Anti-Intuitionistic Fuzzy Soft a-Ideals Applied to BCI-Algebras

Abstract

The notion of anti-intuitionistic fuzzy soft a-ideals of $BCI$-algebras is introduced and several related properties are investigated. Furthermore, the operations, namely; AND, extended intersection, restricted intersection, and union on anti-intuitionistic fuzzy soft a-ideals are discussed. Finally, characterizations of anti-intuitionistic fuzzy soft a-ideals of $BCI$-algebras are given.

1. Introduction

The theory of fuzzy set, intuitionistic fuzzy sets, soft set, and more other theories were introduced to deal with uncertainty. In [1], Zadeh introduced the concept of a fuzzy subset of a set. Later on, a number of generalizations of this fundamental notion have been studied by many authors in different directions. The notion of an intuitionistic fuzzy set defined in [2] is a generalization of a fuzzy set. It gives more opportunity to be accurate when dealing with uncertain objects. Soft set theory was initially suggested by Molodstov in [3], then Maji et al. in [4] combined the soft set theory and the intuitionistic fuzzy set theory, and introduced the notion intuitionistic fuzzy soft sets.

Algebra is the language in which combinatorics are usually expressed. Combinatorics is the study of discrete structures that arise not only in areas of pure mathematics, but in other areas of science, for example, computer science, statistical physics and genetics. From ancient beginnings, this subject truly rose to prominence from the mid-20th century, when scientific discoveries (most notably of DNA) showed that combinatorics is key to understanding the world around us, whilst many of the great advances in computing were built on combinatorial foundations. These concepts were widely studied over different classes of logical algebras as the essential classes of $BCK$/$BCI$-algebras presented by Iseki [5]. The concepts intuitionistic fuzzy ideals of $BCK$-algebras were studied in [6]. Bej et al. [7] declared the concept of doubt intuitionistic fuzzy subalgebra and doubt intuitionistic fuzzy ideal in $BCK$/$BCI$-algebras. Muhiuddin et al. studied various concepts on fuzzy sets and applied them to $BCK$/$BCI$-algebras, and other related notions (see for e.g., [8,9,10,11,12,13,14,15,16,17,18]). Also, some new generalizations of fuzzy sets and other related concepts in different algebras have been studied in (see for e.g., [6,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35]). Additionally, Balamurugan et al. [36] introduced the concepts of intuitionistic fuzzy soft subalgebras, intuitionistic fuzzy soft ideals, and intuitionistic fuzzy soft a-ideals of B-algebra and studied several properties of these notions.

In the present paper, we introduce the notion of anti-intuitionistic fuzzy soft a-ideals in $BCI$-algebras. The results of present paper are organized, as follows: Section 2 summarizes some basic definitions and properties that are needed to develop our main results while in Section 3, we introduce the notion of anti-intuitionistic fuzzy soft a-ideals of $BCI$-algebras and investigate related properties. In Section 4, we give characterizations of anti-intuitionistic fuzzy soft a-ideals of $BCI$-algebras while using the concept of a soft level set.

2. Preliminaries

In this section, we recall basic definitions and results that are related to the subject of the paper.

Definition 1.

[5] An algebra $(\Omega ;\odot ,0)$ of type $(2,0)$ is called a $BCI$-algebra if it satisfies the following conditions:

(1) 

$\left(\right(l\odot m)\odot (l\odot n\left)\right)\odot (n\odot m)=0$,

(2) 

$(l\odot (l\odot m\left)\right)\odot m=0$,

(3) 

$l\odot l=0$;

(4) 

$l\odot m=0$ and $m\odot l=0$⇒$l=m$, for all $l,m,n\in \Omega$.

Any $BCI$-algebra $\Omega$, satisfies the following axioms:

(I)

$l\odot 0=l,$

(II)

$l\le m\Rightarrow l\odot n\le m\odot n$ and $n\odot m\le n\odot l$,

(III)

$(l\odot n)\odot (m\odot n)\le l\odot m$,

(IV)

$0\odot (0\odot (l\odot m\left)\right)=(0\odot m)\odot (0\odot l),$

(V)

$(l\odot m)\odot n=(l\odot n)\odot m,$

where $l\le m$⇔$l\odot m=0$, for any $l,m,n\in \Omega$.

A non-empty subset $\Delta$ of a $BCK$-algebra $\Omega$ is called an ideal of $\Omega$ if it satisfies

(1)

$0\in \Delta$,

(1)

$\forall l,m\in \Omega ,l*m\in \Delta ,m\in \Delta \Rightarrow l\in \Delta$.

A non-empty subset $\Delta$ of a $BCK$-algebra $\Omega$ is called an a-ideal of $\Omega$ if it satisfies $\left(1\right)$ and $\left(3\right)\forall l,m\in \Omega ,(l\odot n)\odot (0\odot m)\in \Delta ,n\in \Delta \Rightarrow m\odot l\in \Delta$.

For an initial set $\Omega$ and a set of parameters $\Delta$, a pair $({\rm Y},\Delta )$ is said to be a soft set over $\Omega$⇔∃${\rm Y}$: $\Delta \to \wp (\Omega )$, where $\wp (\Omega )$ is a family of subsets of $\Omega$. (see [30] for more details on soft set theory).

Definition 2.

[4] Let Π be a collection of parameters and let ${\rm Y}(\Omega )$ indicate the collection of all fuzzy sets in Ω. Then $({\rm Y},\Delta )$ is called a fuzzy soft set over Ω, where $\Delta \subseteq \Pi$ and ${\rm Y}:\Delta \to {\rm Y}(\Omega )$.

Definition 3.

[36] Let $({\rm Y},\Delta )$ be a fuzzy soft set (abbr. $FSS$). Then $({\rm Y},\Delta )$ is an anti-fuzzy soft ideal (abbr. $AFSID)$ of Ω if ${\rm Y}\left[\varpi \right]=\{\left({\xi }_{{\rm Y}\left[\varpi \right]}\left(l\right)\right):l\in \Omega$ and $\varpi \in \Delta \}$ is an $AFID$ of Ω satisfies the following assertions:

(i) 

${\xi }_{{\rm Y}\left[\varpi \right]}\left(0\right)\le {\xi }_{{\rm Y}\left[\varpi \right]}\left(l\right)$,

(ii) 

${\xi }_{{\rm Y}\left[\varpi \right]}\left(l\right)\le {\xi }_{{\rm Y}\left[\varpi \right]}(l\odot m)\vee {\xi }_{{\rm Y}\left[\varpi \right]}\left(m\right)$,

for all $l,m,n\in \Omega$ and $\varpi \in \Delta .$

Definition 4.

[36] Let $({\rm Y},\Delta )$ be a fuzzy soft set (abbr. $FSS$). Then $({\rm Y},\Delta )$ is an anti-fuzzy soft a-ideal (abbr. $AFSID)$ of Ω if ${\rm Y}\left[\varpi \right]=\{\left({\xi }_{{\rm Y}\left[\varpi \right]}\left(l\right)\right):l\in \Omega$ and $\varpi \in \Delta \}$ is an $AFID$ of Ω satisfies the following assertions:

(i) 

${\xi }_{{\rm Y}\left[\varpi \right]}\left(0\right)\le {\xi }_{{\rm Y}\left[\varpi \right]}\left(l\right)$,

(ii) 

${\xi }_{{\rm Y}\left[\varpi \right]}(m\odot l)\le {\xi }_{{\rm Y}\left[\varpi \right]}((l\odot n)\odot (0\odot m))\vee {\xi }_{{\rm Y}\left[\varpi \right]}\left(n\right)$,

for all $l,m,n\in \Omega$ and $\varpi \in \Delta .$

Definition 5.

[4] Let Π be a collection of parameters and let $\mathbb{I}{\rm Y}(\Omega )$ indicate the collection of all intuitionistic fuzzy sets in Ω. Subsequently, $({\rm Y},\Delta )$ is called an intuitionistic fuzzy soft set over Ω, where $\Delta \subseteq \Pi$ and ${\rm Y}:\Delta \to \mathbb{I}{\rm Y}(\Omega )$.

3. Anti-Intuitionistic Fuzzy Soft a-Ideal

In what follows, we write $\Omega$ to denote a $BCI$-algebra $(\Omega ;\odot ,0)$ and $IFSs$ for intuitionistic fuzzy sets and we will introduce an abbreviation for the notions in the following definitions to be used in the rest of the paper.

Definition 6.

Let $({\rm Y},\Delta )$ be an intuitionistic fuzzy soft set (abbr. $IFSS$). Afterwards, $({\rm Y},\Delta )$ is an anti-intuitionistic fuzzy soft ideal (abbr. $AIFSID)$ of Ω if ${\rm Y}\left[\varpi \right]=\{({\xi }_{{\rm Y}\left[\varpi \right]}\left(l\right),{\zeta }_{{\rm Y}\left[\varpi \right]}\left(l\right)):l\in \Omega$ and $\varpi \in \Delta \}$ is an $AIFID$ of Ω satisfies the following assertions:

(i) 

${\xi }_{{\rm Y}\left[\varpi \right]}\left(0\right)\le {\xi }_{{\rm Y}\left[\varpi \right]}\left(l\right)$ and ${\zeta }_{{\rm Y}\left[\varpi \right]}\left(0\right)\ge {\zeta }_{{\rm Y}\left[\varpi \right]}\left(l\right)$,

(ii) 

${\xi }_{{\rm Y}\left[\varpi \right]}\left(l\right)\le {\xi }_{{\rm Y}\left[\varpi \right]}(l\odot m)\vee {\xi }_{{\rm Y}\left[\varpi \right]}\left(m\right)$,

(iii) 

${\zeta }_{{\rm Y}\left[\varpi \right]}\left(l\right)\ge {\zeta }_{{\rm Y}\left[\varpi \right]}(l\odot m)\wedge {\zeta }_{{\rm Y}\left[\varpi \right]}\left(m\right)$,

for all $l,m,n\in \Omega$ and $\varpi \in \Delta .$

Definition 7.

An $IFSS$$({\rm Y},\Delta )$ is called an anti-intuitionistic fuzzy soft a-ideal (abbr. $AIFSAID$) of Ω if ${\rm Y}\left[\varpi \right]=\{({\xi }_{{\rm Y}\left[\varpi \right]}\left(l\right),{\zeta }_{{\rm Y}\left[\varpi \right]}\left(l\right)):l\in \Omega$ and $\varpi \in \Delta \}$ is an $AIFAID$ of Ω satisfies the following assertions:

(i) 

${\xi }_{{\rm Y}\left[\varpi \right]}\left(0\right)\le {\xi }_{{\rm Y}\left[\varpi \right]}\left(l\right)$ and ${\zeta }_{{\rm Y}\left[\varpi \right]}\left(0\right)\ge {\zeta }_{{\rm Y}\left[\varpi \right]}\left(l\right)$,

(ii) 

${\xi }_{{\rm Y}\left[\varpi \right]}(m\odot l)\le {\xi }_{{\rm Y}\left[\varpi \right]}((l\odot n)\odot (0\odot m))\vee {\xi }_{{\rm Y}\left[\varpi \right]}\left(n\right)$,

(iii) 

${\zeta }_{{\rm Y}\left[\varpi \right]}(m\odot l)\ge {\zeta }_{{\rm Y}\left[\varpi \right]}((l\odot n)\odot (0\odot m))\wedge {\zeta }_{{\rm Y}\left[\varpi \right]}\left(n\right)$,

for all $l,m,n\in \Omega$ and $\varpi \in \Delta .$

Example 1.

Suppose that there are four patients in the initial universe set $\Omega =\{p_1,p_2,p_3,p_4\}$ given by

⊙	$p_1$	$p_2$	$p_3$	$p_4$
$p_1$	$p_1$	$p_2$	$p_3$	$p_4$
$p_2$	$p_2$	$p_1$	$p_4$	$p_3$
$p_3$	$p_3$	$p_4$	$p_1$	$p_2$
$p_4$	$p_4$	$p_3$	$p_2$	$p_1$

Afterwards, $(\Omega ;\odot ,p_1)$ is a $BCI$-algebra.

Let a set of parameters, we consider $\Delta =\{f,s,n\}$ be a status of patients, in which

• f stands for the parameter "fever" can be treated by antibiotic,
  s stands for the parameter "sneezing" can be treated by antiallergic,
  n stands for the parameter "nosal block" can be treated by nosal drops.

Subsequently, ${\rm Y}\left[f\right],{\rm Y}\left[s\right]$, and ${\rm Y}\left[n\right]$ are $IFSs$ over Ω represented by:

${\rm Y}$	$p_1$	$p_2$	$p_3$	$p_4$
f	[0.1, 0.8]	[0.1, 0.8]	[0.2, 0.6]	[0.2, 0.6]
s	[0.0, 0.9]	[0.0, 0.9]	[0.3, 0.7]	[0.3, 0.7]
n	[0.2, 0.7]	[0.2, 0.7]	[0.4, 0.6]	[0.4, 0.6]

Therefore, ${\rm Y}\left[f\right],{\rm Y}\left[s\right]$, and ${\rm Y}\left[n\right]$ are an $AIFAID$ of Ω with respect to $f,s$, and n, respectively.

Hence, $({\rm Y},\Delta )$ is an $AIFSAID$ of Ω.

Proposition 1.

For any $AIFSAID$$({\rm Y},\Delta )$ of Ω, the following inequalities hold: ${\xi }_{{\rm Y}\left[\varpi \right]}(m\odot l)\le {\xi }_{{\rm Y}\left[\varpi \right]}(l\odot (0\odot m))$ and ${\zeta }_{{\rm Y}\left[\varpi \right]}(m\odot l)\ge {\zeta }_{{\rm Y}\left[\varpi \right]}(l\odot (0\odot m))$, for any $\varpi \in \Delta$ and $l,m\in \Omega$.

Proof. 

Let $({\rm Y},\Delta )$ be an $AIFSAID$ of $\Omega$.

Subsequently, ${\rm Y}\left[\varpi \right]=\{({\xi }_{{\rm Y}\left[\varpi \right]}\left(l\right),{\zeta }_{{\rm Y}\left[\varpi \right]}\left(l\right)):l\in \Omega$ and $\varpi \in \Delta \}$ is an $AIFAID$ of $\Omega$.

Thus, for every $l,m,n\in \Omega$ and $\varpi \in \Delta$,

  ${\xi }_{{\rm Y}\left[\varpi \right]}(m\odot l)\le {\xi }_{{\rm Y}\left[\varpi \right]}((l\odot n)\odot (0\odot m))\vee {\xi }_{{\rm Y}\left[\varpi \right]}\left(n\right)$

and

  ${\zeta }_{{\rm Y}\left[\varpi \right]}(m\odot l)\ge {\zeta }_{{\rm Y}\left[\varpi \right]}((l\odot n)\odot (0\odot m))\wedge {\zeta }_{{\rm Y}\left[\varpi \right]}\left(n\right)$.

By substituting $n=0$, we get,

  ${\xi }_{{\rm Y}\left[\varpi \right]}(m\odot l)\le {\xi }_{{\rm Y}\left[\varpi \right]}((l\odot 0)\odot (0\odot m))\vee {\xi }_{{\rm Y}\left[\varpi \right]}\left(0\right)$

          $={\xi }_{{\rm Y}\left[\varpi \right]}(l\odot (0\odot m))\vee {\xi }_{{\rm Y}\left[\varpi \right]}\left(0\right)$

  ${\xi }_{{\rm Y}\left[\varpi \right]}(m\odot l)\le {\xi }_{{\rm Y}\left[\varpi \right]}(l\odot (0\odot m))$

and

  ${\zeta }_{{\rm Y}\left[\varpi \right]}(m\odot l)\ge {\zeta }_{{\rm Y}\left[\varpi \right]}((l\odot 0)\odot (0\odot m))\wedge {\zeta }_{{\rm Y}\left[\varpi \right]}\left(0\right)$

          $={\zeta }_{{\rm Y}\left[\varpi \right]}(l\odot (0\odot m))\wedge {\zeta }_{{\rm Y}\left[\varpi \right]}\left(0\right)$

  ${\zeta }_{{\rm Y}\left[\varpi \right]}(m\odot l)\ge {\zeta }_{{\rm Y}\left[\varpi \right]}(l\odot (0\odot m))$. □

Theorem 1.

Over Ω, any $AIFSAID$ is an $AIFSID$.

Proof. 

Let $({\rm Y},\Delta )$ be an $AIFSAID$ of $\Omega$.

Subsequently, ${\rm Y}\left[\varpi \right]=\{({\xi }_{{\rm Y}\left[\varpi \right]}\left(l\right),{\zeta }_{{\rm Y}\left[\varpi \right]}\left(l\right)):l\in \Omega$ and $\varpi \in \Delta \}$ is an $AIFAID$ of $\Omega$.

Thus, for every $l,m,n\in \Omega$ and $\varpi \in \Delta$,

  ${\xi }_{{\rm Y}\left[\varpi \right]}(m\odot l)\le {\xi }_{{\rm Y}\left[\varpi \right]}((l\odot n)\odot (0\odot m))\vee {\xi }_{{\rm Y}\left[\varpi \right]}\left(n\right)$

and

  ${\zeta }_{{\rm Y}\left[\varpi \right]}(m\odot l)\ge {\zeta }_{{\rm Y}\left[\varpi \right]}((l\odot n)\odot (0\odot m))\wedge {\zeta }_{{\rm Y}\left[\varpi \right]}\left(n\right)$.

By substituting $l=0$ we obtain,

  ${\xi }_{{\rm Y}\left[\varpi \right]}(m\odot 0)\le {\xi }_{{\rm Y}\left[\varpi \right]}((0\odot n)\odot (0\odot m))\vee {\xi }_{{\rm Y}\left[\varpi \right]}\left(n\right)$

and

  ${\zeta }_{{\rm Y}\left[\varpi \right]}(m\odot 0)\ge {\zeta }_{{\rm Y}\left[\varpi \right]}((0\odot n)\odot (0\odot m))\wedge {\zeta }_{{\rm Y}\left[\varpi \right]}\left(n\right)$.

Because we know that $(0\odot n)\odot (0\odot m)\le m\odot n$, therefore

  ${\xi }_{{\rm Y}\left[\varpi \right]}((0\odot n)\odot (0\odot m))\le {\xi }_{{\rm Y}\left[\varpi \right]}(m\odot n)$

and

  ${\zeta }_{{\rm Y}\left[\varpi \right]}((0\odot n)\odot (0\odot m))\ge {\zeta }_{{\rm Y}\left[\varpi \right]}(m\odot n)$.

Thus,

  ${\xi }_{{\rm Y}\left[\varpi \right]}\left(m\right)\le {\xi }_{{\rm Y}\left[\varpi \right]}((0\odot n)\odot (0\odot m))\vee {\xi }_{{\rm Y}\left[\varpi \right]}\left(n\right)\le {\xi }_{{\rm Y}\left[\varpi \right]}(m\odot n)\vee {\xi }_{{\rm Y}\left[\varpi \right]}\left(n\right)$

and

  ${\zeta }_{{\rm Y}\left[\varpi \right]}\left(m\right)\ge {\zeta }_{{\rm Y}\left[\varpi \right]}((0\odot n)\odot (0\odot m))\wedge {\zeta }_{{\rm Y}\left[\varpi \right]}\left(n\right)\ge {\zeta }_{{\rm Y}\left[\varpi \right]}(m\odot n)\wedge {\zeta }_{{\rm Y}\left[\varpi \right]}\left(n\right)$,

  i.e., ${\rm Y}\left[\varpi \right]=\{({\xi }_{{\rm Y}\left[\varpi \right]}\left(l\right),{\zeta }_{{\rm Y}\left[\varpi \right]}\left(l\right)):l\in \Omega$ and $\varpi \in \Delta \}$ is an $AIFID$ of $\Omega$.

Hence $({\rm Y},\Delta )$ is an $AIFSID$ of $\Omega$.□

The converse of Theorem 1 is not true in general i.e., an $AIFSID$ might not be an $AIFSAID$, as shown in the next example and we will give in the latter theorem a condition for this converse to be true.

Example 2.

Let $\Omega =\left\{0,p,q,r,s\right\}$ with Cayley table:

⊙	0	p	q	r	s
0	0	0	s	r	q
p	p	0	s	r	q
q	q	q	0	s	r
r	r	r	q	0	s
s	s	s	r	q	0

Subsequently, $(\Omega ;\odot ,0)$ is a $BCI$-algebra.

Let $\Delta =\left\{\theta ,\vartheta ,\kappa \right\}$ be a set of parameters and consider the $IFSS$$({\rm Y},\Delta )$ over Ω. Then ${\rm Y}\left[\theta \right],{\rm Y}\left[\vartheta \right]$, and ${\rm Y}\left[\kappa \right]$ are $IFSs$ over Ω represented by:

${\rm Y}$	0	p	q	r	s
$\theta$	[0.1, 0.9]	[0.4, 0.4]	[0.3, 0.6]	[0.2, 0.8]	[0.5, 0.1]
$\vartheta$	[0, 0.9]	[0.1, 0.7]	[0.4, 0.4]	[0.3, 0.5]	[0.2, 0.6]
$\kappa$	[0, 1]	[0.2, 0.6]	[0.3, 0.5]	[0.4, 0.3]	[0.1, 0.7]

Afterwards, $({\rm Y},\Delta )$ is an $AIFSID$ of Ω, but since

${\xi }_{{\rm Y}\left[\vartheta \right]}(p\odot s)={\xi }_{{\rm Y}\left[\vartheta \right]}\left(q\right)=0.4\nleqq 0.2={\xi }_{{\rm Y}\left[\vartheta \right]}((s\odot 0)\odot (0\odot p))\vee {\xi }_{{\rm Y}\left[\vartheta \right]}\left(0\right)$

and

${\zeta }_{{\rm Y}\left[\vartheta \right]}(p\odot s)={\zeta }_{{\rm Y}\left[\vartheta \right]}\left(q\right)=0.4\ngeqq 0.6={\zeta }_{{\rm Y}\left[\vartheta \right]}((s\odot 0)\odot (0\odot p))\wedge {\zeta }_{{\rm Y}\left[\vartheta \right]}\left(0\right)$,

i.e., ${\rm Y}\left[\vartheta \right]=\{({\xi }_{{\rm Y}\left[\vartheta \right]}\left(l\right),{\zeta }_{{\rm Y}\left[\vartheta \right]}\left(l\right)):l\in \Omega$ and $\vartheta \in \Delta \}$ is not an $AIFAID$ of Ω.

Therefore $({\rm Y},\Delta )$ is not an $AIFSAID$ of Ω with respect to ϑ.

Hence $({\rm Y},\Delta )$ is not an $AIFSAID$ of Ω.

Theorem 2.

Let $({\rm Y},\Delta )$ be an $AIFSID$ over Ω. If for any $\varpi \in \Delta$ and $l,m\in \Omega$, ${\xi }_{{\rm Y}\left[\varpi \right]}(m\odot l)\le {\xi }_{{\rm Y}\left[\varpi \right]}(l\odot (0\odot m))$ and ${\zeta }_{{\rm Y}\left[\varpi \right]}(m\odot l)\ge {\zeta }_{{\rm Y}\left[\varpi \right]}(l\odot (0\odot m))$, then $({\rm Y},\Delta )$ is an $AIFSAID$ over Ω.

Proof. 

Let $({\rm Y},\Delta )$ be an $AIFSID$ over $\Omega$.

Therefore, ${\rm Y}\left[\varpi \right]=\{({\xi }_{{\rm Y}\left[\varpi \right]}\left(l\right),{\zeta }_{{\rm Y}\left[\varpi \right]}\left(l\right)):l\in \Omega$ and $\varpi \in \Delta \}$ is an $AIFID$ of $\Omega$.

Thus, for any $\varpi \in \Delta$ and $l,m,n\in \Omega$,

${\xi }_{{\rm Y}\left[\varpi \right]}(m\odot l)\le {\xi }_{{\rm Y}\left[\varpi \right]}(l\odot (0\odot m))$

       $\le {\xi }_{{\rm Y}\left[\varpi \right]}((l\odot (0\odot m))\odot n)\vee {\xi }_{{\rm Y}\left[\varpi \right]}\left(n\right)$

${\xi }_{{\rm Y}\left[\varpi \right]}(m\odot l)\le {\xi }_{{\rm Y}\left[\varpi \right]}((l\odot n)\odot (0\odot m))\vee {\xi }_{{\rm Y}\left[\varpi \right]}\left(n\right)$

and

${\zeta }_{{\rm Y}\left[\varpi \right]}(m\odot l)\ge {\zeta }_{{\rm Y}\left[\varpi \right]}(l\odot (0\odot m))$

       $\ge {\zeta }_{{\rm Y}\left[\varpi \right]}((l\odot (0\odot m))\odot n)\wedge {\zeta }_{{\rm Y}\left[\varpi \right]}\left(n\right)$

${\zeta }_{{\rm Y}\left[\varpi \right]}(m\odot l)\ge {\zeta }_{{\rm Y}\left[\varpi \right]}((l\odot n)\odot (0\odot m))\wedge {\zeta }_{{\rm Y}\left[\varpi \right]}\left(n\right)$

${\rm Y}\left[\varpi \right]=\{({\xi }_{{\rm Y}\left[\varpi \right]}\left(l\right),{\zeta }_{{\rm Y}\left[\varpi \right]}\left(l\right)):l\in \Omega$ and $\varpi \in \Delta \}$ is an $AIFAID$ of $\Omega$.

Hence $({\rm Y},\Delta )$ is an $AIFSAID$ over $\Omega$.

Theorem 3.

If $({\rm Y},\Delta )$ is an $AIFSAID$ of Ω, then for any parameter $\varpi \in \Delta$ and $l,m,n\in \Omega$, ${\xi }_{{\rm Y}\left[\varpi \right]}((l\odot n)\odot (0\odot m))\le {\xi }_{{\rm Y}\left[\varpi \right]}(l\odot (n\odot m))$ and ${\zeta }_{{\rm Y}\left[\varpi \right]}((l\odot n)\odot (0\odot m))\ge {\zeta }_{{\rm Y}\left[\varpi \right]}(l\odot (n\odot m))$.

Proof. 

Let $({\rm Y},\Delta )$ be an $AIFSAID$ of $\Omega$.

Because $(l\odot n)\odot (0\odot m)=(l\odot n)\odot \left(\right(n\odot m)\odot n)\le l\odot (n\odot m)$.

Therefore, $(l\odot n)\odot (0\odot m)\odot (l\odot (n\odot m\left)\right)=0$.

By Theorem 1, $({\rm Y},\Delta )$ is an $AIFSID$ of $\Omega$.

Thus,${\rm Y}\left[\varpi \right]=\{({\xi }_{{\rm Y}\left[\varpi \right]}\left(l\right),{\zeta }_{{\rm Y}\left[\varpi \right]}\left(l\right)):l\in \Omega$ and $\varpi \in \Delta \}$ is an $AIFID$ of $\Omega$.

Thus, for every $l,m,n\in \Omega$ and $\varpi \in \Delta$,

$\begin{array}{ll}{\xi }_{{\rm Y}\left[\varpi \right]}((l\odot n)\odot (0\odot m))& \le {\xi }_{{\rm Y}\left[\varpi \right]}(((l\odot n)\odot (0\odot m))\odot (l\odot (n\odot m)))\vee {\xi }_{{\rm Y}\left[\varpi \right]}(l\odot (n\odot m))\\ & ={\xi }_{{\rm Y}\left[\varpi \right]}\left(0\right)\vee {\xi }_{{\rm Y}\left[\varpi \right]}(l\odot (n\odot m))\\ & \le {\xi }_{{\rm Y}\left[\varpi \right]}(l\odot (n\odot m))\end{array}$

and

$\begin{array}{ll}{\zeta }_{{\rm Y}\left[\varpi \right]}((l\odot n)\odot (0\odot m))\ge & {\zeta }_{{\rm Y}\left[\varpi \right]}(((l\odot n)\odot (0\odot m))\odot (l\odot (n\odot m)))\wedge {\zeta }_{{\rm Y}\left[\varpi \right]}(l\odot (n\odot m))\\ & ={\zeta }_{{\rm Y}\left[\varpi \right]}\left(0\right)\wedge {\zeta }_{{\rm Y}\left[\varpi \right]}(l\odot (n\odot m))\\ & \ge {\zeta }_{{\rm Y}\left[\varpi \right]}(l\odot (n\odot m)).\end{array}$ □

Definition 8.

Let $({\rm Y},\Delta )$ and $(\Gamma ,\Psi )$ be two $IFSSs$ over Ω. Then $({\rm Y},\Delta )$ “AND” $(\Gamma ,\Psi )$ written as $({\rm Y},\Delta )\tilde{\wedge }(\Gamma ,\Psi )$ is $(\Pi ,\Delta \times \Psi )$ of Ω, where $\Pi [\varpi ,\omega ]={\rm Y}\left[\varpi \right]\cap \Gamma \left[\omega \right]$ for all $(\varpi ,\omega )\in \Delta \times \Psi$.

Theorem 4.

If $({\rm Y},\Delta )$ and $(\Gamma ,\Psi )$ are two $AIFSAIDs$ of Ω, then $(\Pi ,\Delta \times \Psi )$ is also an $AIFSAID$ of Ω.

Proof. 

By definition, $({\rm Y},\Delta )\tilde{\wedge }(\Gamma ,\Psi )=(\Pi ,\Delta \times \Psi )$, where

$\Pi [\varpi ,\omega ]={\rm Y}\left[\varpi \right]\cap \Gamma \left[\omega \right]=\{({\xi }_{{\rm Y}\left[\varpi \right]\cap \Gamma \left[\omega \right]}\left(l\right),{\zeta }_{{\rm Y}\left[\varpi \right]\cap \Gamma \left[\omega \right]}\left(l\right)):l\in \Omega and(\varpi ,\omega )\in \Delta \times \Psi \}$

For any $l\in \Omega$ and $(\varpi ,\omega )\in \Delta \times \Psi$,

  $\begin{array}{ll}{\xi }_{\Pi [\varpi ,\omega ]}\left(0\right)=& {\xi }_{{\rm Y}\left[\varpi \right]\cap \Gamma \left[\omega \right]}\left(0\right)\\ & ={\xi }_{{\rm Y}\left[\varpi \right]}\left(0\right)\vee {\xi }_{\Gamma \left[\omega \right]}\left(0\right)\\ & \le {\xi }_{{\rm Y}\left[\varpi \right]}\left(l\right)\vee {\xi }_{\Gamma \left[\omega \right]}\left(l\right)\\ & ={\xi }_{{\rm Y}\left[\varpi \right]\cap \Gamma \left[\omega \right]}\left(l\right)\\ {\xi }_{\Pi [\varpi ,\omega ]}\left(0\right)\le & {\xi }_{\Pi [\varpi ,\omega ]}\left(l\right)\end{array}$

and

  $\begin{array}{ll}{\zeta }_{\Pi [\varpi ,\omega ]}\left(0\right)=& {\zeta }_{{\rm Y}\left[\varpi \right]\cap \Gamma \left[\omega \right]}\left(0\right)\\ & ={\zeta }_{{\rm Y}\left[\varpi \right]}\left(0\right)\wedge {\zeta }_{\Gamma \left[\omega \right]}\left(0\right)\\ & \ge {\varsigma }_{{\rm Y}\left[\varpi \right]}\left(l\right)\wedge {\zeta }_{\Gamma \left[\omega \right]}\left(l\right)\\ & ={\zeta }_{{\rm Y}\left[\varpi \right]\cap \Gamma \left[\omega \right]}\left(l\right)\\ {\zeta }_{\Pi [\varpi ,\omega ]}\left(0\right)\ge & {\zeta }_{\Pi [\varpi ,\omega ]}\left(l\right)\end{array}$.

For any $l,m,n\in \Omega$, and $(\varpi ,\omega )\in \Delta \times \Psi$,

${\xi }_{\Pi [\varpi ,\omega ]}(m\odot l)={\xi }_{{\rm Y}\left[\varpi \right]\cap \Gamma \left[\omega \right]}(m\odot l)={\xi }_{{\rm Y}\left[\varpi \right]}(m\odot l)\vee {\xi }_{\Gamma \left[\omega \right]}(m\odot l)$

$\le ({\xi }_{{\rm Y}\left[\varpi \right]}((l\odot n)\odot (0\odot m))\vee {\xi }_{{\rm Y}\left[\varpi \right]}\left(n\right))\vee ({\xi }_{\Gamma \left[\omega \right]}((l\odot n)\odot (0\odot m))\vee {\xi }_{\Gamma \left[\omega \right]}\left(n\right))$

$=({\xi }_{{\rm Y}\left[\varpi \right]}((l\odot n)\odot (0\odot m))\vee {\xi }_{\Gamma \left[\omega \right]}((l\odot n)\odot (0\odot m)))\vee ({\xi }_{{\rm Y}\left[\varpi \right]}\left(n\right)\vee {\xi }_{\Gamma \left[\omega \right]}\left(n\right))$

$=\left({\xi }_{{\rm Y}\left[\varpi \right]\cap \Gamma \left[\omega \right]}((l\odot n)\odot (0\odot m))\right)\vee \left({\xi }_{{\rm Y}\left[\varpi \right]\cap \Gamma \left[\omega \right]}\left(n\right)\right)$

and

${\zeta }_{\Pi [\varpi ,\omega ]}(m\odot l)={\zeta }_{{\rm Y}\left[\varpi \right]\cap \Gamma \left[\omega \right]}(m\odot l)={\zeta }_{{\rm Y}\left[\varpi \right]}(m\odot l)\wedge {\varsigma }_{\Gamma \left[\omega \right]}(m\odot l)$

$\ge ({\zeta }_{{\rm Y}\left[\varpi \right]}((l\odot n)\odot (0\odot m))\wedge {\zeta }_{{\rm Y}\left[\varpi \right]}\left(n\right))\wedge ({\zeta }_{\Gamma \left[\omega \right]}((l\odot n)\odot (0\odot m))\wedge {\zeta }_{\Gamma \left[\omega \right]}\left(n\right))$

= $({\zeta }_{{\rm Y}\left[\varpi \right]}((l\odot n)\odot (0\odot m))\wedge {\zeta }_{\Gamma \left[\omega \right]}((l\odot n)\odot (0\odot m)))\wedge ({\zeta }_{{\rm Y}\left[\varpi \right]}\left(n\right)\wedge {\zeta }_{\Gamma \left[\omega \right]}\left(n\right))$

= $\left({\zeta }_{{\rm Y}\left[\varpi \right]\cap \Gamma \left[\omega \right]}((l\odot n)\odot (0\odot m))\right)\wedge \left({\zeta }_{{\rm Y}\left[\varpi \right]\cap \Gamma \left[\omega \right]}\left(n\right)\right)$.

Thus, $\Pi [\varpi ,\omega ]={\rm Y}\left[\varpi \right]\cap \Gamma \left[\omega \right]$ is an $AIFAID$ of $\Omega$ for any $(\varpi ,\omega )\in \Delta \times \Psi$.

Hence $(\Pi ,\Delta \times \Psi )$ is an $AIFSAID$ of $\Omega$ for any $(\varpi ,\omega )\in \Delta \times \Psi$.□

Definition 9.

The "extended intersection" of two $IFSSs$$({\rm Y},\Delta )$ and $(\Gamma ,\Psi )$ denoted by $({\rm Y},\Delta ){\sqcap }_E(\Gamma ,\Psi )$ is $(\Pi ,\Theta )$, where $\Theta =\Delta \cup \Psi$ and for every $\varpi \in \Theta$,

$$\begin{array}{c}\Pi \left(\varpi \right)=\left\{\begin{array}{cc}{\rm Y}\left[\varpi \right],& \varpi \in \Delta -\Psi ,\\ \Gamma \left[\varpi \right],& \varpi \in \Psi -\Delta ,\\ {\rm Y}\left[\varpi \right]\cap \Gamma \left[\varpi \right],& \varpi \in \Delta \cap \Psi .\end{array}\right.\end{array}$$

Theorem 5.

If $({\rm Y},\Delta )$ and $(\Gamma ,\Psi )$ are $AIFSAIDs$ of Ω, then $({\rm Y},\Delta ){\sqcap }_E(\Gamma ,\Psi )$ is an $AIFSAID$ of Ω.

Proof. 

We know that $({\rm Y},\Delta ){\sqcap }_E(\Gamma ,\Psi )=(\Pi ,\Theta )$, where $\Theta =\Delta \cup \Psi$ and for every $\varpi \in \Theta$,

$$\begin{array}{c}\Pi \left(\varpi \right)=\left\{\begin{array}{cc}{\rm Y}\left[\varpi \right],& \varpi \in \Delta -\Psi ,\\ \Gamma \left[\varpi \right],& \varpi \in \Psi -\Delta ,\\ {\rm Y}\left[\varpi \right]\cap \Gamma \left[\varpi \right],& \varpi \in \Delta \cap \Psi .\end{array}\right.\end{array}$$

For any $\varpi \in \Theta$, if $\varpi \in \Delta -\Psi$, then $\Pi \left(\varpi \right)={\rm Y}\left(\varpi \right)$ is an $AIFAID$ of $\Omega$.

Likewise, if $\varpi \in \Psi -\Delta$, $\Pi \left(\varpi \right)=\Gamma \left(\varpi \right)$, which is an $AIFAID$ of $\Omega$.

Moreover if $\varpi \in \Theta$, such that $\varpi \in \Delta \cap \Psi$, then $\Pi \left(\varpi \right)={\rm Y}\left[\varpi \right]\cap \Gamma \left[\varpi \right]$ is also an $AIFAID$ of $\Omega$.

Therefore, $\Pi \left(\varpi \right)$ is an $AIFAID$ of $\Omega$.

Hence, $(\Pi ,\Theta )$ is an $AIFSAID$ of $\Omega$.

We deduce the following Corollary.

Corollary 1.

The “restricted intersection” of two $AIFSAIDs$ is an $AIFSAID$.

Definition 10.

Let $({\rm Y},\Delta )$ and $(\Gamma ,\Psi )$ be two $IFSSs$ over Ω. Subsequently, the "union" denoted by $({\rm Y},\Delta )\tilde{\cup }(\Gamma ,\Psi )$ is $(\Pi ,\Theta )$, where $\Theta =\Delta \cup \Psi$ and for every $\varpi \in \Theta$,

$$\begin{array}{c}\Pi \left(\varpi \right)=\left\{\begin{array}{cc}{\rm Y}\left[\varpi \right],& \varpi \in \Delta -\Psi ,\\ \Gamma \left[\varpi \right],& \varpi \in \Psi -\Delta ,\\ {\rm Y}\left[\varpi \right]\cup \Gamma \left[\varpi \right],& \varpi \in \Delta \cap \Psi .\end{array}\right.\end{array}$$

(1)

The union of two $AIFSAIDs$ is not necessarily an $AIFSAID$, as shown in the next example.

Example 3.

Let $\Omega =\{0,p,q,r,s\}$ with Cayley table given by:

⊙	0	p	q	r	s
0	0	0	q	r	s
p	p	0	q	r	s
q	q	q	0	s	r
r	r	r	s	0	q
s	s	s	r	q	0

Subsequently, $(\Omega ;\odot ,0)$ is a $BCI$-algebra.

Let $\Delta =\{\theta ,\vartheta ,\kappa ,\delta \}$ and $\Psi =\{\kappa ,\delta ,\eta \}$ be two collections of parameters and consider the $IFSS$$({\rm Y},\Delta )$ over Ω. Afterwards, ${\rm Y}\left[\theta \right],{\rm Y}\left[\vartheta \right],{\rm Y}\left[\kappa \right]$ and ${\rm Y}\left[\delta \right]$ are $IFSs$ over Ω given by:

${\rm Y}$	0	p	q	r	s
$\theta$	[0, 0.9]	[0, 0.9]	[0.3, 0.4]	[0.1, 0.4]	[0.3, 0.4]
$\vartheta$	[0.2, 0.6]	[0.2, 0.6]	[0.4, 0.3]	[0.4, 0.3]	[0.3, 0.5]
$\kappa$	[0.1, 0.8]	[0.1, 0.8]	[0.5, 0.2]	[0.3, 0.5]	[0.5, 0.2]
$\delta$	[0.2, 0.7]	[0.2, 0.7]	[0.3, 0.5]	[0.5, 0.3]	[0.5, 0.3]

Then ${\rm Y}\left[\varpi \right]$ is an $AIFAID$ of Ω with respect to $\theta ,\vartheta ,\kappa$, and δ.

Thus $({\rm Y},\Delta )$ is an $AIFSAID$ of Ω.

Now let $(\Gamma ,\Psi )$ be an $IFSS$ over Ω. Then $\Gamma \left[\kappa \right],\Gamma \left[\delta \right]$ and $\Gamma \left[\eta \right]$ are $IFSs$ over Ω given by:

$\Gamma$	0	p	q	r	s
$\kappa$	[0, 0.7]	[0, 0.7]	[0.3, 0.5]	[0.5, 0.2]	[0.5, 0.2]
$\delta$	[0.2, 0.6]	[0.2, 0.6]	[0.5, 0.2]	[0.5, 0.2]	[0.3, 0.4]
$\eta$	[0, 0.9]	[0, 0.9]	[0.3, 0.4]	[0.1, 0.6]	[0.3, 0.4]

Subsequently, $\Gamma \left[\varpi \right]$ is an $AIFAID$ of Ω with respect to κ, δ, and η.

Thus, $(\Gamma ,\Psi )$ is an $AIFSAID$ of Ω.

Note that $({\rm Y},\Delta )\tilde{\cup }(\Gamma ,\Psi )=(\Pi ,\Theta )$ is not an $AIFSAID$ of Ω based on $\kappa \in \Delta \cap \Psi .$ If $\Delta \cap \Psi =\varnothing$, then the union is an $AIFSAID$ of Ω proved in the next theorem.

Theorem 6.

Let $({\rm Y},\Delta )$ and $(\Gamma ,\Psi )$ be two $AIFSAIDs$ of Ω. If $\Delta \cap \Psi =\varnothing$, then $({\rm Y},\Delta )\tilde{\cup }(\Gamma ,\Psi )=(\Pi ,\Theta )$ is an $AIFSAID$ of Ω.

Proof. 

We know that $({\rm Y},\Delta )\tilde{\cup }(\Gamma ,\Psi )=(\Pi ,\Theta )$, where $\Theta =\Delta \cup \Psi$ and for every $\varpi \in \Theta$,

$$\begin{array}{c}\Pi \left(\varpi \right)=\left\{\begin{array}{cc}{\rm Y}\left[\varpi \right],& \varpi \in \Delta -\Psi ,\\ \Gamma \left[\varpi \right],& \varpi \in \Psi -\Delta ,\\ {\rm Y}\left[\varpi \right]\cup \Gamma \left[\varpi \right],& \varpi \in \Delta \cap \Psi .\end{array}\right.\end{array}$$

Because $\Delta \cap \Psi =\varnothing$, then either $\varpi \in \Delta -\Psi$ or $\varpi \in \Psi -\Delta$ for all $\varpi \in \Theta$.

If $\varpi \in \Delta -\Psi$, then $\Pi \left(\varpi \right)={\rm Y}\left(\varpi \right)$, which is an $AIFAID$ of $\Omega$.

Thus, $({\rm Y},\Delta )$ is an $AIFSAID$ of $\Omega$.

Similarly $\varpi \in \Psi -\Delta$, then $\Pi \left(\varpi \right)=\Gamma \left(\varpi \right)$ is an $AIFAID$ of $\Omega$.

Thus, $(\Gamma ,\Psi )$ is an $AIFSAID$ of $\Omega$.

Hence, $({\rm Y},\Delta )\tilde{\cup }(\Gamma ,\Psi )$ is an $AIFSAID$ of $\Omega$.

Definition 11.

Let $({\rm Y},\Delta )$ be an anti-soft $BCI$-algebra (abbr. $A{S}_{BCI}A$) over Ω. An $IFSS$$(\Gamma ,\Psi )$ over Ω is an $AIFSID$ of $({\rm Y},\Delta )$, written as $(\Gamma ,\Psi )\widetilde{▴}({\rm Y},\Delta )$, if $\Psi \subset \Delta$ and for any $\varpi \in \Psi$,

$\Gamma \left[\varpi \right]=\{({\xi }_{\Gamma \left[\varpi \right]}\left(l\right),{\zeta }_{\Gamma \left[\varpi \right]}\left(l\right)):l\in \Omega \}▴{\rm Y}\left[\varpi \right]$.

Definition 12.

Let $({\rm Y},\Delta )$ be an $A{S}_{BCI}A$ over Ω. An $IFSS$$(\Gamma ,\Psi )$ over Ω is an $AIFSAID$ of $({\rm Y},\Delta )$, denoted by $(\Gamma ,\Psi ){\widetilde{▴}}_a({\rm Y},\Delta )$, if $\Psi \subset \Delta$ and for any $\varpi \in \Psi$,

$\Gamma \left[\varpi \right]=\{({\xi }_{\Gamma \left[\varpi \right]}\left(l\right),{\zeta }_{\Gamma \left[\varpi \right]}\left(l\right)):l\in \Omega \}{▴}_a{\rm Y}\left[\varpi \right]$.

Example 4.

Let $\Omega =\{0,p,q,r,s\}$ with Cayley table:

⊙	0	p	q	r	s
0	0	0	q	r	s
p	p	0	q	r	s
q	q	q	0	s	r
r	r	r	s	0	q
s	s	s	r	q	0

Subsequently, $(\Omega ;\odot ,0)$ is a $BCI$-algebra.

Let $\Delta =\{\theta ,\vartheta ,\kappa \}$ be a set of parameters and let $({\rm Y},\Delta )$ be a soft set over Ω and so let ${\rm Y}\left[\theta \right]={\rm Y}\left[\vartheta \right]=\{0,q,r,s\}$, ${\rm Y}\left[\kappa \right]=\{0,q\}$, that are all sub-algebras of Ω.

Hence, $({\rm Y},\Delta )$ is an $A{S}_{BCI}A$ over Ω.

Let $(\Gamma ,\Psi )$ be an $IFSS$ over Ω, where $\Psi =\{\theta ,\vartheta \}$$\subset \Delta$. Afterwards, $\Gamma \left[\theta \right]$ and $\Gamma \left[\vartheta \right]$ are $IFSs$ in Ω defined by:

$\Gamma$	0	p	q	r	s
$\theta$	[0.2, 0.7]	[0.2, 0.7]	[0.2, 0.7]	[0.4, 0.1]	[0.4, 0.1]
$\vartheta$	[0.3, 0.7]	[0.3, 0.7]	[0.3, 0.7]	[0.5, 0.4]	[0.5, 0.4]

Afterwards, $\Gamma \left[\theta \right]=\{({\xi }_{\Gamma \left[\theta \right]}\left(l\right),{\zeta }_{\Gamma \left[\theta \right]}\left(l\right)):l\in \Omega \}$ and $\Gamma \left[\vartheta \right]=\{({\xi }_{\Gamma \left[\vartheta \right]}\left(l\right),{\zeta }_{\Gamma \left[\vartheta \right]}\left(l\right)):l\in \Omega \}$ are $AIFAIDs$ of Ω related to $\Gamma \left[\theta \right]$ and $\Gamma \left[\vartheta \right]$, respectively.

Hence, $(\Gamma ,\Psi ){\widetilde{▴}}_a({\rm Y},\Delta )$.

Any $AIFSAID$$(\Gamma ,\Psi )$ of an $A{S}_{BCI}A$$({\rm Y},\Delta )$ is an $AIFSID$ of $({\rm Y},\Delta )$, but the converse is not true, as proved by the next example.

Example 5.

Let $\Omega =\{0,p,q,r,s\}$ with Cayley table.

⊙	0	p	q	r	s
0	0	0	0	0	0
p	p	0	0	0	0
q	q	q	0	q	0
r	r	r	r	0	0
s	s	s	r	q	0

Subsequently, $(\Omega ;\odot ,0)$ is a “$BCK$-algebra” and, thus, a “$BCI$-algebra”.

Let $\Delta =\{\theta ,\vartheta ,\kappa ,\delta ,\eta \}$ be a set of parameters.

Let $({\rm Y},\Delta )$ be a soft set over Ω and so we let ${\rm Y}\left[\theta \right]=\Omega$, ${\rm Y}\left[\vartheta \right]={\rm Y}\left[\kappa \right]=\{0,q,r,s\}$ and ${\rm Y}\left[\delta \right]={\rm Y}\left[\eta \right]=\{0,q\}$, that are all subalgebras of Ω.

Hence, $({\rm Y},\Delta )$ is a $A{S}_{BCI}A$ over Ω.

Suppose that $(\Gamma ,\Psi )$ is an $IFSS$ over Ω, where $\Psi =\{\kappa ,\delta ,\eta \}\subset \Delta$. Afterwards, $\Gamma \left[\kappa \right],\Gamma \left[\delta \right]$ and $\Gamma \left[\eta \right]$ are an $IFSs$ in Ω represented by:

$\Gamma$	0	p	q	r	s
$\kappa$	[0, 0.7]	[0.1, 0.6]	[0.2, 0.5]	[0.3, 0.3]	[0.3, 0.3]
$\delta$	[0.1, 0.8]	[0.2, 0.7]	[0.3, 0.6]	[0.4, 0.4]	[0.4, 0.4]
$\eta$	[0.1, 0.5]	[0.2, 0.4]	[0.3, 0.3]	[0.4, 0.1]	[0.4, 0.1]

Subsequently, $(\Gamma ,\Psi )$ is an $AIFSID$ of $({\rm Y},\Delta )$, but since

${\xi }_{{\rm Y}\left[\kappa \right]}(r\odot q)={\xi }_{{\rm Y}\left[\kappa \right]}\left(r\right)=0.3\le 0.2={\xi }_{{\rm Y}\left[\kappa \right]}((q\odot 0)\odot (0\odot r))\vee {\xi }_{{\rm Y}\left[\varpi \right]}\left(0\right)$

and

${\zeta }_{{\rm Y}\left[\kappa \right]}(r\odot q)={\zeta }_{{\rm Y}\left[\kappa \right]}\left(r\right)=0.3\ge 0.5={\xi }_{{\rm Y}\left[\kappa \right]}((q\odot 0)\odot (0\odot r))\wedge {\zeta }_{{\rm Y}\left[\varpi \right]}\left(0\right)$.

i.e., $\Gamma \left[\kappa \right]=\{({\xi }_{\Gamma \left[\kappa \right]}\left(l\right),{\zeta }_{\Gamma \left[\kappa \right]}\left(l\right)):l\in \Omega \}$ is not an $AIFAID$ of Ω related to ${\rm Y}\left[\kappa \right]$.

Therefore $(\Gamma ,\Psi )$ is not an $AIFSAID$ of $A{S}_{BCI}A({\rm Y},\Delta )$.

Theorem 7.

Let $({\rm Y},\Delta )$ be an $A{S}_{BCI}A$ over Ω. If $(\Gamma ,\Psi )$ and $(\Pi ,\Lambda )$ are $AIFSAIDs$ of $({\rm Y},\Delta )$, then the “extended intersection" of $(\Gamma ,\Psi )$ and $(\Pi ,\Lambda )$ is an $AIFSAIDs$ of $({\rm Y},\Delta )$.

Proof. 

We know that $(\Gamma ,\Psi ){\sqcap }_E(\Pi ,\Lambda )=(\Xi ,\Theta )$, where $\Theta =\Psi \cup \Lambda \subset \Delta$ and for every $\varpi \in \Theta$,

$$\begin{array}{c}\Xi \left(\varpi \right)=\left\{\begin{array}{cc}\Gamma \left[\varpi \right],& \varpi \in \Psi -\Lambda ,\\ \Pi \left[\varpi \right],& \varpi \in \Lambda -\Psi ,\\ \Gamma \left[\varpi \right]\cap \Pi \left[\varpi \right],& \varpi \in \Psi \cap \Lambda .\end{array}\right.\end{array}$$

For any $\varpi \in \Theta$, if $\varpi \in \Psi -\Lambda$, then $\Xi \left[\varpi \right]=\Gamma \left[\varpi \right]=\{({\xi }_{\Gamma \left[\varpi \right]}\left(l\right),{\zeta }_{\Gamma \left[\varpi \right]}\left(l\right)):l\in \Omega \}{▴}_a{\rm Y}\left[\varpi \right]$, since $(\Gamma ,\Psi ){\widetilde{▴}}_a({\rm Y},\Delta )$.

Likewise, if $\varpi \in \Lambda -\Psi$, then $\Xi \left[\varpi \right]=\Pi \left[\varpi \right]=\{({\xi }_{\Pi \left[\varpi \right]}\left(l\right),{\zeta }_{\Pi \left[\varpi \right]}\left(l\right)):l\in \Omega \}{▴}_a{\rm Y}\left[\varpi \right]$, since $(\Pi ,\Lambda ){\widetilde{▴}}_a({\rm Y},\Delta )$.

Moreover if $\varpi \in \Theta$, such that $\varpi \in \Psi \cap \Lambda$, then $\Xi \left(\varpi \right)=\Gamma \left[\varpi \right]\cap \Pi \left[\varpi \right]=\{({\xi }_{\Gamma \left[\varpi \right]}\left(l\right)\vee {\xi }_{\Pi \left[\varpi \right]}\left(l\right)),({\zeta }_{\Gamma \left[\varpi \right]}\left(l\right)\wedge {\zeta }_{\Pi \left[\varpi \right]}\left(l\right))\}{▴}_a{\rm Y}\left[\varpi \right]$.

Therefore, $\Xi \left(\varpi \right){▴}_a{\rm Y}\left[\varpi \right]$ for any $\varpi \in \Theta$.

Hence, $(\Xi ,\Theta )=(\Gamma ,\Psi ){\sqcap }_E(\Pi ,\Lambda ){\widetilde{▴}}_a({\rm Y},\Delta )$.□

Next corollary follows directly.

Corollary 2.

Let $(\Gamma ,\Psi )$ and $(\Pi ,\Lambda )$ be two $AIFSAIDs$ of an $A{S}_{BCI}A$$({\rm Y},\Delta )$. If $\Psi \cap \Lambda =\varnothing$, then the “union” $(\Gamma ,\Psi )\tilde{\cup }(\Pi ,\Lambda )$ is an $AIFSAID$ of $({\rm Y},\Delta )$.

4. Characterization of Anti-Intuitionistic Fuzzy Soft a-Ideals

In this section, we give characterizations of an $AIFSAID$$({\rm Y},\Delta )$ over $\Omega$ while using the idea of a soft $(\gamma ,\nu )$-level set, $L({\rm Y}\left[\varpi \right];\gamma ;\nu )=\{l\in \Omega |{\xi }_{{\rm Y}\left[\varpi \right]}\left(l\right)\le \gamma \mathrm{and}{\zeta }_{{\rm Y}\left[\varpi \right]}\left(l\right)\ge \nu \}$, for any $\varpi \in \Delta$ and $\gamma ,\nu \in [0,1]$.

Theorem 8.

An $AIFSS$$({\rm Y},\Delta )$ over Ω is an $AIFSAID$ over $\Omega ⟺$ the non-empty soft $(\gamma ,\nu )$-level set, $L({\rm Y}\left[\varpi \right];\gamma ;\nu )=\{l\in \Omega |{\xi }_{{\rm Y}\left[\varpi \right]}\left(l\right)\le \gamma \mathrm{and}{\zeta }_{{\rm Y}\left[\varpi \right]}\left(l\right)\ge \nu \}$ is an a-ideal of Ω, for any $\varpi \in \Delta$ and $\gamma ,\nu \in [0,1]$.

Proof. 

Let $({\rm Y},\Delta )$ be an $AIFSAID$ over $\Omega$.

Afterwards, ${\rm Y}\left[\varpi \right]=\left\{({\xi }_{{\rm Y}\left[\varpi \right]}\left(l\right),{\zeta }_{{\rm Y}\left[\varpi \right]}\left(l\right))\right|l\in \Omega \}$ is an $AIFAID$ of $\Omega$, for any $\varpi \in \Delta$.

Let $L({\rm Y}\left[\varpi \right];\gamma ;\nu )=\{l\in \Omega |{\xi }_{{\rm Y}\left[\varpi \right]}\left(l\right)\le \gamma and{\zeta }_{{\rm Y}\left[\varpi \right]}\left(l\right)\ge \nu \}\ne \varnothing$, for any $\varpi \in \Delta$ and $\gamma ,\nu \in [0,1]$.

Subsequently, for any $l\in L({\rm Y}[\varpi ];\gamma ;\nu )$,

${\xi }_{{\rm Y}\left[\varpi \right]}\left(0\right)\le {\xi }_{{\rm Y}\left[\varpi \right]}\left(l\right)\le \gamma$ and ${\zeta }_{{\rm Y}\left[\varpi \right]}\left(0\right)\ge {\zeta }_{{\rm Y}\left[\varpi \right]}\left(l\right)\ge \nu$,

i.e., 0 $\in L({\rm Y}[\varpi ];\gamma ;\nu )$.

Now, let $(l\odot n)\odot (0\odot m)\in L({\rm Y}[\varpi ];\gamma ;\nu )$ and $n\in L({\rm Y}[\varpi ];\gamma ;\nu )$, for any $l,m,n\in \Omega$.

Subsequently, ${\xi }_{{\rm Y}\left[\varpi \right]}((l\odot n)\odot (0\odot m))\le \gamma ,{\xi }_{{\rm Y}\left[\varpi \right]}\left(n\right)\le \gamma$

and

${\zeta }_{{\rm Y}\left[\varpi \right]}((l\odot n)\odot (0\odot m))\ge \nu ,{\zeta }_{{\rm Y}\left[\varpi \right]}\left(n\right)\ge \nu$.

Thus, for any $l,m,n\in \Omega$,

${\xi }_{{\rm Y}\left[\varpi \right]}(m\odot l)\le {\xi }_{{\rm Y}\left[\varpi \right]}((l\odot n)\odot (0\odot m))\vee {\xi }_{{\rm Y}\left[\varpi \right]}\left(n\right)\le \gamma$.

${\zeta }_{{\rm Y}\left[\varpi \right]}(m\odot l)\ge {\zeta }_{{\rm Y}\left[\varpi \right]}((l\odot n)\odot (0\odot m))\wedge {\zeta }_{{\rm Y}\left[\varpi \right]}\left(n\right)\ge \nu$.

i.e., $m\odot l\in L({\rm Y}[\varpi ];\gamma ;\nu )$.

Hence, $L({\rm Y}[\varpi ];\gamma ;\nu )\ne \varnothing$ is an a-ideal of $\Omega$, for any $\varpi \in \Delta$ and $\gamma ,\nu \in [0,1]$.

Conversely assume that $L({\rm Y}[\varpi ];\gamma ;\nu )$ is an a-ideal of $\Omega$, for any $\varpi \in \Delta$ and $\gamma ,\nu \in [0,1]$.

If for some $l_0\in \Omega$ and ${\varpi }_0\in \Delta$, ${\xi }_{{\rm Y}\left[{\varpi }_0\right]}\left(0\right)>{\xi }_{{\rm Y}\left[{\varpi }_0\right]}\left(l_0\right)$ and ${\zeta }_{{\rm Y}\left[{\varpi }_0\right]}\left(0\right)<{\zeta }_{{\rm Y}\left[{\varpi }_0\right]}\left(l_0\right)$, then ${\xi }_{{\rm Y}\left[{\varpi }_0\right]}\left(0\right)>{\gamma }_0\ge {\xi }_{{\rm Y}\left[{\varpi }_0\right]}\left(l_0\right)$ and ${\zeta }_{{\rm Y}\left[{\varpi }_0\right]}\left(0\right)<{\nu }_0\le {\zeta }_{{\rm Y}\left[{\varpi }_0\right]}\left(l_0\right)$, for some ${\gamma }_0,{\nu }_0\in [0,1]$.

This implies that $l_0\in L({\rm Y}\left[{\varpi }_0\right];{\gamma }_0;{\nu }_0)$ and that $0\notin L({\rm Y}\left[{\varpi }_0\right];{\gamma }_0;{\nu }_0)$, this contradicts the hypothesis that $L({\rm Y}\left[{\varpi }_0\right];{\gamma }_0;{\nu }_0)$ is an a-ideal of $\Omega$.

Thus ${\xi }_{{\rm Y}\left[\varpi \right]}\left(0\right)\le {\xi }_{{\rm Y}\left[\varpi \right]}\left(l\right)$ and ${\zeta }_{{\rm Y}\left[\varpi \right]}\left(0\right)\ge {\zeta }_{{\rm Y}\left[\varpi \right]}\left(l\right)$, for any $\varpi \in \Delta$ and $l\in \Omega$.

Moreover, if there are elements $l_0,m_0,n_0\in \Omega$ and ${\varpi }_0\in \Delta$, such that

${\xi }_{{\rm Y}\left[{\varpi }_0\right]}(m_0\odot l_0)>{\xi }_{{\rm Y}\left[{\varpi }_0\right]}((l_0\odot n_0)\odot (0\odot m_0))\vee {\xi }_{{\rm Y}\left[{\varpi }_0\right]}\left(n_0\right)$

and

${\zeta }_{{\rm Y}\left[{\varpi }_0\right]}(m_0\odot l_0)<{\zeta }_{{\rm Y}\left[{\varpi }_0\right]}((l_0\odot n_0)\odot (0\odot m_0))\wedge {\zeta }_{{\rm Y}\left[\varpi \right]}\left(n_0\right)$.

Afterwards, for some ${\gamma }_0,{\nu }_0\in [0,1]$,

${\xi }_{{\rm Y}\left[{\varpi }_0\right]}(m_0\odot l_0)>{\gamma }_0\ge {\xi }_{{\rm Y}\left[{\varpi }_0\right]}((l_0\odot n_0)\odot (0\odot m_0))\vee {\xi }_{{\rm Y}\left[{\varpi }_0\right]}\left(n_0\right)$

and

${\zeta }_{{\rm Y}\left[{\varpi }_0\right]}(m_0\odot l_0)<{\nu }_0\le {\zeta }_{{\rm Y}\left[{\varpi }_0\right]}((l_0\odot n_0)\odot (0\odot m_0))\wedge {\zeta }_{{\rm Y}\left[\varpi \right]}\left(n_0\right)$.

i.e., $m_0\odot l_0\notin L({\rm Y}\left[{\varpi }_0\right];{\gamma }_0;{\nu }_0)$, again a contradiction.

Thus, for any $l,m,n\in \Omega$ and for any $\varpi \in \Delta$,

${\xi }_{{\rm Y}\left[\varpi \right]}(m\odot l)\le {\xi }_{{\rm Y}\left[\varpi \right]}((l\odot n)\odot (0\odot m))\vee {\xi }_{{\rm Y}\left[\varpi \right]}\left(n\right)$

and

${\zeta }_{{\rm Y}\left[\varpi \right]}(m\odot l)\ge {\zeta }_{{\rm Y}\left[\varpi \right]}((l\odot n)\odot (0\odot m))\wedge {\zeta }_{{\rm Y}\left[\varpi \right]}\left(n\right)$

i.e., ${\rm Y}\left[\varpi \right]=\left\{({\xi }_{{\rm Y}\left[\varpi \right]}\left(l\right),{\zeta }_{{\rm Y}\left[\varpi \right]}\left(l\right))\right|l\in \Omega \}$ is an $AIFAID$ of $\Omega$, for any $\varpi \in \Delta$.

Hence, $({\rm Y},\Delta )$ is an $AIFSAID$ over $\Omega$.□

From the above Theorem we get the following corollary.

Corollary 3.

An $AIFSS$$({\rm Y},\Delta )$ over Ω is an $AIFSAID$ over $\Omega ⟺$ the non-empty soft $(\gamma ,\nu )$-level set, $L({\rm Y}\left[\varpi \right];\gamma ;\nu )=\{l\in \Omega |{\xi }_{{\rm Y}\left[\varpi \right]}\left(l\right)\le \gamma \mathrm{and}{\zeta }_{{\rm Y}\left[\varpi \right]}\left(l\right)\ge \nu \}$, is an a-ideal of Ω, for any $\varpi \in \Delta$ and $\gamma ,\nu \in (1/2,1]$.

Theorem 9.

A non-empty soft $(\gamma ,\nu )$-level set, $L({\rm Y}\left[\varpi \right];\gamma ;\nu )=\{l\in \Omega |{\xi }_{{\rm Y}\left[\varpi \right]}\left(l\right)\le \gamma \mathrm{and}{\zeta }_{{\rm Y}\left[\varpi \right]}\left(l\right)\ge \nu \}$, is an a-ideal of Ω, for any $\varpi \in \Delta$ and $\gamma ,\nu \in (1/2,1]⟺$ the following conditions hold:

(i) $({\xi }_{{\rm Y}\left[\varpi \right]}\left(0\right)\vee 1/2)\le {\xi }_{{\rm Y}\left[\varpi \right]}\left(l\right)$ and $({\zeta }_{{\rm Y}\left[\varpi \right]}\left(0\right)\vee 1/2)\ge {\zeta }_{{\rm Y}\left[\varpi \right]}\left(l\right)$,

(ii) $({\xi }_{{\rm Y}\left[\varpi \right]}(m\odot l)\vee 1/2)\le {\xi }_{{\rm Y}\left[\varpi \right]}((l\odot n)\odot (0\odot m))\vee {\xi }_{{\rm Y}\left[\varpi \right]}\left(n\right)$,

(iii) $({\zeta }_{{\rm Y}\left[\varpi \right]}(m\odot l)\vee 1/2)\ge {\zeta }_{{\rm Y}\left[\varpi \right]}((l\odot n)\odot (0\odot m))\wedge {\zeta }_{{\rm Y}\left[\varpi \right]}\left(n\right)$,

for any $\varpi \in \Delta$ and $l,m,n\in \Omega$.

Proof. 

Let the non-empty soft $(\gamma ,\nu )$-level set, $L({\rm Y}\left[\varpi \right];\gamma ;\nu )=\{l\in \Omega |{\xi }_{{\rm Y}\left[\varpi \right]}\left(l\right)\le \gamma \mathrm{and}{\zeta }_{{\rm Y}\left[\varpi \right]}\left(l\right)\ge \nu \}$ be an a-ideal of $\Omega$, for any $\varpi \in \Delta$ and $\gamma ,\nu \in (1/2,1]$.

If for some $l_0\in \Omega$ and ${\varpi }_0\in \Delta$,

$({\xi }_{{\rm Y}\left[{\varpi }_0\right]}\left(0\right)\vee 1/2)>{\xi }_{{\rm Y}\left[{\varpi }_0\right]}\left(l_0\right)$ and $({\xi }_{{\rm Y}\left[{\varpi }_0\right]}\left(0\right)\vee 1/2)<{\xi }_{{\rm Y}\left[{\varpi }_0\right]}\left(l_0\right)$.

Then there are ${\gamma }_0,{\nu }_0\in (1/2,1]$, such that

$({\xi }_{{\rm Y}\left[{\varpi }_0\right]}\left(0\right)\vee 1/2)>{\gamma }_0\ge {\xi }_{{\rm Y}\left[{\varpi }_0\right]}\left(l_0\right)$ and $({\zeta }_{{\rm Y}\left[{\varpi }_0\right]}\left(0\right)\vee 1/2)<{\nu }_0\le {\zeta }_{{\rm Y}\left[{\varpi }_0\right]}\left(l_0\right)$.

This implies that ${\xi }_{{\rm Y}\left[{\varpi }_0\right]}\left(0\right)>{\gamma }_0\ge {\xi }_{{\rm Y}\left[{\varpi }_0\right]}\left(l_0\right)$ and ${\zeta }_{{\rm Y}\left[{\varpi }_0\right]}\left(0\right)<{\nu }_0\le {\zeta }_{{\rm Y}\left[{\varpi }_0\right]}\left(l_0\right)$.

i.e., $l_0\in L({\rm Y}\left[{\varpi }_0\right];{\gamma }_0;{\nu }_0)$ but $0\notin L({\rm Y}\left[{\varpi }_0\right];{\gamma }_0;{\nu }_0)$, which gives a contradiction to the assumption that $L({\rm Y}\left[{\varpi }_0\right];{\gamma }_0;{\nu }_0)$ is an a-ideal of $\Omega$, for any ${\varpi }_0\in \Delta$ and ${\gamma }_0,{\nu }_0\in (1/2,1]$.

Thus, (i) is valid.

Moreover, if there are elements $l_0,m_0,n_0\in \Omega$ and ${\varpi }_0\in \Delta$, such that

$({\xi }_{{\rm Y}\left[{\varpi }_0\right]}(m_0\odot l_0)\vee 1/2)>{\xi }_{{\rm Y}\left[{\varpi }_0\right]}((l_0\odot n_0)\odot (0\odot m_0))\vee {\xi }_{{\rm Y}\left[{\varpi }_0\right]}\left(n_0\right)$

and

$({\zeta }_{{\rm Y}\left[{\varpi }_0\right]}(m_0\odot l_0)\vee 1/2)<{\zeta }_{{\rm Y}\left[{\varpi }_0\right]}((l_0\odot n_0)\odot (0\odot m_0))\wedge {\zeta }_{{\rm Y}\left[\varpi \right]}\left(n_0\right)$.

Subsequently, for some ${\gamma }_0,{\nu }_0\in (1/2,1]$,

$({\xi }_{{\rm Y}\left[{\varpi }_0\right]}(m_0\odot l_0)\vee 1/2)>{\gamma }_0\ge {\xi }_{{\rm Y}\left[{\varpi }_0\right]}((l_0\odot n_0)\odot (0\odot m_0))\vee {\xi }_{{\rm Y}\left[{\varpi }_0\right]}\left(n_0\right)$

and

$({\zeta }_{{\rm Y}\left[{\varpi }_0\right]}(m_0\odot l_0)\vee 1/2)<{\nu }_0\le {\zeta }_{{\rm Y}\left[{\varpi }_0\right]}((l_0\odot n_0)\odot (0\odot m_0))\wedge {\zeta }_{{\rm Y}\left[\varpi \right]}\left(n_0\right)$.

i.e., ${\xi }_{{\rm Y}\left[{\varpi }_0\right]}(m_0\odot l_0)>{\gamma }_0\ge {\xi }_{{\rm Y}\left[{\varpi }_0\right]}((l_0\odot n_0)\odot (0\odot m_0))\vee {\xi }_{{\rm Y}\left[{\varpi }_0\right]}\left(n_0\right)$

and

${\zeta }_{{\rm Y}\left[{\varpi }_0\right]}(m_0\odot l_0)<{\nu }_0\le {\zeta }_{{\rm Y}\left[{\varpi }_0\right]}((l_0\odot n_0)\odot (0\odot m_0))\wedge {\zeta }_{{\rm Y}\left[\varpi \right]}\left(n_0\right)$.

i.e., $((l_0\odot n_0)\odot (0\odot m_0))\in L({\rm Y}\left[{\varpi }_0\right];{\gamma }_0;{\nu }_0)$

and

$n_0\in L({\rm Y}\left[{\varpi }_0\right];{\gamma }_0;{\nu }_0)$ but $(m_0\odot l_0)\notin L({\rm Y}\left[{\varpi }_0\right];{\gamma }_0;{\nu }_0)$, which—again—contradicts the assumption that $L({\rm Y}\left[{\varpi }_0\right];{\gamma }_0;{\nu }_0)$ is an a-ideal of $\Omega$, for any ${\varpi }_0\in \Delta$ and ${\gamma }_0,{\nu }_0\in (1/2,1]$.

Hence, (ii) and (iii) are valid.

Conversely, suppose that the conditions (i), (ii), and (iii) are valid.

Let $L({\rm Y}\left[\varpi \right];\gamma ;\nu )=\{l\in \Omega |{\xi }_{{\rm Y}\left[\varpi \right]}\left(l\right)\le \gamma and{\zeta }_{{\rm Y}\left[\varpi \right]}\left(l\right)\ge \nu \}\ne \varnothing$, for any $\varpi \in \Delta$ and $\gamma ,\nu \in (1/2,1]$.

Subsequently, for any $l\in L({\rm Y}[\varpi ];\gamma ;\nu )$,

$({\xi }_{{\rm Y}\left[\varpi \right]}\left(0\right)\vee 1/2)\le {\xi }_{{\rm Y}\left[\varpi \right]}\left(l\right)\le \gamma$ and

$({\zeta }_{{\rm Y}\left[\varpi \right]}\left(0\right)\vee 1/2)\ge {\zeta }_{{\rm Y}\left[\varpi \right]}\left(l\right)\ge \nu$

which implies ${\xi }_{{\rm Y}\left[\varpi \right]}\left(0\right)\le \gamma$ and ${\zeta }_{{\rm Y}\left[\varpi \right]}\left(0\right)\ge \nu$.

Thus, $0\in L({\rm Y}[\varpi ];\gamma ;\nu )$.

Now let $(l\odot n)\odot (0\odot m)\in L({\rm Y}[\varpi ];\gamma ;\nu )$ and $n\in L({\rm Y}[\varpi ];\gamma ;\nu )$, for any $l,m,n\in \Omega$.

Subsequently, ${\xi }_{{\rm Y}\left[\varpi \right]}((l\odot n)\odot (0\odot m))\le \gamma$, ${\xi }_{{\rm Y}\left[\varpi \right]}\left(n\right)\le \gamma$

and

${\zeta }_{{\rm Y}\left[\varpi \right]}((l\odot n)\odot (0\odot m))\ge \nu$, ${\zeta }_{{\rm Y}\left[\varpi \right]}\left(n\right)\ge \nu$.

Thus, from (ii), we get,

$({\xi }_{{\rm Y}\left[\varpi \right]}(m\odot l)\vee 1/2)\le {\xi }_{{\rm Y}\left[\varpi \right]}((l\odot n)\odot (0\odot m))\vee {\xi }_{{\rm Y}\left[\varpi \right]}\left(n\right)\le \gamma$

and

$({\zeta }_{{\rm Y}\left[\varpi \right]}(m\odot l)\vee 1/2)\ge {\zeta }_{{\rm Y}\left[\varpi \right]}((l\odot n)\odot (0\odot m))\wedge {\zeta }_{{\rm Y}\left[\varpi \right]}\left(n\right)\ge \nu$.

This implies, ${\xi }_{{\rm Y}\left[\varpi \right]}(m\odot l)\le \gamma$ and ${\zeta }_{{\rm Y}\left[\varpi \right]}(m\odot l)\ge \nu$.

Thus, $m\odot l\in L({\rm Y}[\varpi ];\gamma ;\nu )$.

Therefore, $L({\rm Y}[\varpi ];\gamma ;\nu )$ is an a-ideal of $\Omega$, for any $\varpi \in \Delta$ and $\gamma ,\nu \in (1/2,1].$

5. Conclusions

The notion of anti-intuitionistic fuzzy soft a-ideal (abbr. $AIFSAID$) is introduced and studied over a $BCI$-algebra $\Omega$. We proved that any $AIFSAID$ is an anti-intuitionistic fuzzy soft ideal (abbr. $AIFSID$) of $\Omega$ and that the converse is not always true. We proved that the operations “AND”, “extended intersection”, and “restricted intersection” between any two $AIFSAIDs$ of $\Omega$, is also an $AIFSAID$ of $\Omega$ whereas the “union” is not necessarily an $AIFSAID$. Moreover, characterizations of $AIFSAID$ using the concept of a soft level set were given.