Anti-Intuitionistic Fuzzy Soft a-Ideals Applied to BC I -Algebras

: 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.


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 Definition 5. [4] Let Π be a collection of parameters and let IΥ(Ω) indicate the collection of all intuitionistic fuzzy sets in Ω. Subsequently, (Υ, ∆) is called an intuitionistic fuzzy soft set over Ω, where ∆ ⊆ Π and Υ : ∆ → IΥ(Ω).

Anti-Intuitionistic Fuzzy Soft a-Ideal
In what follows, we write Ω to denote a BCI-algebra (Ω; , 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 (Υ, ∆) be an intuitionistic fuzzy soft set (abbr. IFSS). Afterwards, (Υ, ∆) is an anti-intuitionistic fuzzy soft ideal (abbr.
∈ ∆} is an AIFID of Ω satisfies the following assertions: for all l, m, n ∈ Ω and ∈ ∆.
Example 1. Suppose that there are four patients in the initial universe set Ω = {p 1 , p 2 , p 3 , p 4 } given by Afterwards, (Ω; , p 1 ) is a BCI-algebra. Let a set of parameters, we consider ∆ = { 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.  Proof. Let (Υ, ∆) be an AIFSAID of Ω. Subsequently, Υ[ ] = {(ξ Υ[ ] (l), ζ Υ[ ] (l)) : l ∈ Ω and ∈ ∆} is an AIFAID of Ω. Thus, for every l, m, n ∈ Ω and ∈ ∆,
Thus, for every l, m, n ∈ Ω and ∈ ∆, 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.
We deduce the following Corollary.

Corollary 1.
The "restricted intersection" of two AIFSAIDs is an AIFSAID.
From the above Theorem we get the following corollary.

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