1 Introduction

In decision making, Zadeh was the first to initiate the concept known as fuzzy set [51] in which the set is characterized by assigning each element in the set to a membership grade that ranges between 0 and 1. Fuzzy set concept was extended to intuitionistic fuzzy set [9], interval valued fuzzy set [52], fuzzy multiset [50], bipolar fuzzy set [53] and m-polar fuzzy set [11] (see [29, 47, 48] for other related notions). Moreover, a novel technique based on m-polar fuzzy set was introduced in [4].

The theory of m-polar fuzzy set and interval valued fuzzy set was proposed to be used in cognitive modeling, particularly in multi-agent decision making, i.e. if there are multi-attributes or more than one attribution with further bifurcated. The elements in both m-polar fuzzy set and interval valued fuzzy set consist of m components but in the first each component is a real number in [0, 1] and in the latter each component is a subset interval of [0, 1].

The m-polar fuzzy set and interval valued fuzzy set theories were studied on different algebraic structures as lie subalgebras, groups, lie ideals of lie algebras, subgroups and rings, ideals and bi-ideals of semigroup, quasi-ideals in semigroups and matroids (see for example [2, 3, 12, 13, 20, 46, 49]) and from a different point of view see [1, 27].

Moreover, the algebraic structure of BCK/BCI-algebras [14, 15] were extensively investigated and a lot of theories has been studied on the theory of BCK/BCI-algebras such as ideal theory (see the references [17,18,19, 21, 23, 28]). The results of BCK/BCI-algebras were developed to m-polar fuzzy set and interval valued fuzzy set frames (see [16, 25, 26] and more recent, [5,6,7,8, 10, 30]). Also, in [22, 24, 30,31,32,33,34], some more general ideas on bipolar fuzzy were studied. In BCK/BCI-algebras and other related algebraic structures, different kinds of related concepts were investigated in various ways (see for example, [35,36,37,38,39,40,41,42,43,44,45]).

In this paper, results are organized sectionwise as follows: in Sect. 3, first we define the notion of IVmPF p-ideals in BCI-algebra. Then the inter-relations between IVmPF p-ideals and IVmPF ideals are considered. In Sect. 4, IVmPF q-ideals in BCI-algebra is defined and characterized and also the inter-relations between IVmPF q-ideals and IVmPF ideals are investigated. In Sect. 5, we define the concept of IVmPF a-ideals in BCI-algebra and relations between IVmPF a-ideals with IVmPF ideals, IVmPF subalgebras and IVmPF p(q)-ideals are investigated. In Sect. 6, correspondence between (fuzzy) p(q and a)-ideals of BCI-algebra and IVmPF p(q and a)-ideals of BCI-algebra is given.

2 Preliminaries

In this section, we recall some basic definitions and notions that will be used throughout this paper.

An algebra \(Z = (Z; *, 0)\) of type (2, 0) is a BCI-algebra if for all \(\upsilon , \kappa , \tau \in Z\),

\((K_1)\):

\(((\upsilon *\kappa ) *(\upsilon *\tau )) *(\tau *\kappa ) = 0\),

\((K_2)\):

\((\upsilon *(\upsilon *\kappa )) *\kappa = 0\),

\((K_3)\):

\(\upsilon *\upsilon = 0\),

\((K_4\)):

\(\upsilon *\kappa = 0\) and \(\kappa *\upsilon = 0 \Rightarrow \upsilon = \kappa\).

Any BCI-algebra Z satisfies:

  • \((P_1) \ \upsilon *0 = \upsilon\),

  • \((P_2) \ (\upsilon *\kappa ) *\tau = (\upsilon *\tau ) *\kappa\),

  • \((P_3) \ \upsilon \le \kappa \Rightarrow \upsilon *\tau \le \kappa *\tau\) and \(\tau *\kappa \le \tau *\upsilon\),

  • \((P_4) \ 0 *(\upsilon *\kappa ) = (0 *\upsilon ) *(0 *\kappa )\),

  • \((P_5) \ 0 *(0 *(\upsilon *\kappa )) = 0 *(\kappa *\upsilon )\),

  • \((P_6) \ (\upsilon *\tau ) *(\kappa *\tau ) \le (\upsilon *\kappa )\),

  • \((P_7) \ \upsilon *(\upsilon *(\upsilon *\kappa )) = \upsilon *\kappa\),

  • \((P_8) \ 0 *(0 *((\upsilon *\tau ) *(\kappa *\tau ))) = (0 *\kappa ) *(0 *\upsilon )\),

  • \((P_9) \ 0 *(0 *(\upsilon *\kappa )) = (0 *\kappa ) *(0 *\upsilon )\),

where \(\upsilon \le \kappa \Leftrightarrow \upsilon *\kappa = 0\). Note that \((Z,\le )\) is a partially ordered set.

A subset \((\emptyset \ne ) A\) of Z is called a subalgebra if for all \(\upsilon ,\kappa \in Z\), \(\upsilon *\kappa \in A\) and is called an ideal of Z if \(0 \in A\) and for all \(\upsilon , \kappa \in Z, \upsilon *\kappa \in A, \kappa \in A\) implies \(\upsilon \in A\). Further, A is called p-ideal (resp. q-ideal and a-ideal) of Z if \(0 \in A\) and for all \(\upsilon , \kappa , \tau \in Z, ((\upsilon *\tau ) *(\kappa *\tau )) \in A, \kappa \in A\) implies \(\upsilon \in A\) (resp. \(\upsilon *(\kappa *\tau ) \in A, \kappa \in A\) implies \(\upsilon *\tau \in A\) and \(((\upsilon *\tau ) *(0 *\kappa )) \in A, \tau \in A\) implies \(\kappa *\upsilon \in A\)).

A mapping \(\mu ~:~Z\rightarrow [0,1]\) is called a fuzzy set of Z. If \(\mu (\upsilon *\kappa ) \ge \mu (\upsilon ) \wedge \mu ( \kappa )\) for all \(\upsilon ,\kappa \in Z\) then \(\mu\) is called a fuzzy subalgebra. If \(\mu (0) \ge \mu (\upsilon )\) and \(\mu (\upsilon ) \ge \mu (\upsilon *\kappa )\wedge \mu ( \kappa )\) for all \(\upsilon , \kappa \in Z\) then \(\mu\) is called a fuzzy ideal. Moreover, if \(\mu (0) \ge \mu (\upsilon )\) and \(\mu (\upsilon ) \ge \mu ((\upsilon *\tau ) *(\kappa *\tau )) \wedge \mu (\kappa )\) (resp. \(\mu (\upsilon *\tau ) \ge \mu (\upsilon *(\kappa *\tau )) \wedge \mu (\kappa )\) and \(\mu (\kappa *\upsilon ) \ge \mu ((\upsilon *\tau ) *(0 *\kappa )) \wedge \mu (\tau ))\) for all \(\upsilon , \kappa , \tau \in Z\) then \(\mu\) is called a fuzzy p-ideal (resp. fuzzy q-ideal and fuzzy a-ideal) of Z.

The interval number \({\hat{t}}\) is the interval \([t^-, t^+]\), where \(0\le t^-\le t^+\le 1\), D[0, 1] is the set of all interval numbers. For the interval numbers \({\hat{t}}_i=[t_{i}^-, t_{i}^+]\), \({\hat{d}}_i=[d_{i}^-, d_{i}^+]\in D[0,1], i\in I\), we describe:

  1. (a)

    \({\hat{t}}_i \wedge {\hat{d}}_i=[t_{i}^- \wedge d_{i}^-, t_{i}^+ \wedge d_{i}^+]\);

  2. (b)

    \({\hat{t}}_1\le {\hat{t}}_2 \Leftrightarrow t_{1}^-\le t_{2}^-\) and \(t_{1}^+\le t_{2}^+\);

  3. (c)

    \({\hat{t}}_1 = {\hat{t}}_2 \Leftrightarrow t_{1}^- = t_{2}^-\) and \(t_{1}^+ = t_{2}^+\).

A mapping \(\widehat{\mathcal {G}}~:~Z\rightarrow D[0,1]\) is called an interval valued fuzzy set of Z, where \(\widehat{\mathcal {G}}(\upsilon )=[{\mathcal {G}}^{-}(\upsilon ), {\mathcal {G}}^{+}(\upsilon )]\) for all \(\upsilon \in Z\), \({\mathcal {G}}^{-}\) and \({\mathcal {G}}^+\) are fuzzy sets of Z with \({\mathcal {G}}^{-}(\upsilon )\le {\mathcal {G}}^{+}(\upsilon )\) for all \(\upsilon \in Z\).

Definition 2.1

[30] A mapping \(\widehat{\mathcal {G}}:Z\rightarrow D[0,1]^m\) is called an interval valued m-polar fuzzy set (briefly, IVmPF set) of Z and is defined as:

$$\begin{aligned} \widehat{\mathcal {G}}(\upsilon )=(\pi _1\circ \widehat{\mathcal {G}}(\upsilon ), \pi _2\circ \widehat{\mathcal {G}}(\upsilon ), \ldots , \pi _m\circ \widehat{\mathcal {G}}(\upsilon )) \end{aligned}$$

where \(\pi _i:D[0,1]^m \rightarrow D[0,1]\) is the ith projection mapping for \(i \in \{1,2,\ldots ,m\}\).

That is,

$$\begin{aligned}&\widehat{\mathcal {G}}(\upsilon )\\&\quad =([\mathcal {G}_1^{-}(\upsilon ), \mathcal {G}_1^{+}(\upsilon )],[\mathcal {G}_2^{-}(\upsilon ), \mathcal {G}_2^{+}(\upsilon )], \ldots , [\mathcal {G}_m^{-}(\upsilon ), \mathcal {G}_m^{+}(\upsilon )]) \end{aligned}$$

for all \(\upsilon \in Z\), \(\mathcal {G}_i^{-}\) and \(\mathcal {G}_i^+\) are fuzzy sets of Z with \(\mathcal {G}_i^{-}(\upsilon )\le \mathcal {G}_i^{+}(\upsilon )\) for all \(\upsilon \in Z\) and \(i \in \{1,2,\ldots ,m\}\).

The \(i{{\rm th}}\) projection map \(\pi _i\) is order preserving and vice versa i.e.,

$$\begin{aligned} \upsilon \le \kappa \Leftrightarrow \pi _i (\upsilon ) \le \pi _i (\kappa ) ~~~\forall ~i \in \{1,2,\ldots ,m\}. \end{aligned}$$

Definition 2.2

[30] An IVmPF set \(\widehat{\mathcal {G}}\) is said to be an IVmPF subalgebra if:

$$\begin{aligned} (~\forall ~ \upsilon ,\kappa \in Z)~ \widehat{\mathcal {G}}(\upsilon *\kappa ) \ge \widehat{\mathcal {G}}(\upsilon ) \wedge \widehat{\mathcal {G}}( \kappa ), \end{aligned}$$

that is,

$$\begin{aligned} \big (~\forall ~ \upsilon ,\kappa \in Z,~i\in & {} \{1,2,\ldots ,m\}\big )~ \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon *\kappa )\\\ge &\, {} \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon )\wedge \widehat{\pi _i} \circ \widehat{\mathcal {G}}( \kappa ). \end{aligned}$$

Definition 2.3

[30] An IVmPF set \(\widehat{\mathcal {G}}\) is said to be an IVmPF ideal if:

  1. (1)

    \(~(\forall ~ \upsilon \in Z)~\widehat{\mathcal {G}}(0) \ge \widehat{\mathcal {G}}(\upsilon )\),

  2. (2)

    \((\forall ~ \upsilon ,\kappa \in Z)~\widehat{\mathcal {G}}(\upsilon ) \ge \widehat{\mathcal {G}}(\upsilon *\kappa )\wedge \widehat{\mathcal {G}}( \kappa )\),

that is,

  1. (1)

    \(\widehat{\pi _i} \circ \widehat{\mathcal {G}}(0) \ge \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon )\),

  2. (2)

    \(\widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon ) \ge \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon *\kappa )\wedge \widehat{\pi _i} \circ \widehat{\mathcal {G}}( \kappa )\),

for all \(\upsilon ,\kappa \in Z\) and \(i \in \{1,2,\ldots ,m\}\).

Lemma 2.4

[30] Let \(\widehat{\mathcal {G}}\) be an IVmPF ideal of Z and \(\upsilon , \kappa \in Z\) such that \(\upsilon \le \kappa\). Then,

$$\begin{aligned} \widehat{\mathcal {G}}(\upsilon ) \ge \widehat{\mathcal {G}}(\kappa ). \end{aligned}$$

Lemma 2.5

[30] Let \(\widehat{\mathcal {G}}\) be an IVmPF ideal of Z and \(\upsilon , \kappa , \tau \in Z\) such that \(\upsilon *\kappa \le \tau\). Then,

$$\begin{aligned} \widehat{\mathcal {G}}(\upsilon ) \ge \widehat{\mathcal {G}}(\kappa )\wedge \widehat{\mathcal {G}}(\tau ). \end{aligned}$$

Throghout following sections, the BCI-algebra will be denoted by Z.

3 IVmPF p-ideal

The notion IVmPF p-ideals of BCI-algebras is described in this section, and relationships are provided between the IVmPF ideals and IVmPF p-ideals.

Definition 3.1

An IVmPF set \(\widehat{\mathcal {G}}\) of Z is called an IVmPF p-ideal of Z if:

  1. (1)

    \(~(\forall ~ \upsilon \in Z)~\widehat{\mathcal {G}}(0) \ge \widehat{\mathcal {G}}(\upsilon )\),

  2. (2)

    \((\forall ~ \upsilon , \kappa , \tau \in Z)\) \(\widehat{\mathcal {G}}(\upsilon ) \ge \widehat{\mathcal {G}}((\upsilon *\tau ) *(\kappa *\tau )) \wedge \widehat{\mathcal {G}}(\kappa )\),

that is,

  1. (1)

    \(\widehat{\pi _i} \circ \widehat{\mathcal {G}}(0) \ge \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon )\),

  2. (2)

    \(\widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon ) \ge \widehat{\pi _i} \circ \widehat{\mathcal {G}}((\upsilon *\tau ) *(\kappa *\tau )) \wedge \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\kappa )\),

for all \(\upsilon , \kappa , \tau \in Z\) and \(i \in \{1,2,\ldots ,m\}\).

Example 1

Consider a BCI-algebra \(Z=\{0,\imath ,\jmath ,\ell \}\) with operation \((*)\) which is defined in Table 1.

Table 1 Cayley table of the binary operation*
Full size table

Define an IV3PF set \(\widehat{\mathcal {G}}\) on Z as:

$$\begin{aligned} \widehat{\mathcal {G}}(x) = \left\{ \begin{array}{ll} \big ([0.6,0.7],[0.5,0.8],[0.2,0.4]\big ) &{} \text{ if } x=0, \\ \big ([0.5,0.6],[0.5,0.5],[0.1,0.4]\big ) &{} \text{ if } x= \imath ,\\ \big ([0.3,0.4],[0.2,0.4],[0.1,0.2]\big ) &{} \text{ if } x \in \{\jmath , \ell \}. \end{array} \right. \end{aligned}$$

It is straightforward to show that \(\widehat{\mathcal {G}}\) is an IV3PF p-ideal of Z.

Theorem 3.2

Any IVmPF p-ideal of Z is an IVmPF ideal.

Proof

Suppose that \(\widehat{\mathcal {G}}\) is an IVmPF p-ideal of Z. Then for each \(\upsilon , \kappa , \tau \in Z\) and \(i \in \{1,2,\ldots ,m\}\), we have

$$\begin{aligned} \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon ) \ge \widehat{\pi _i} \circ \widehat{\mathcal {G}}((\upsilon *\tau ) *(\kappa *\tau )) \wedge \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\kappa ). \end{aligned}$$

Substitute \(\tau\) by 0 in above inequality, to get

$$\begin{aligned} \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon )\ge &\, {} \widehat{\pi _i} \circ \widehat{\mathcal {G}}((\upsilon *0) *(\kappa *0)) \wedge \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\kappa )\\= &\, {} \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon *\kappa ) \wedge \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\kappa ). \end{aligned}$$

Hence, \(\widehat{\mathcal {G}}\) is an IVmPF ideal. \(\square\)

Generally, the converse of Theorem 3.2 need not be true. This is illustrated by the example below.

Example 2

Consider \(Z=\{0,\imath ,\jmath ,\kappa ,\ell \}\) a BCI-algebra under the operation (\(*\)) which is defined by Table 2.

Table 2 Cayley table of the binary operation*
Full size table

Define an IV4PF set \(\widehat{\mathcal {G}}\) on Z as:

$$\begin{aligned} \widehat{\mathcal {G}}(x) = \left\{ \begin{array}{ll} \big ([0.6,0.7],[0.5,0.8],[0.4,0.6],[0.7,0.9]\big ) &{} \text{ if } x=0, \\ \big ([0.1,0.2],[0.2,0.3],[0.1,0.2],[0.1,0.2]\big ) &{} \text{ if } x \in \{\imath , \ell \},\\ \big ([0.3,0.6],[0.3,0.7],[0.2,0.4],[0.4,0.5]\big ) &{} \text{ if } x=\jmath ,\\ \big ([0.2,0.3],[0.3,0.4],[0.1,0.3],[0.2,0.4]\big ) &{} \text{ if } x=\kappa .\\ \end{array} \right. \end{aligned}$$

It is straightforward to check that \(\widehat{\mathcal {G}}\) is an IV4PF ideal of Z but not an IV4PF p-ideal because \(\big ([0.3,0.6],[0.3,0.7],[0.2,0.4],[0.4,0.5]\big )={\widehat{G}}(\jmath ) \ngeq {\widehat{G}}((\jmath *\ell )*(0 *\ell )) \wedge {\widehat{G}}( 0) = {\widehat{G}}( 0)=\big ([0.6,0.7],[0.5,0.8],[0.4,0.6],[0.7,0.9]\big )\).

Theorem 3.3

If \(\widehat{\mathcal {G}}\) is an IVmPF p-ideal of Z, then

  1. (a)

    \(\widehat{\mathcal {G}}(\upsilon ) \ge \widehat{\mathcal {G}}(0 *(0 *\upsilon ))\) \(\forall ~\upsilon \in Z\).

  2. (b)

    \(\widehat{\mathcal {G}}(\upsilon ) = \widehat{\mathcal {G}}(0 *(0 *\upsilon ))\) \(\forall ~\upsilon \in Z\).

Proof

(a):

. Let \(\widehat{\mathcal {G}}\) be an IVmPF p-ideal of Z. Then,

$$\begin{aligned} \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon ) \ge \widehat{\pi _i} \circ \widehat{\mathcal {G}}((\upsilon *\tau ) *(\kappa *\tau )) \wedge \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\kappa ) \end{aligned}$$

for all \(\upsilon , \kappa , \tau \in Z\) and \(i \in \{1,2,\ldots ,m\}\). Substitute \(\upsilon\) for \(\tau\) and 0 for \(\kappa\) to get

$$\begin{aligned} \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon )&\ge \widehat{\pi _i} \circ \widehat{\mathcal {G}}((\upsilon *\upsilon ) *(0 *\upsilon )) \wedge \widehat{\pi _i} \circ \widehat{\mathcal {G}}(0)\\&= \widehat{\pi _i} \circ \widehat{\mathcal {G}}(0*(0 *\upsilon )) \wedge \widehat{\pi _i} \circ \widehat{\mathcal {G}}(0)\\&= \widehat{\pi _i} \circ \widehat{\mathcal {G}}(0*(0 *\upsilon )). \end{aligned}$$
(b):

. Since \(0 *(0 *\upsilon ) \le \upsilon\) for any \(\upsilon \in Z\). Therefore, by Lemma 2.4 we have

$$\begin{aligned} \widehat{\pi _i} \circ \widehat{\mathcal {G}}(0 *(0 *\upsilon )) \ge \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon )~~~~~~(\forall ~ i \in \{1,2,\ldots ,m\}). \end{aligned}$$

From (a), \(\widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon ) \ge \widehat{\pi _i} \circ \widehat{\mathcal {G}}(0 *(0 *\upsilon ))\). Thus,

$$\begin{aligned} \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon ) = \widehat{\pi _i} \circ \widehat{\mathcal {G}}(0 *(0 *\upsilon )). \end{aligned}$$

\(\square\)

A condition for IVmPF ideal to be IVmPF p-ideal is provided in the following two results.

Theorem 3.4

Let \(\widehat{\mathcal {G}}\) be an IVmPF ideal of Z satisfying the condition:

$$\begin{aligned} (\forall ~ \upsilon , \kappa , \tau \in Z)~\widehat{\mathcal {G}}(\upsilon *\kappa ) \ge \widehat{\mathcal {G}}((\upsilon *\tau ) *(\kappa *\tau )). \end{aligned}$$
(1)

Then, \(\widehat{\mathcal {G}}\) is an IVmPF p-ideal of Z.

Proof

Suppose that \(\widehat{\mathcal {G}}\) is an IVmPF ideal of Z satisfying (1). Then,

$$\begin{aligned} \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon )&\ge \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon *\kappa ) \wedge \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\kappa )\\&\ge \widehat{\pi _i} \circ \widehat{\mathcal {G}}((\upsilon *\tau ) *(\kappa *\tau )) \wedge \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\kappa ), \end{aligned}$$

\(\forall ~\upsilon , \kappa , \tau \in Z\) and \(i \in \{1,2,\ldots ,m\}\). Hence, \(\widehat{\mathcal {G}}\) is an IVmPF p-ideal of Z. \(\square\)

Theorem 3.5

Let \(\widehat{\mathcal {G}}\) be an IVmPF ideal of Z satisfying the identity:

$$\begin{aligned} (\forall ~\upsilon \in Z)~ \widehat{\mathcal {G}}(\upsilon )\ge \widehat{\mathcal {G}}(0 *(0 *\upsilon )). \end{aligned}$$
(2)

Then, \(\widehat{\mathcal {G}}\) is an IVmPF p-ideal of Z.

Proof

Let \(\upsilon ,\kappa ,\tau \in Z\). Then for all \(i \in \{1,2,\ldots ,m\}\), we have

$$\begin{aligned}&\widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon *\kappa )\\&\quad \ge \widehat{\pi _i} \circ \widehat{\mathcal {G}}(0 *(0 *(\upsilon *\kappa ))) ~~~~~~~~~~~~~~~~~{\text{ by }~(2)}\\&\quad = \widehat{\pi _i} \circ \widehat{\mathcal {G}}((0 *\kappa ) *(0 *\upsilon )) ~~~~~~~~~~~~~~~~~{\text{ by }~(P_9)}\\&\quad = \widehat{\pi _i} \circ \widehat{\mathcal {G}}(0 *(0 *((\upsilon *\tau )*(\kappa *\tau )))) ~~~~~~{\text{ by }~(P_8)}\\&\quad = \widehat{\pi _i} \circ \widehat{\mathcal {G}}((\upsilon *\tau )*(\kappa *\tau )). ~~~~~~~~~~~~~~~~{\text{ by } \text{ Theorem }~3.3} \end{aligned}$$

Hence from Theorem 3.4, \(\widehat{\mathcal {G}}\) is an IVmPF p-ideal of Z. \(\square\)

4 IVmPF q-ideal

In this section, we define IVmPF q-ideals of BCI-algebras and associated properties of IVmPF ideals and IVmPF q-ideals are considered.

Definition 4.1

An IVmPF set \(\widehat{\mathcal {G}}\) of Z is called an IVmPF q-ideal of Z if:

  1. (1)

    \(~(\forall ~ \upsilon \in Z)~\widehat{\mathcal {G}}(0) \ge \widehat{\mathcal {G}}(\upsilon )\),

  2. (2)

    \(~(\forall ~ \upsilon , \kappa , \tau \in Z)~\widehat{\mathcal {G}}(\upsilon *\tau ) \ge \widehat{\mathcal {G}}(\upsilon *(\kappa *\tau )) \wedge \widehat{\mathcal {G}}(\kappa )\),

that is,

  1. (1)

    \(\widehat{\pi _i} \circ \widehat{\mathcal {G}}(0) \ge \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon )\),

  2. (2)

    \(\widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon *\tau ) \ge \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon *(\kappa *\tau )) \wedge \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\kappa ),\)

for all \(\upsilon , \kappa , \tau \in Z\) and \(i \in \{1,2,\ldots ,m\}\).

Example 3

Consider the BCI-algebra Z and the IV4PF set \(\widehat{\mathcal {G}}\) on Z defined in Example 2. It is straightforward to check that \(\widehat{\mathcal {G}}\) is an IV4PF q-ideal of Z.

Theorem 4.2

Every IVmPF q-ideal of Z is an IVmPF ideal.

Proof

Let \(\widehat{\mathcal {G}}\) be an IVmPF q-ideal of Z. Then for all \(\upsilon , \kappa , \tau \in Z\) and \(i \in \{1,2,\ldots ,m\}\), we have

$$\begin{aligned} \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon *\tau ) \ge \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon *(\kappa *\tau )) \wedge \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\kappa ). \end{aligned}$$

Substitute \(\tau\) by 0, to have

$$\begin{aligned} \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon *0) \ge \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon *(\kappa *0)) \wedge \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\kappa ). \end{aligned}$$

It follows that \(\widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon ) \ge \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon *\kappa ) \wedge \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\kappa )\), as required. \(\square\)

In general, the converse of Theorem 4.2 is not valid, as illustrated in the following example.

Example 4

Consider a BCI-algebra \(Z=\{0,\imath ,\jmath ,\kappa ,\ell \}\) with Table 3.

Table 3 Cayley table of the binary operation*
Full size table

Define an IV3PF set \(\widehat{\mathcal {G}}\) on Z as:

$$\begin{aligned} \widehat{\mathcal {G}}(x) = \left\{ \begin{array}{ll} \big ([0.6,0.7],[0.6,0.9],[0.3,0.5]\big ) &{} \text{ if } x=0, \\ \big ([0.4,0.5],[0.5,0.7],[0.3,0.4]\big ) &{} \text{ if } x=\imath ,\\ \big ([0.2,0.3],[0.3,0.4],[0.2,0.3]\big ) &{} \text{ if } x \in \{\jmath , \kappa , \ell \}. \end{array} \right. \end{aligned}$$

It is straightforward to check that \(\widehat{\mathcal {G}}\) is an IV3PF ideal of Z but not an IV3PF q-ideal because \(\big ([0.2,0.3],[0.3,0.4],[0.2,0.3]\big )= {\widehat{G}}(\ell *\jmath )\ngeq {\widehat{G}}( \ell *(0 *\jmath )) \wedge {\widehat{G}}( 0)= {\widehat{G}}( 0) =\big ([0.6,0.7],[0.6,0.9],[0.3,0.5]\big )\).

Theorem 4.3

If \(\widehat{\mathcal {G}}\) is an IVmPF ideal of Z, then the following statements are equivalent:

(1):

\(\widehat{\mathcal {G}}\) is an IVmPF q-ideal of Z.

(2):

\(\widehat{\mathcal {G}}(\upsilon *\kappa ) \ge \widehat{\mathcal {G}}(\upsilon *(0 *\kappa ))\) \(\forall ~\upsilon ,\kappa \in Z\).

(3):

\(\widehat{\mathcal {G}}((\upsilon *\kappa ) *\tau ) \ge \widehat{\mathcal {G}}(\upsilon *(\kappa *\tau ))\) \(\forall ~\upsilon ,\kappa ,\tau \in Z\).

Proof

(\(1) \Rightarrow (2\)). Let \(\widehat{\mathcal {G}}\) be an IVmPF q-ideal of Z. Then for all \(\upsilon , \kappa , \tau \in Z\) and \(i \in \{1,2,\ldots ,m\}\), we have

$$\begin{aligned} \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon *\tau ) \ge \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon *(\kappa *\tau )) \wedge \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\kappa ) . \end{aligned}$$

Replacing \(\tau\) by \(\kappa\) and \(\kappa\) by \(\tau\), we get

$$\begin{aligned} \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon *\kappa ) \ge \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon *(\tau *\kappa )) \wedge \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\tau ). \end{aligned}$$

Substitute \(\tau\) by 0,

$$\begin{aligned} \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon *\kappa )&\ge \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon *(0 *\kappa )) \wedge \widehat{\pi _i} \circ \widehat{\mathcal {G}}(0) \\&=\widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon *(0 *\kappa )). \end{aligned}$$

(\(2) \Rightarrow (3\)). Let \(\upsilon ,\kappa ,\tau \in Z\). Then,

$$\begin{aligned} \widehat{\pi _i} \circ \widehat{\mathcal {G}}((\upsilon *\kappa ) *\tau ) \ge \widehat{\pi _i} \circ \widehat{\mathcal {G}}((\upsilon *\kappa ) *(0*\tau )) ~~~~~~~(\forall ~i \in \{1,2,\ldots ,m\}). \end{aligned}$$
(3)

Now we have

$$\begin{aligned}&((\upsilon *\kappa ) *(0*\tau ))*(\upsilon *(\kappa *\tau ))\\&\quad = ((\upsilon *\kappa ) *(\upsilon *(\kappa *\tau )) )*(0*\tau )\\&\quad \le ((\kappa *\tau )*\kappa ) *(0 *\tau )\\&\quad = ((\kappa *\kappa )*\tau ) *(0 *\tau )\\&\quad = (0*\tau ) *(0 *\tau )\\&\quad = 0. \end{aligned}$$

By Lemma 2.5, we have

$$\begin{aligned} \widehat{\pi _i} \circ \widehat{\mathcal {G}}((\upsilon *\kappa ) *(0*\tau )) \ge \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon *(\kappa *\tau )). \end{aligned}$$
(4)

It follows from (3) and (4) that \(\widehat{\mathcal {G}}((\upsilon *\kappa ) *\tau ) \ge \widehat{\mathcal {G}}(\upsilon *(\kappa *\tau ))\).

(\(3) \Rightarrow (1\)). As \(\widehat{\mathcal {G}}\) is an IVmPF ideal of Z, so for all \(\upsilon , \kappa , \tau \in Z\) and \(i \in \{1,2,\ldots ,m\}\), we have

$$\begin{aligned} \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon *\tau )&\ge \widehat{\pi _i} \circ \widehat{\mathcal {G}}((\upsilon *\tau ) *\kappa ) \wedge \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\kappa )\\&\ge \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon *(\kappa *\tau )) \wedge \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\kappa ). \end{aligned}$$

\(\square\)

Lemma 4.4

Let \(\widehat{\mathcal {G}}\) be an IVmPF q-ideal of Z. Then \(\widehat{\mathcal {G}}(0 *\tau ) \ge \widehat{\mathcal {G}}(\tau )\) for all \(\tau \in Z\).

Proof

Suppose that \(\widehat{\mathcal {G}}\) is an IVmPF q-ideal of Z. Then for all \(\upsilon , \kappa , \tau \in Z\) and \(i \in \{1,2,\ldots ,m\}\), we have

$$\begin{aligned} \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon *\tau ) \ge \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon *(\kappa *\tau )) \wedge \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\kappa ). \end{aligned}$$

Substitute \(\upsilon\) by 0 and \(\kappa\) by \(\tau\), to have

$$\begin{aligned} \widehat{\pi _i} \circ \widehat{\mathcal {G}}(0 *\tau ) \ge \widehat{\pi _i} \circ \widehat{\mathcal {G}}(0 *(\tau *\tau )) \wedge \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\tau )= \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\tau ). \end{aligned}$$

\(\square\)

5 IVmPF a-ideal

In this section, the notion of IVmPF a-ideals of BCI-algebra is defined and associated properties are investigated. Inter related properties of IVmPF a-ideals with IVmPF q-ideals and IVmPF p-ideals are discussed.

Definition 5.1

An IVmPF set \(\widehat{\mathcal {G}}\) of Z is called an IVmPF a-ideal of Z if:

  1. (1)

    \(\widehat{\mathcal {G}}(0) \ge \widehat{\mathcal {G}}(\upsilon )\),

  2. (2)

    \(\widehat{\mathcal {G}}(\upsilon *\tau ) \ge \widehat{\mathcal {G}}((\tau *\kappa ) *(0 *\upsilon )) \wedge \widehat{\mathcal {G}}(\kappa )\),

that is,

  1. (1)

    \(\widehat{\pi _i} \circ \widehat{\mathcal {G}}(0) \ge \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon )\),

  2. (2)

    \(\widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon *\tau ) \ge \widehat{\pi _i} \circ \widehat{\mathcal {G}}((\tau *\kappa ) *(0 *\upsilon )) \wedge \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\kappa ),\)

for all \(\upsilon , \kappa , \tau \in Z\) and \(i \in \{1,2,\ldots ,m\}\).

Example 5

Consider the BCI-algebra Z and the IV3PF set \({\widehat{G}}\) on Z defined in Example 1. It is straightforward to verify that \({\widehat{G}}\) is an IV3PF a-ideal.

Theorem 5.2

Every IVmPF a-ideal of Z is both an IVmPF ideal and an IVmPF subalgebra.

Proof

Let \(\widehat{\mathcal {G}}\) be an IVmPF a-ideal of Z. Then for all \(\upsilon , \kappa , \tau \in Z\) and \(i \in \{1,2,\ldots ,m\}\), we have

$$\begin{aligned} \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon *\tau ) \ge \widehat{\pi _i} \circ \widehat{\mathcal {G}}((\tau *\kappa ) *(0 *\upsilon )) \wedge \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\kappa ). \end{aligned}$$
(5)

Substitute \(\kappa\) and \(\upsilon\) by 0, to get

$$\begin{aligned} \widehat{\pi _i} \circ \widehat{\mathcal {G}}(0 *\tau ) \ge \widehat{\pi _i} \circ \widehat{\mathcal {G}}((\tau *0) *(0 *0)) \wedge \widehat{\pi _i} \circ \widehat{\mathcal {G}}(0)= \widehat{\mathcal {G}}(\tau ). \end{aligned}$$
(6)

Again in (5), substitute \(\tau\) and \(\kappa\) by 0 and use (6), to get

$$\begin{aligned} \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon *0) \ge \widehat{\pi _i} \circ \widehat{\mathcal {G}}((0 *0) *(0 *\upsilon )) \wedge \widehat{\pi _i} \circ \widehat{\mathcal {G}}(0) \ge \widehat{\mathcal {G}}(0 *\upsilon ). \end{aligned}$$

As \(\upsilon *0=\upsilon\), Therefore,

$$\begin{aligned} \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon )&\ge \widehat{\pi _i} \circ \widehat{\mathcal {G}}(0 *\upsilon ) \\&\ge \widehat{\pi _i} \circ \widehat{\mathcal {G}}((\upsilon *\tau ) *(0 *0)) \wedge \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\tau )\\&= \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon *\tau ) \wedge \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\tau ). \end{aligned}$$

Hence, \(\widehat{\mathcal {G}}\) is an IVmPF ideal.

Next, take any \(\upsilon , \tau \in Z\). By above inequality and (6), we have

$$\begin{aligned} \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon *\tau )&\ge \widehat{\pi _i} \circ \widehat{\mathcal {G}}((\upsilon *\tau ) *\upsilon ) \wedge \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon )\\&=\widehat{\pi _i} \circ \widehat{\mathcal {G}}(0 *\tau ) \wedge \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon )\\&\ge \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon ) \wedge \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\tau ). \end{aligned}$$

Hence, \(\widehat{\mathcal {G}}\) is an IVmPF subalgebra. \(\square\)

The following example shows that the converse of Theorem 5.2 is not valid.

Example 6

Consider a BCI-algebra \(Z=\{0,\imath ,\jmath \}\) under the operation (\(*\)) defined in Table 4.

Table 4 Cayley table of the binary operation*
Full size table

Define an IV4PF set \(\widehat{\mathcal {G}}\) on Z as:

$$\begin{aligned} \widehat{\mathcal {G}}(x) = \left\{ \begin{array}{ll} \big ([0.6,0.7],[0.5,0.8],[0.4,0.6],[0.7,0.9]\big ) &{} \text{ if } x=0, \\ \big ([0.2,0.3],[0.4,0.7],[0.1,0.2],[0.1,0.2]\big ) &{} \text{ if } x \in \{\imath , \jmath \}.\\ \end{array} \right. \end{aligned}$$

It is straightforward to check that \(\widehat{\mathcal {G}}\) is both an IV4PF ideal and an IV4PF subalgebra but not an IV4PF a-ideal because \(\big ([0.2,0.3],[0.4,0.7],[0.1,0.2],[0.1,0.2]\big )= {\widehat{G}}(\imath *0) \ngeq {\widehat{G}}((0 *0)*(0 *\imath )) \wedge {\widehat{G}}( 0) = {\widehat{G}}( 0) =\big ([0.6,0.7],[0.5,0.8],[0.4,0.6],[0.7,0.9]\big )\).

The following result provides a condition under which every IVmPF ideal of Z is an IVmPF a-ideal.

Theorem 5.3

Let \(\widehat{\mathcal {G}}\) be an IVmPF ideal of Z satisfying:

$$\begin{aligned} (\forall ~ \tau , \kappa \in Z)~\widehat{\mathcal {G}}(\kappa *\tau ) \ge \widehat{\mathcal {G}}(\tau *(0 *\kappa )). \end{aligned}$$

Then, \(\widehat{\mathcal {G}}\) is an IVmPF a-ideal.

Proof

As \(\widehat{\mathcal {G}}\) is an IVmPF ideal of Z, so for all \(\upsilon , \kappa , \tau \in Z\) and \(i \in \{1,2,\ldots ,m\}\), we have

$$\begin{aligned} \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\kappa *\tau )&\ge \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\tau *(0 *\kappa ))\\&\ge \widehat{\pi _i} \circ \widehat{\mathcal {G}}((\tau *(0 *\kappa )) *\upsilon ) \wedge \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon )\\&\ge \widehat{\pi _i} \circ \widehat{\mathcal {G}}((\tau *\upsilon ) *(0 *\kappa )) \wedge \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon ). \end{aligned}$$

Hence, \(\widehat{\mathcal {G}}\) is an IVmPF a-ideal of Z. \(\square\)

Theorem 5.4

IVmPF a-ideals are IVmPF p-ideals in Z, conversely IVmPF p-ideals need not be IVmPF a-ideals.

Proof

Let \(\widehat{\mathcal {G}}\) be an IVmPF a-ideal of Z. Then for all \(\upsilon , \kappa , \tau \in Z\) and \(i \in \{1,2,\ldots ,m\}\), we have

$$\begin{aligned} \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon *\tau ) \ge \widehat{\pi _i} \circ \widehat{\mathcal {G}}((\tau *\kappa ) *(0 *\upsilon )) \wedge \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\kappa ). \end{aligned}$$

Substitute \(\kappa\) and \(\tau\) by 0, we get

$$\begin{aligned} \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon *0)\ge &\, {} \widehat{\pi _i} \circ \widehat{\mathcal {G}}((0 *0) *(0 *\upsilon )) \wedge \widehat{\pi _i} \circ \widehat{\mathcal {G}}(0)\nonumber \\= &\, {} \widehat{\mathcal {G}}(0 *(0 *\upsilon )). \end{aligned}$$
(7)

As \(\upsilon *0=\upsilon\), so from (7)

$$\begin{aligned} \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon )\ge \widehat{\mathcal {G}}(0 *(0 *\upsilon )). \end{aligned}$$
(8)

From (8) and Theorem 3.5, \(\widehat{\mathcal {G}}\) is an IVmPF p-ideal.

To show the converse, consider the BCI-algebra of Example 4 defined in Table 3. Choose \(\widehat{[\theta ,\lambda ]}=\big ([\theta _1,\lambda _1],[\theta _2,\lambda _2],\ldots ,[\theta _m,\lambda _m]\big )\) and \(\widehat{[\rho ,\sigma ]}=\big ([\rho _1,\sigma _1],[\rho _2,\sigma _2], \ldots , [\rho _m,\sigma _m]\big )\in D[0,1]^m\) such that \(\widehat{[\theta ,\lambda ]} \ge \widehat{[\rho ,\sigma ]}\). Define an IVmPF set \(\widehat{\mathcal {G}}\) on Z as:

$$\begin{aligned} \widehat{\mathcal {G}}(x) = \left\{ \begin{array}{ll} \widehat{[\theta ,\lambda ]}= \big ([\theta _1,\lambda _1],[\theta _2,\lambda _2],\ldots , [\theta _m,\lambda _m]\big ) &{} \text{ if } x \in \{0, \imath \}, \\ \widehat{[\rho ,\sigma ]}= \big ([\rho _1,\sigma _1],[\rho _2,\sigma _2],\ldots , [\rho _m,\sigma _m]\big ) &{} \text{ if } x\in \{\jmath ,\kappa ,\ell \}. \end{array} \right. \end{aligned}$$

It is straightforward to check that \(\widehat{\mathcal {G}}\) is an IVmPF p-ideal of Z but not an IVmPF a-ideal because \(\widehat{[\rho ,\sigma ]}= {\widehat{G}}(\ell *\jmath ) \ngeq {\widehat{G}}((\jmath *0)*(0 *\ell )) \wedge {\widehat{G}}(0) \wedge {\widehat{G}}( 0) = {\widehat{G}}( 0) =\widehat{[\theta ,\lambda ]}\). \(\square\)

Theorem 5.5

IVmPF a-ideals are IVmPF q-ideals in Z, conversely IVmPF q-ideals need not be IVmPF a-ideals.

Proof

Let \(\widehat{\mathcal {G}}\) be an IVmPF a-ideal of Z. By Theorems 5.4 and 3.3, we have

$$\begin{aligned} \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon *\kappa ) \ge \widehat{\pi _i} \circ \widehat{\mathcal {G}}(0*(0 *(\upsilon *\kappa ))). \end{aligned}$$
(9)

By properties of BCI-algebra Z, we have

$$\begin{aligned}&(0*(0 *(\upsilon *\kappa ))) *(\upsilon *(0*\kappa ))\\&\quad = \big (0*((0 *\upsilon )*(0*\kappa ))\big ) *(\upsilon *(0*\kappa ))\\&\quad =\big ( (0*(0 *\upsilon )) *(0 *(0*\kappa ))\big ) *(\upsilon *(0*\kappa ))\\&\quad \le (\upsilon *\kappa ) *(\upsilon *(0*\kappa )) \end{aligned}$$

By Lemma 2.5, we have

$$\begin{aligned}&\widehat{\pi _i} \circ \widehat{\mathcal {G}}(0*(0 *(\upsilon *\kappa ))) \\&\ge \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon *(0*\kappa ))\wedge \widehat{\pi _i} \circ \widehat{\mathcal {G}}( (\upsilon *\kappa ) *(\upsilon *(0*\kappa )))\\&\ge \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon *(0*\kappa )) \wedge \widehat{\pi _i} \circ \widehat{\mathcal {G}} (\upsilon *\kappa ) \wedge \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon *(0*\kappa ))\\&\ge \widehat{\pi _i} \circ \widehat{\mathcal {G}} (\upsilon *\kappa ) \wedge \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon *(0*\kappa )). \end{aligned}$$

Thus, \(\widehat{\pi _i} \circ \widehat{\mathcal {G}}(0*(0 *(\upsilon *\kappa ))) \ge \widehat{\pi _i} \circ \widehat{\mathcal {G}} (\upsilon *\kappa ) \wedge \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon *(0*\kappa )).\) So from (9), we get

$$\begin{aligned} \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon *\kappa ) \ge \widehat{\pi _i} \circ \widehat{\mathcal {G}} (\upsilon *\kappa ) \wedge \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon *(0*\kappa )). \end{aligned}$$

It follows that \(\widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon *\kappa ) \ge \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon *(0*\kappa )).\) Hence by Theorem 4.3, \(\widehat{\mathcal {G}}\) is an IVmPF q-ideal.

To show the converse, consider the BCI-algebra defined in Example 2 by Table 2. Let \(\widehat{[\theta ,\lambda ]}=([\theta _1,\lambda _1],[\theta _2,\lambda _2], \ldots ,[\theta _m,\lambda _m]), \widehat{[\psi ,\phi ]}=([\psi _1,\phi _1],[\psi _2,\phi _2],\ldots ,[\psi _m,\phi _m]), \widehat{[\rho ,\sigma ]}= ([\rho _1,\sigma _1],[\rho _2,\sigma _2],\ldots ,[\rho _m,\sigma _m])\) and \(\widehat{[\alpha ,\beta ]}=([\alpha _1,\beta _1],[\alpha _2,\beta _2],\ldots ,[\alpha _m,\beta _m])\) \(\in D[0,1]^m\) such that \(\widehat{[\theta ,\lambda ]}\ge \widehat{[\alpha ,\beta ]}\ge \widehat{[\rho ,\sigma ]}\ge \widehat{[\psi ,\phi ]}\). Define an IVmPF set \(\widehat{\mathcal {G}}\) on Z as:

$$\begin{aligned} \widehat{\mathcal {G}}(x) = \left\{ \begin{array}{ll} \widehat{[\theta ,\lambda ]}= \big ([\theta _1,\lambda _1],[\theta _2,\lambda _2],\ldots , [\theta _m,\lambda _m]\big ) &{} \text{ if } x=0, \\ \widehat{[\psi ,\phi ]}= \big ([\psi _1,\phi _1],[\psi _2,\phi _2],\ldots , [\psi _m,\phi _m]\big ) &{} \text{ if } x\in \{\imath , \ell \},\\ \widehat{[\alpha ,\beta ]}= \big ([\alpha _1,\beta _1],[\alpha _2,\beta _2],\ldots , [\alpha _m,\beta _m]\big ) &{} \text{ if } x=\jmath ,\\ \widehat{[\rho ,\sigma ]}=\big ([\rho _1,\sigma _1],[\rho _2,\sigma _2],\ldots , [\rho _m,\sigma _m]\big ) &{} \text{ if } x=\kappa . \end{array} \right. \end{aligned}$$

It is straightforward to verify that \(\widehat{\mathcal {G}}\) is an IVmPF q-ideal of Z but not an IVmPF a-ideal because \(\widehat{[\psi ,\phi ]}= {\widehat{G}}(\ell *\jmath )\ngeq {\widehat{G}}((\jmath *0)*(0 *\ell )) \wedge {\widehat{G}}(0) = {\widehat{G}}(\ell ) =\widehat{[\rho ,\sigma ]}\). \(\square\)

6 Correspondence Between p(q and a)-Ideals and IVmPF p(qand a)-Ideals

In this section, we discuss correspondence between p(q and a)-ideals and IVmPF p(q and a)-ideals.

Definition 6.1

Let \(\widehat{\mathcal {G}}\) be any IVmPF set on Z. For \(\widehat{[\alpha ,\beta ]}=([\alpha _1,\beta _1],[\alpha _2,\beta _2],\ldots ,[\alpha _m,\beta _m]) \in D[0,1]^m\) define a level subset \(U(\widehat{\mathcal {G}};\widehat{[\alpha ,\beta ]})\) as follows:

$$\begin{aligned} U(\widehat{\mathcal {G}};\widehat{[\alpha ,\beta ]})=\{\upsilon \in Z~|~ \widehat{\pi _i}\circ \widehat{\mathcal {G}}(x)\ge [\alpha _i,\beta _i] ~\text{ for } \text{ all }~i \in \{1,2,\ldots ,m\}\}. \end{aligned}$$

Theorem 6.2

An IVmPF set \(\widehat{\mathcal {G}}\) is an IVmPF p-ideal (resp. IVmPF q-ideal and IVmPF a-ideal) of Z \(\Leftrightarrow\) each level subset \((\emptyset \ne )U(\widehat{\mathcal {G}};\widehat{[\alpha ,\beta ]})\) is a p-ideal (resp. q-ideal and a-ideal) of Z \(\forall ~\widehat{[\alpha ,\beta ]}=([\alpha _1,\beta _1],[\alpha _2,\beta _2],\ldots ,[\alpha _m,\beta _m]) \in D[0,1]^m\).

Proof

(\(\Rightarrow\)) Let us suppose that \(\widehat{\mathcal {G}}\) is an IVmPF p-ideal of Z and \(\upsilon \in U(\widehat{\mathcal {G}};\widehat{[\alpha ,\beta ]})\). Then \(\widehat{\pi _i} \circ \widehat{\mathcal {G}} (\upsilon )\ge [\alpha _i,\beta _i]\). By hypothesis, \(\widehat{\pi _i} \circ \widehat{\mathcal {G}}(0) \ge \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon ) \ge [\alpha _i,\beta _i]\). Thus, \(0 \in U(\widehat{\mathcal {G}};\widehat{[\alpha ,\beta ]})\). Next, take any \(\upsilon ,\kappa , \tau \in Z\) such that \((\upsilon *\tau ) *(\kappa *\tau ) \in U(\widehat{\mathcal {G}};\widehat{[\alpha ,\beta ]})\) and \(\kappa \in U(\widehat{\mathcal {G}};\widehat{[\alpha ,\beta ]})\). Then \(\widehat{\pi _i} \circ \widehat{\mathcal {G}} ((\upsilon *\tau ) *(\kappa *\tau ) )\ge [\alpha _i,\beta _i]\) and \(\widehat{\pi _i} \circ \widehat{\mathcal {G}} (\kappa )\ge [\alpha _i,\beta _i]\). As \(\widehat{\mathcal {G}}\) is an IVmPF p-ideal, we have

$$\begin{aligned} \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon )\ge &\, {} \widehat{\pi _i} \circ \widehat{\mathcal {G}}((\upsilon *\tau ) *(\kappa *\tau ))\wedge \widehat{\pi _i} \circ \widehat{\mathcal {G}}( \kappa )\\\ge &\, {} [\alpha _i,\beta _i] \wedge [\alpha _i,\beta _i]\\= &\, {} [\alpha _i,\beta _i] \end{aligned}$$

Therefore, \(\upsilon \in U(\widehat{\mathcal {G}};\widehat{[\alpha ,\beta ]})\). Hence, \(U(\widehat{\mathcal {G}};\widehat{[\alpha ,\beta ]})\) is a p-ideal of Z.

(\(\Leftarrow\)) Assume that \(U(\widehat{\mathcal {G}};\widehat{[\alpha ,\beta ]})\) is a p-ideal of Z \(\forall ~\widehat{[\alpha ,\beta ]} \in D[0,1]^m\). If \(\widehat{\pi _i} \circ \widehat{\mathcal {G}} (0) < \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon )\) for some \(\upsilon \in Z\). So \(\exists ~ \widehat{[\gamma ,\lambda ]}= ([\gamma _1,\lambda _1],[\gamma _2,\lambda _2],\ldots ,[\gamma _m,\lambda _m]) \in D[0,1]^m\) such that \(\widehat{\pi _i} \circ \widehat{\mathcal {G}}(0) < [\gamma _i,\lambda _i] \le \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon )\) for each \(i \in \{1,2,\ldots ,m\}\) implies \(\upsilon \in U(\widehat{\mathcal {G}};\widehat{[\gamma ,\lambda ]})\) but \(0 \notin U(\widehat{\mathcal {G}};\widehat{[\gamma ,\lambda ]})\), which is a contradiction. Therefore, \(\widehat{\pi _i} \circ \widehat{\mathcal {G}}(0) \ge \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon )\) for all \(\upsilon \in Z\) and \(i \in \{1,2,\ldots ,m\}\). Again, if \(\widehat{\pi _i} \circ \widehat{\mathcal {G}} (\upsilon ) < \widehat{\pi _i} \circ \widehat{\mathcal {G}}((\upsilon *\tau ) *(\kappa *\tau ))\wedge \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\kappa )\) for some \(\upsilon ,\kappa , \tau \in Z\). So \(\exists ~ \widehat{[\gamma ,\lambda ]}= ([\gamma _1,\lambda _1],[\gamma _2,\lambda _2],\ldots ,[\gamma _m,\lambda _m]) \in D[0,1]^m\) such that \(\widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon ) < [\gamma _i,\lambda _i] \le \widehat{\pi _i} \circ \widehat{\mathcal {G}}((\upsilon *\tau ) *(\kappa *\tau )) \wedge \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\kappa )\) for each \(i \in \{1,2,\ldots ,m\}\) implies \((\upsilon *\tau ) *(\kappa *\tau ) \in U(\widehat{\mathcal {G}};\widehat{[\gamma ,\lambda ]})\) and \(\kappa \in U(\widehat{\mathcal {G}};\widehat{[\gamma ,\lambda ]})\) but \(\upsilon \notin U(\widehat{\mathcal {G}};\widehat{[\gamma ,\lambda ]})\), gives a contradiction. Therefore, \(\widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon ) \ge \widehat{\pi _i} \circ \widehat{\mathcal {G}}((\upsilon *\tau ) *(\kappa *\tau )) \wedge \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\kappa )\) for all \(\upsilon , \kappa , \tau \in Z\) and \(i \in \{1,2,\ldots ,m\}\).

A similar argument is used with respect to IVmPF q-ideal and IVmPF a-ideal. \(\square\)

Theorem 6.3

An IVmPF set \(\widehat{\mathcal {G}}=\big ([\mathcal {G}_1^{-}, \mathcal {G}_1^{+}],[\mathcal {G}_2^{-}, \mathcal {G}_2^{+}], \ldots , [\mathcal {G}_m^{-}, \mathcal {G}_m^{+}]\big )\) in Z is an IVmPF p-ideal (resp. IVmPF q-ideal and IVmPF a-ideal) of Z \(\Leftrightarrow\) \(\mathcal {G}_i^{-}\) and \(\mathcal {G}_i^{+}\) for all \(i \in \{1,2,\ldots ,m\}\) are fuzzy p-ideal (resp. fuzzy q-ideal and fuzzy a-ideal) of Z.

Proof

(\(\Rightarrow\)) Suppose that \(\widehat{\mathcal {G}}(\upsilon )=\big ([\mathcal {G}_1^{-}(\upsilon ), \mathcal {G}_1^{+}(\upsilon )],[\mathcal {G}_2^{-}(\upsilon ), \mathcal {G}_2^{+}(\upsilon )], \ldots , [\mathcal {G}_m^{-}(\upsilon ), \mathcal {G}_m^{+}(\upsilon )]\big )\) in Z is an IVmPF p-ideal. For any \(\upsilon \in Z\), we have

$$\begin{aligned} \widehat{\pi _i} \circ \widehat{\mathcal {G}}(0) \ge \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon )~\forall ~ i \in \{1,2,\ldots ,m\} \end{aligned}$$

implies for each \(i \in \{1,2,\ldots ,m\}\),

$$\begin{aligned} {[}\mathcal {G}_i^{-}(0), \mathcal {G}_i^{+}(0)] \ge [\mathcal {G}_i^{-}(\upsilon ), \mathcal {G}_i^{+}(\upsilon )]. \end{aligned}$$

It follows that \(\mathcal {G}_i^{-}(0) \ge \mathcal {G}_i^{-}(\upsilon )\) and \(\mathcal {G}_i^{+}(0) \ge \mathcal {G}_i^{+}(\upsilon )\). Take any \(\upsilon , \kappa , \tau \in Z\). By hypothesis, we have for all \(i \in \{1,2,\ldots ,m\}\)

$$\begin{aligned} \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon ) \ge \widehat{\pi _i} \circ \widehat{\mathcal {G}}((\upsilon *\tau ) *(\kappa *\tau ))\wedge \widehat{\pi _i} \circ \widehat{\mathcal {G}}( \kappa ) \end{aligned}$$

implies

$$\begin{aligned}&{[}\mathcal {G}_i^{-}(\upsilon ), \mathcal {G}_i^{+}(\upsilon )]\\&\quad \ge [\mathcal {G}_i^{-}((\upsilon *\tau ) *(\kappa *\tau )), \mathcal {G}_i^{+}((\upsilon *\tau ) *(\kappa *\tau ))]\wedge [\mathcal {G}_i^{-}(\kappa ), \mathcal {G}_i^{+}(\kappa )]\\&\quad = [\mathcal {G}_i^{-}((\upsilon *\tau ) *(\kappa *\tau ))\wedge \mathcal {G}_i^{-}(\kappa ), \mathcal {G}_i^{+}((\upsilon *\tau ) *(\kappa *\tau ))\wedge \mathcal {G}_i^{+}(\kappa )]. \end{aligned}$$

Therefore, \(\mathcal {G}_i^{-}(\upsilon ) \ge \mathcal {G}_i^{-}((\upsilon *\tau ) *(\kappa *\tau ))\wedge \mathcal {G}_i^{-}(\kappa )\) and \(\mathcal {G}_i^{+}(\upsilon ) \ge \mathcal {G}_i^{+}((\upsilon *\tau ) *(\kappa *\tau ))\wedge \mathcal {G}_i^{+}(\kappa )\). Hence, \(\mathcal {G}_i^{-}\) and \(\mathcal {G}_i^{+}\) are fuzzy p-ideals of Z.

(\(\Leftarrow\)) Suppose that for all \(i \in \{1,2,\ldots ,m\}\), \(\mathcal {G}_i^{-}\) and \(\mathcal {G}_i^{+}\) are fuzzy p-ideals of Z.

$$\begin{aligned} \widehat{\pi _i} \circ \widehat{\mathcal {G}}(0)&= [{\mathcal {G}}_i^{-}(0), {\mathcal {G}}_i^{+}(0 )]\\&\ge [{\mathcal {G}}_i^{-}(\upsilon ), {\mathcal {G}}_i^{+}(\upsilon )]\\&=\widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon ) \end{aligned}$$

and

$$\begin{aligned}&\widehat{\pi _i} \circ \widehat{\mathcal {G}}(\upsilon )\\&\quad = [{\mathcal {G}}_i^{-}(\upsilon ), {\mathcal {G}}_i^{+}(\upsilon )]\\&\quad \ge [{\mathcal {G}}_i^{-}((\upsilon *\tau ) *(\kappa *\tau ))\wedge {\mathcal {G}}_i^{-}(\kappa ), {\mathcal {G}}_i^{+}((\upsilon *\tau ) *(\kappa *\tau ))\wedge {\mathcal {G}}_i^{+}(\kappa )]\\&\quad = [{\mathcal {G}}_i^{-}((\upsilon *\tau ) *(\kappa *\tau )), {\mathcal {G}}_i^{+}((\upsilon *\tau ) *(\kappa *\tau ))]\wedge [{\mathcal {G}}_i^{-}(\kappa ), {\mathcal {G}}_i^{+}(\kappa )]\\&\quad =\widehat{\pi _i} \circ \widehat{\mathcal {G}}((\upsilon *\tau ) *(\kappa *\tau ))\wedge \widehat{\pi _i} \circ \widehat{\mathcal {G}}(\kappa ). \end{aligned}$$

Hence, \(\widehat{\mathcal {G}}\) is an IVmPF p-ideal.

A similar argument is used with respect to IVmPF q-ideal and IVmPF a-ideal. \(\square\)

7 Conclusion

Based on IVmPF structures, we developed the ideal theory of BCI-algebra. We introduced the concepts of IVmPF p-ideals, IVmPF q-ideals and IVmPF a-ideals in BCI-algebra. Then, we have shown that IVmPF p(q and a) ideals are IVmPF ideals but not the converse. We provided a condition under which IVmPF ideals become IVmPF p(q and a) ideals. Also, we proved that an IVmPF a-ideal is both an IVmPF p-ideal and an IVmPF q-ideal and that the converse implications are not true. Further, we characterize IVmPF p(q and a) ideals in terms of fuzzy p(q and a) ideals of BCI-algebra. Moreover, correspondence between IVmPF p(q and a) ideals of BCI-algebra and p(q and a)-ideals of BCI-algebra is investigated.

In our future work, we intend to apply the presented notions of this present paper to different algebras such as MTL-algebras, BL-algebras, R0-algebras, EQ-algebras, MV-algebras and lattice implication algebras etc.