy ideal of X is an (∈, ∈ ∨ q)-fuzzy q-ideal;

     
  5. (5)

    Every fuzzy ideal of X is an \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy q-ideal.

     

Corollary 4.9

For any BCI-algebra X, the following are equivalent:
  1. (1)

    X is associative;

     
  2. (2)

    X is both quasi-associative and p-semisimple;

     
  3. (3)

    Every ideal of X is an a-ideal;

     
  4. (4)

    Every closed ideal of X is both a p-ideal and a q-ideal;

     
  5. (5)

    Every fuzzy ideal of X is a fuzzy a-ideal;

     
  6. (6)

    Every fuzzy ideal of X is both a fuzzy p-ideal and a fuzzy q-ideal;

     
  7. (7)

    Every fuzzy ideal of X is an (∈, ∈ ∨ q)-fuzzy a-ideal;

     
  8. (8)

    Every fuzzy ideal of X is both an ( ∈ ,  ∈ ∨q)-fuzzy p-ideal and an (∈, ∈ ∨ q)-fuzzy q-ideal;

     
  9. (9)

    Every fuzzy ideal of X is an \((\overline{\in},\overline{\in} {\vee}\overline{q})\)-fuzzy a-ideal;

     
  10. (10)

    Every fuzzy ideal of X is both an \((\overline{\in},\overline{\in} \vee \overline{q})\)-fuzzy p-ideal and an (∈, ∈ ∨ q)-fuzzy q-ideal.

     

5 Conclusions

To investigate the structure of an algebraic system, it is clear that (fuzzy) ideals with special properties play an important role. In [20], we introduced the concepts of (∈, ∈ ∨ q)-fuzzy (p-, q- and a-) ideals of BCI-algebras and investigated some of their related properties. As a continuation of [20], we further discuss this topic in the present paper. We describe the relationships among ordinary fuzzy (p-, q-, a-) ideals, (∈, ∈ ∨ q)-fuzzy (p-, q-, a-) ideals and \((\overline{\in}, \overline{\in} {\vee} \overline{q})\)-fuzzy (p-, q-, a-) ideals of BCI-algebras. Finally, we give some characterizations of three particular cases of BCI-algebras by these generalized fuzzy ideals. It is our hope that this work would offer foundations for further study of the theory of BCK/BCI-algebras.

In our future study of fuzzy structure of BCI-algebras, may be the following topics should be considered:
  1. (1)

    To establish a fuzzy spectrum of BCI-algebras;

     
  2. (2)

    To consider the structure of quotient BCI-algebras by using these generalized fuzzy ideals;

     
  3. (3)

    To describe the fuzzy soft BCI-algebras and its applications.

     

Acknowledgements

This research is partially supported by a grant of the National Natural Science Foundation of China (60875034); a grant of the Natural Science Foundation of Education Committee of Hubei Province, China (D20092901) and also the support of the Natural Science Foundation of Hubei Province, China (2009CDB340).

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