Int. J. Intelligent Systems Technologies and Applications, Vol. 16, No. 3, 2017
Application of (α, β) -soft intersectional sets on BCK/BCI-algebras Chiranjibe Jana* and Madhumangal Pal Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore 721102, India Email:
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[email protected] *Corresponding author Abstract: In this paper, the notion of (α, β)-soft intersectional set is introduced and explained with some examples. Then, some properties of (α, β)-soft intersectional BCK/BCI-algebras and (α, β)-soft intersectional BCK/BCIideals are investigated. Strong (α, β)- soft intersectional BCI-algebra is defined on the basis of BCI-subalgebras. The concept of an (α, β)-intersectional tsupport of a soft set is introduced and proposed a condition for an (α, β)intersectional t-support to be a soft BCK/BCI-subalgebras and soft ideal of a BCK/BCI-algebras. Also, it is shown that the image and inverse image of a BCK/BCI-algebras are also (α, β)-soft intersectional BCK/BCI-algebras. Again, (α, β)-soft intersectional BCK/BCI-ideal is developed on the basis of a homomorphism of BCK/BCI-algebras. Keywords: BCK/BCI-algebras, soft set; (α, β)-soft intersectional set; (α, β)-soft intersectional BCK/BCI-algebras; (α, β)-intersectional t-support; soft (pre-)image. Reference to this paper should be made as follows: Jana, C. and Pal, M. (2017) ‘Application of (α, β)-soft intersectional sets on BCK/BCI-algebras’, Int. J. Intelligent Systems Technologies and Applications, Vol. 16, No. 3, pp.269–288. Biographical notes: Chiranjibe Jana received his Bachelor of Science degree with honours in Mathematics in 2007 from Midnapore College, Pashim Medinipur, West Bengal, India and Master of Science degree in Mathematics in 2009 from Vidyasagar University, West Bengal, India. His research interests include fuzzy sets, fuzzy algebra, soft algebraic structures and decision making theory. Madhumangal Pal is a Professor of Applied Mathematics, Vidyasagar University, West Bengal, India. He has published more than 250 research papers in international and national journals. He has produced 26 PhD. His specialisations include computational and fuzzy graph theory, genetic algorithms and parallel algorithms, fuzzy matrices and fuzzy algebra. He is the author of eight books published from India and UK and these are written for undergraduate and postgraduate students and other professionals. He is the Editor-in-Chief of the
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C. Jana and M. Pal Journal of Physical Sciences and Annals of Pure and Applied Mathematics. He is the Member of the editorial board of several reputed journals. He has visited China, London, Greece, Hong Kong, Thailand, Malaysia, UAE, Bangladesh for academic purpose.
1 Introduction Theory of algebraic structure has a vital role in applications of many disciplines such as automata theory, graph theory, computer science, signal processing, quantum physics, control engineering, discrete mathematics and so on. In algebraic point of view, the ideal theory is a very useful algebraic structure in many fields. The study of BCK/BCI-algebras was initiated by Imai and Iseki (1966); Iseki (1966) in 1966 as a generalisation of the concept of set-theoretic difference and propositional calculus. Chaudhry (1990) introduced the notions of positive implicative and weakly implicative ideals on BCI-algebra. Jana et al. (2015a, 2015b, 2015c, 2016), Jana and Pal (2016), Jana (2015), Jana and Senapati (2015) and Senapati et al. (2015) have done lot of works on BCK/BCI and B/BG/G-algebras which is related to these algebras. The characterisation of various problems in system identification are essentially non-probabilistic in nature (Zadeh, 1962). In response to this situation, Zadeh (1965) introduced fuzzy set theory as an alternative to probability theory. Uncertainty is an attribute of information. The classical methods cannot be dealt modelling of uncertain data in economics, engineering, environmental science, sociology and information sciences successfully because they have inherent difficulties and have troubled the usual theoretical approaches. To overcome these difficulties, Molodtsov (1999) introduced the concept of soft set theory as a new mathematical tool for dealing with uncertainties and also, pointed out for the applications of soft sets in several directions. At present, works on the soft set theory are progressing rapidly and have many applications in real life problems. Maji et al. (2002, 2003) described the application of soft set theory in a decision-making problem and studied several operations on the theory of soft sets. Ali et al. (2008) studied some new operations on soft set theory. The new definition of soft parameterisations reduction, and compared this definition with the related concept of attributes reduction in rough set theory introduced by Chen et al. (2005). Ali et al. (2008) introduced fuzzy parameterised (F P ) soft sets and their related properties. They proposed a decision-making method based on F P -soft set theory, and provided an example which shows that the method can be successfully applied to the problems that contain uncertainties. Çagman (2007) described soft groups, Acar et al. (2010) and Acar and Özürk (2015) introduced the notions of soft rings and irreducible and prime soft ideals of BCK/BCI-algebras, Çagman et al. (2012) studied soft-int groups, Feng et al. (2008) described soft semirings and Sezgin et al. (2012) introduced the notion of soft intersection near-rings. Now, it has been applied to the discipline of information sciences, decision support systems, knowledge systems, decision making, and so on (Çagman and Enginoglu, 2010, 2010b; Feng et al., 2008; Zou and Xiao, 2008). It is well known that BCK/BCI-algebras are two classes of algebras of logic. After the initiation of BCK/BCI-algebras proposed by Imai and Iseki (1966) and Iseki (1966) in 1966, many researchers studied it in different disciplines dealing with uncertainty. Jun (2008) introduced the notions of soft BCK/BCI-algebras. Jun et al. (2009a, 2009b) proposed a soft set theory to study ideals in d-algebras and pseudo d-algebras.
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Jun et al. (2008) described applications of soft sets in ideal theory of BCK/BCI-algebras. Jun et al. (2009c) studied soft p-ideal of soft BCI-algebras. Recently, Jun et al. (2012, 2013) studied intersectional soft sets and applications to BCK/BCI-algebra and intersectional soft BCK/BCI-ideals and Zhan et al. (2014, 2015) and Zhan and Yu (2016) studied concept of (M, N )-soft union sets in hemirings and characterises in hemiring details. Motivated by the above works and best of our knowledge there is no work available on (α, β)-soft intersectional BCK/BCI-algebras. For this reason, we developed theories for (α, β)-soft intersectional BCK/BCI-algebras. ln this paper, we introduced the concept of (α, β)-soft intersectional sets and given several examples. We obtained some useful results by applying this concept in BCK/BCIalgebras and BCK/BCI-ideals.
2 Preliminaries In this section, we introduced some elementary aspects that are necessary for this paper. For more information regarding BCK/BCI-algebras reader may see the books (Huang et al., 2016; Jun et al., 2012). By a BCI-algebras we mean an algebra (X, ∗, 0) of type (2, 0) satisfying the following axioms for all x, y, z ∈ X: (C1 ) ((x ∗ y) ∗ (x ∗ z)) ∗ (z ∗ y) = 0 (C2 ) (x ∗ (x ∗ y)) ∗ y = 0 (C3 ) x ∗ x = 0 (C4 ) x ∗ y = 0 and y ∗ x = 0 imply x = y. If a BCI-algebra X satisfies the following identity: (C5 ) 0 ∗ x = 0. then X is called a BCK-algebra. Any BCK/BCI-algebras satisfies the following axioms: for all x, y, z ∈ X (C6 ) x ∗ 0 = x (C7 ) x ≤ y ⇒ x ∗ z ≤ y ∗ z, z ∗ y ≤ z ∗ x (C8 ) (x ∗ y) ∗ z = (x ∗ z) ∗ (C9 ) (x ∗ z) ∗ (y ∗ z) ≤ x ∗ y. The partial ordering is defined as x ≤ y if and only if x ∗ y = 0. A non-empty subset S of a BCK/BCI-algebras X is called BCK/BCI-subalgebras of X if x ∗ y ∈ S for all x, y ∈ X. A mapping f : X → Y of BCK/BCI-algebras is called a homomorphism if f (x ∗ y) = f (x) ∗ f (y) for all x, y ∈ X. A subset A of a BCK/BCI-algebras X is called a BCK/BCI-ideal of X if it satisfies the following for all x, y ∈ X (C10 ) 0 ∈ A (C11 ) y ∈ A and x ∗ y ∈ A ⇒ x ∈ A.
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3 Soft sets and soft intersectional BCK/BCI -algebras In this section, we review some concepts on soft sets and soft intersectional BCK/BCIalgebras which are required to develop the proposed work. In what follows, let U be the universal set and E be the set of parameters. Let P(U ) be the power set of U and A ⊂ E. Definition 3.1 (Molodtsov, 1999): A pair (F, E) is called a soft set over U if F is a mapping given by F : E → P(U ). In other words, a soft set over the universe U is a parameterised family of subsets of the universal set U . For ε ∈ A, F(ε) may be considered as the set of ε-elements of the soft set (F, A), or as the set of ε-approximate elements of the soft set. Definition 3.2 (Çagman and Enginoglu, 2010a; Molodtsov, 1999): For a subset A of E, a soft set (F, E) over U satisfying the following condition:
non-empty
F(x) = ∅ for all x ̸∈ A is called A-soft set over U and is denoted by FA so, an A-soft set FA over U is a function FA : E → P(U ) such that FA (x) = ∅ for all x ̸∈ A. A soft set over U can be represented by the set of ordered pair: FA = {(x, FA (x)) : x ∈ E, FA (x) ∈ P (U )}. It is noted that a soft set is a parameterised family of subsets of the set U . A soft set FA (x) may be arbitrary. Some of them may be empty, and some may have nonempty intersection. We denote the set of all soft sets over U by S(U ). Definition 3.3: Let FA ∈ S(U ). If FA (x) = ∅ for all x ∈ E, then FA is called an empty soft set and denoted by ΦA . If FA (x) = U for all x ∈ A, then FA is called an A-universal soft set and denoted by FAe. If FA (x) = U and A = E for all x ∈ E, then FAe is called a universal soft set and denoted by FEe . Definition 3.4 (Çagman and Enginoglu, 2010a): Let FA , FB ∈ S(U ). Then FA is a soft e B , if FA (x) ⊆ FB (x) for all x ∈ E. The soft sets subset of FB and is denoted by FA ⊆F FA and FB are called soft equal if FA (x) = FB (x) for all x ∈ E, denoted by FA = FB . Definition 3.5 (Çagman and Enginoglu, 2010a): The intersection of two soft sets FA and ˜ FB = FA∩B , where FA∩B (x) = FA (x) ∩ FB (x) for all x ∈ E. FA is defined by FA ∩ ˜ FB = FA∪B , where FA∪B (x) = The union of two soft sets FA and FB is defined by FA ∪ FA (x) ∪ FB (x) for all x ∈ E. Proposition 3.6 (Çagman and Enginoglu, 2010a): Let FA ∈ S(U ). Then e FA = FA , FA ∩ e FA = FA (i) FA ∪ e ΦA = FA , FA ∩ e ΦA = ΦA (ii) FA ∪ e FE = FE , FA ∩ e FE = FA (iii) FA ∪ c ce c e FA (iv) FA ∪ = FE , FA ∪ ˜ FA = ΦA , where ΦA is an empty set.
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Definition 3.7 (Jun, 2008): Let (F, A) be a soft set over X. Then (F, A) is called a soft BCK/BCI-algebras over X if F(x) is a BCK/BCI-subalgebra of X for all x ∈ X. Definition 3.8 (Jun et al., 2012): Let X be a BCK/BCI-algebras. For a given subalgebra A of X, let FA be an A-soft set over U . Then FA is called a A-soft intersectional BCK/BCI-algebras over U if it satisfies the following condition: (1) FA (x) ∩ FA (y) ⊆ FA (x ∗ y) for all x, y ∈ X. Definition 3.9 (Jun et al., 2012): Let X be a BCK/BCI-algebras. For a given subalgebra A of X, let FA be an A-soft set over U . Then FA is called a A-soft intersectional BCK/BCI-ideal over U if it satisfies the following conditions: (1) FA (x) ⊆ FA (0) (2) FA (x ∗ y) ∩ FA (y) ⊆ FA (x) for all x, y ∈ X. Definition 3.10 (Jun et al., 2013): For any BCK/BCI-algebras X and Y , let FX and FY be soft sets over the common universe U and Ψ is a function from X to Y . (1) Then soft image of FX under Ψ, denoted by Ψ(FX ) is a soft set over U defined by Ψ(FX ) = {(y, Ψ(FX )(y) : y ∈ Y, Ψ(FX )(y) ∈ P(U )}, where {∪ {FX (x)| x ∈ X, Ψ(x) = y} if Ψ−1 (y) ̸= ∅ Ψ(FX )(y) = ∅ otherwise. (2) The soft set Ψ−1 (FY ) = {(x, Ψ−1 (FY )(x)) : x ∈ X, Ψ−1 (FY )(x) ∈ P(U )}, where Ψ−1 (FY )(x) = FY (Ψ(x)), is called soft pre-image of FY under Ψ.
4 (α, β)-soft intersectional set In this section, let U be the initial universe set and E be the set of parameters, and ‘ ′ be the binary operation. We take S(U ) be the set of all soft sets. From now let, ∅ ⊆ α ⊂ β ⊆ U . Definition 4.1: For any non-empty subset A of E, the soft set FA ∈ S(U ). Then, for all x, y ∈ A, the soft set FA is called an (α, β)-soft intersectional set over U if it satisfies the condition: FA (x y) ∪ α ⊇ FA (x) ∩ FA (y) ∩ β. Example 4.2: We consider five houses in the initial universal set U which is given by U = {h1 , h2 , h3 , h4 , h5 }.
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Let the set of parameters E = {ξ1 , ξ2 , ξ3 , ξ4 } be the status of the set of houses which follows for the parameters ‘cheap’, ‘expensive’, ‘in the flooded area’ and ‘in urban area’ respectively, with following binary operation ξ1 ξ2 ξ3 ξ4
ξ1 ξ1 ξ2 ξ3 ξ4
ξ2 ξ2 ξ1 ξ4 ξ3
ξ3 ξ3 ξ4 ξ3 ξ4
ξ4 ξ4 ξ3 ξ4 ξ3
1
For a subset A = {ξ1 , ξ3 , ξ4 } of E, consider a soft set (FA , A) over U defined as follows: FA (ξ1 ) = {h1 , h3 , h4 , h5 }, FA (ξ3 ) = {h1 , h3 , h5 } and FA (ξ4 ) = {h1 , h2 , h4 , h5 }. Then it is examine that (FA , A) is an (α, β)-soft intersectional set over U for β = {h1 , h2 , h3 , h5 } and α = {h1 , h2 , h3 }.
2
Let B = {ξ1 , ξ2 , ξ3 }, then soft set (FB , B) over U is defined as FB (ξ1 ) = {h1 , h2 , h3 , h4 , h5 }, FB (ξ2 ) = {h2 , h5 }, FB (ξ3 ) = {h1 , h4 , h5 } and FB (ξ4 ) = ∅ is an (α, β)-soft intersectional set over U for β = {h1 , h2 , h3 , h5 } and α = {h1 , h2 , h5 }.
2
The soft set (FH , H), where H = {ξ1 , ξ2 , ξ3 } is a subset of E defined by FH (ξ1 ) = {h1 , h2 , h3 , h4 , h5 }, FH (ξ2 ) = {h2 , h3 , h5 }, FH (ξ3 ) = {h1 , h3 , h4 , h5 } and FA (ξ4 ) = ∅ is not an (α, β)-soft intersectional set over U , where β = {h2 , h3 , h4 , h5 } and α = {h2 , h4 , h5 }, because FH (ξ2 ) ∩ FH (ξ3 ) ∩ β = {h3 , h5 } * {h2 , h4 , h5 } = FH (ξ2 ξ3 ) ∪ α.
Theorem 4.3: For soft sets FA , FB ∈ S(U ) such that FA is a soft subset of FB . If FB is an (α, β)-soft intersectional set over U , then FA is so. Proof: Let x, y ∈ A such that x y ∈ A. Then, x y ∈ B as A ⊆ B. Thus, FA (x) ∩ FA (y) ∩ β = FB (x) ∩ FB (y) ∩ β ⊆ FB (x y) ∪ α = FB (x) ∪ FB (y)FA (x y) ∪ α = FA (x y) ∪ α. Therefore, FA is an (α, β)-soft intersectional set over U . The converse of Theorem 5.7 is not true in general as seen in the following example. Example 4.4: Let U be the universal set and E be the set of parameters which is given in Example 4.2, considering the (α, β)-soft intersectional set (FB , B) illustrate in Example 4.2(2). Let (UE , E) be a soft set given as UE (ξ1 ) = {h1 , h2 , h3 , h4 , h5 }, UE (ξ2 ) = {h2 , h3 , h5 }, UE (ξ3 ) = {h1 , h3 , h4 } and UE (ξ4 ) = {h1 , h2 , h4 } is not an (α, β)-soft intersectional set over U , where β = {h1 , h2 , h3 , h5 } and α = {h1 , h2 , h5 }, because UE (ξ2 ) ∩ UE (ξ3 ) ∩ β = {h3 } * {h1 , h2 , h4 , h5 } = UE (ξ2 ξ3 ) ∪ α. But, (FB , B) is a soft subset of (UE , E). Theorem 4.5: Every soft intersectional set over U is an (α, β)-soft intersectional set over U for arbitrary ∅ ⊆ α ⊂ β ⊆ U , but the converse is not true.
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The proof of Theorem 4.5 is established by the following example. Example 4.6: Let U be the universal set and E be the set of parameters which is given in Example 4.2, considering the soft set (UE , E) defined as UE (ξ1 ) = {h1 , h2 , h3 , h4 , h5 }, UE (ξ2 ) = {h2 , h3 , h4 , h5 }, UE (ξ3 ) = {h1 , h2 , h3 , h4 } and UE (ξ4 ) = {h1 , h2 , h4 } is an (α, β)-soft intersectional set over U , where β = {h1 , h2 , h4 , h5 } and α = {h1 , h2 , h3 }, but it is not an soft intersectional set over U , because UE (ξ2 ) ∩ UE (ξ3 ) = {h2 , h3 , h4 } * {h1 , h2 , h4 } = UE (ξ2 ξ3 ).
5 (α, β)-soft intersectional sets on BCK/BCI -algebras In this section, we introduced the concept of (α, β)-soft intersectional BCK/BCIsubalgebra of BCK/BCI-algebras and investigated some of its characterisation. Throughout this work, E = X always means a BCK/BCI-algebras without any specification. We take S(U ) be the set of all (α, β)-soft intersectional BCK/BCI-algebras. Definition 5.1: Let E be a BCK/BCI-algebras. A soft set FA over U is called an (α, β)soft intersectional BCK/BCI-algebras if it satisfies the condition: FA (x) ∩ FA (y) ∩ β ⊆ FA (x ∗ y) ∪ α for all x, y ∈ E. ′
e (α,β) on S(U ) as a manner: for any We consider the order relation ‘⊆ e (α,β) GE ⇔ FE ∩ β ⊆G e E ∪ α. FE , GE ∈ S(U ) and ∅ ⊆ α ⊂ β ⊆ U , we define FE ⊆ ′ e (α,β) GE and GE ⊆ e (α,β) FE . We define a relation ‘ = such as FE =(α,β) GE ⇔ FE ⊆ (α,β)
Using the above notion, (α, β)-soft intersectional BCK/BCI-algebras defined as follows Definition 5.2: Let E be a BCK/BCI-algebras. A soft set FA over U is called an (α, β)soft intersectional BCK/BCI-algebras if it satisfies the condition: e (α,β) FA (x ∗ y) FA (x) ∩ FA (y)⊆ for all x, y ∈ E. Example 5.3: Let X = {0, a, b, c, d} be a BCK-algebra with the following Caley table: ∗ 0 a b c d
0 0 a b c d
a 0 0 b c d
b 0 a 0 c d
c 0 a b 0 d
d 0 a b c 0
Let (FA , A) be a soft over U = X, where E = A = X and FA : A → P (X) is a setvalued function defined by FA (x) = {y ∈ X|xRy ⇔ y ∈ x−1 I}, where I = {0, a} and for
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all x ∈ A, and x−1 I = {y ∈ X|x ∧ y ∈ I}. Then FA (0) = F(a) = {0, a, b, c, d}, F(b) = {0, a, c, d}, F(c) = {0, a, b, d} and F(d) = {0, a, b, c} is an (α, β)-soft intersectional BCK-algebras over X, where β = {0, b, c, d} and α = {0, b, d}. Example 5.4: Let us consider a BCI-algebra X = {0, a, b, c, d, e, f, g} with the following Caley table: ∗ 0 a b c d e f g
0 0 a b c d e f g
a 0 0 b b d d f f
b 0 0 0 a d d d e
c d 0 d 0 e 0 f 0 g d 0 d a d b d c
e d d f f 0 0 b b
f d d d e 0 0 0 a
g d d d d 0 0 0 0
Let (FA , A) be a soft set over U = X, where E = A = X and FA : A → P (X) be a setvalued function defined as FA (x) = {0} ∪ {y ∈ X|xRy ⇔ o(x) = o(y)} for all x ∈ A. Then FA (0) = FA (a) = FA (b) = FA (c) = {0, a, b, c} and FA (d) = FA (e) = FA (f ) = FA (g) = {0, d, e, f, g} is not an (α, β)-soft intersectional BCI-algebra over X, where β = {0, b, c, d, f, g} and α = {0, b, c, d, f }, because FA (d) ∩ FA (e) ∩ β = {0, d, f, g} * {0, a, b, c, d, f } = FA (d ∗ e). Remark 5.5: A soft BC/BCI-algebras may not be a soft intersectional BCK/BCIalgebras and a soft intersectional BCK/BCI-algebras may not be a soft BCK/BCIalgebras. We have shown that a soft BCK/BCI-algebras is an (α, β)-soft intersectional BCK/BCI-algebras over U for arbitrary ∅ ⊆ α ⊂ β ⊆ U , but not soft intersectional BCK/BCI-algebras by the following example. Example 5.6: Let X = {0, a, b, c} be BCI-algebra with the following Caley table: ∗ 0 a b c
0 a b c 0 a b c a 0 c b b c 0 a c b a 0
Let (FA , A) be a soft set over U = X, where E = A = X and FA : A → P (X) be a set valued function defined as follows FA (x) = {y ∈ X|y = xn , n ∈ N } for all x ∈ A. The soft set FA (0) = {0}, FA (a) = {0, a}, FA (b) = {0, b} and FA (c) = {0, c} is a soft BCI-algebra (see Jun (2008)) and which is an (α, β)-soft intersectional BCI-algebras for β = {0, a, b} and α = {a, b}, but not a soft intersectional BCI-algebra (see Jun et al. (2013)).
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Theorem 5.7: For a subset A of X, if FA is an (α, β)-soft intersectional BCK/BCIalgebras over U , then FA (0) ∪ α ⊇ FA (x) ∩ β for all x ∈ E. Proof: For any x ∈ A, we have FA (0) ∪ α = FA (x ∗ x) ∪ α ⊇ FA (x) ∩ FA (x) ∩ β = e (α,β) FA (0) for all x ∈ E. FA (x) ∩ β. Therefore, FA (x)⊆ Theorem 5.8: If a soft set FA over U is an (α, β)-soft intersectional algebras of A, then (FA (0) ∩ β) ∪ α ⊇ (FA (x) ∩ β) ∪ α, for all x ∈ A. Proof: The proof of the theorem is obvious.
Theorem 5.9: Let FA be an (α, β)-soft intersectional BCK/BCI-algebras over U , where A is a subset of E. For any given x ∈ A, the following are equivalent: (1) FA (x) =(α,β) FA (0) e (α,β) FA (x ∗ y) for all y ∈ A with x ∗ y ∈ A. (2) FA (y)⊆ Proof: Assume that (2) holds. Take y = 0 and using (C6 ), we get FA (0) ∩ β ⊆ FA (x ∗ 0) ∪ α = FA (x) ∪ α and from Theorem 5.7, FA (x) ∩ β = FA (0) ∪ α. Conversely, suppose that FA (x) ∩ β = FA (0) ∪ α. Then FA (y) ∩ β = FA (0) ∩ FA (y) ∩ β = FA (x) ∩ FA (y) ∩ β ⊆ FA (x ∗ y) ∪ α. 2
Definition 5.10: For a given subset A of E, soft intersectional BCI-algebra (FA , A) over U is said to be (α, β)-strong if for 0 ∗ x ∈ A satisfies the condition e (α,β) FA (0 ∗ x). FA (x)⊆ Theorem 5.11: For a given subset A of E, every strong (α, β)-soft intersectional BCIalgebra (FA , A) over U satisfies the following condition: e (α,β) FA (x ∗ (0 ∗ y)) FA (x) ∩ FA (y)⊆ for 0 ∗ y ∈ A. Proof: We assume that (FA , A) be an (α, β)-soft intersectional BCI-algebra over U is strong. Let x, y ∈ A be such that 0 ∗ y ∈ A. Then by using Definition 3.7, FA (x) ∩ FA (y) ∩ β = FA (x) ∩ FA (0 ∗ y) ∩ β ⊆ FA (x ∗ (0 ∗ y)) ∪ α. e (α,β) FA (x ∗ (0 ∗ y)) for 0 ∗ y ∈ A. Therefore, FA (x) ∩ FA (y)⊆
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Theorem 5.12: For a subset A of a p-semi-simple BCI-algebra E, for an (α, β)-soft intersectional BCI-algebra (FA , A) over U satisfies the condition of Theorem 5.11, then it is strong. Proof: Let x ∈ A be such that 0 ∗ x ∈ A. Then we have FA (x) ∩ β = FA (0) ∩ FA (x) ∩ β = FA (0) ∩ FA (0 ∗ (0 ∗ x) ∩ β ⊆ FA (0 ∗ (0 ∗ (0 ∗ x))) ∪ α = FA (0 ∗ x) ∩ α. Since (0 ∗ (0 ∗ x)) ∈ A, then it follows that FA (0 ∗ x) ∪ α ⊆ FA (0 ∗ (0 ∗ x)) ∩ β = FA (x) ∩ β. Hence, (FA , A) satisfies the condition of Definition 3.7, so it is (α, β)-strong. 2 Definition 5.13: Let (F, E) be a soft set over U and t ∈ E. The set F(E t ) is defined by F(E t ) = {t ∈ E|F(t) ⊆ F(x) ∩ β}. Then F(E t ) is the (α, β)-t support of (F, E) over U . Theorem 5.14: Let E = X be a BCK/BCI-algebras and (F, A) be an (α, β)-soft intersectional BCK/BCI-algebras over U , where A is a subset of E. If A is a subalgebra of E, then F(E t ) is a subalgebra of E for any t ∈ E. Proof: Let x, y ∈ F (E t ). Then x, y ∈ A, F(t) ⊆ F(x) ∩ β and F(t) ⊆ F (y) ∩ β. Since A is a subalgebra, it follows that x ∗ y ∈ A and F(t) ⊆ F(x) ∩ F(y) ∩ β ⊆ F (x ∗ y) ∪ α. So, that x ∗ y ∈ F (E t ). Therefore, F(E t ) is a subalgebra of E. We have the following corollary.
2
Corollary 5.15: Let E be a BCK/BCI-algebras and (F, A) be an (α, β)-soft intersectional BCK/BCI-algebras over U with A = E. Then (α, β)-t support of (F, A) is a subalgebra of E for any t ∈ E. For a soft set (F, A) over E, let us consider the set X0 = {x ∈ A|F(x) = F(0)}. Theorem 5.16: Let E be a BCK/BCI-algebras and A be a subalgebra of E. Let (F, A) be an (α, β)-soft intersectional BCK/BCI-algebras over E. Then the set X0 is a subalgebra of E. Proof: Let x, y ∈ X0 . Then x, y ∈ A and F(x) = F(0) = F(y). Since A is a subalgebra of E, which imply x ∗ y ∈ A. It follows from Definitions 3.7 and 5.2 that F(0) = F(x) ∩ F(y) ∩ β ⊆ F (x ∗ y) ∪ α, so from Definition 5.10, F(0) = F(x ∗ y). Hence, x ∗ y ∈ X0 . Therefore, X0 is a subalgebra of E. Theorem 5.17: Let E be a BCK/BCI-algebras and A be a subalgebra of E. Let (FA , A) be an (α, β)-soft intersectional BCK/BCI-algebras over E. Then the set X0∗ = {x ∈ A|(FA (x) ∩ β) ∪ α = (FA (0) ∩ β) ∪ α} is a subalgebra of E.
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Proof: If FA is an (α, β)-soft intersectional BCK/BCI-algebras of A over U , then x, y ∈ X0∗ we have, (FA (x) ∩ β) ∪ α = (FA (0) ∩ β) ∪ α = (FA (y) ∩ β) ∪ α. Then from Theorem 5.8, we have (FA (0) ∩ β) ∪ α ⊇ (FA (x ∗ y) ∩ β) ∪ α for all x, y ∈ A. This also takes the form, (FA (x ∗ y) ∩ β) ∪ α = ((FA (x ∗ y) ∪ α) ∩ β) ∪ α ⊇ (FA (x) ∩ FA (y) ∩ β) ∪ α = ((FA (x) ∩ β) ∪ α) ∩ (FA (y) ∩ β) ∪ α) = (FA (0) ∩ β) ∪ α. Hence, (FA (x ∗ y) ∩ β) ∪ α = (FA (0) ∩ β) ∪ α, and so, x ∗ y ∈ X0∗ . Thus, X0∗ is a subalgebras of A.
2
Theorem 5.18: Let Ψ : E1 → E2 be a homomorphism of BCK/BCI-algebras and FE2 ∈ S(U ). If FE2 is an (α, β)-soft intersectional BCK/BCI-algebras, then soft preimage Ψ−1 (FE2 ) of FE2 under Ψ is also an (α, β)-soft intersectional BCK/BCI-algebras over U . Then Ψ−1 (FE2 )(x1 ) ∩ Ψ−1 (FE2 )(x2 ) ⊆(α,β) Ψ−1 (FE2 )(x1 ∗ x2 ) hold. Proof: For any x1 , x2 ∈ X, we have Ψ−1 (FE2 )(x1 ) ∩ Ψ−1 (FE2 )(x2 ) ∩ β = FE2 (Ψ(x1 )) ∩ FE2 (Ψ(x2 )) ⊆ FE2 (Ψ(x1 ) ∗ Ψ(x2 )) ∪ α = FE2 (Ψ(x1 ∗ x2 )) ∪ α = Ψ−1 (FE2 )(x1 ∗ x2 ) ∪ α. This implies that Ψ−1 (FE2 )(x1 ) ∩ Ψ−1 (FE2 )(x2 ) ⊆(α,β) Ψ−1 (FE2 )(x1 ∗ x2 ) hold. Thus, Ψ−1 (FE2 ) is an (α, β)-soft intersectional BCK/BCI-algebras over U .
Theorem 5.19: Let Ψ : E1 → E2 be a homomorphism of BCK/BCI-algebras and FE1 ∈ S(U ). If FE1 is an (α, β)-soft intersectional BCK/BCI-algebras and Ψ is injective, then the soft image of Ψ(FE1 ) under Ψ is also an (α, β)-soft intersectional BCK/BCI-algebras over U . Then Ψ(FE1 )(y1 ) ∩ Ψ(FE1 )(y2 ) ⊆(α,β) Ψ(FE1 (y1 ∗ y2 ) hold. Proof: Let y1 , y2 ∈ E2 . If at least one of Ψ−1 (y1 ) and Ψ−1 (y2 ) is empty then Ψ(FE1 )(y1 ) ∩ Ψ(FE1 )(y2 ) ∩ β ⊆ Ψ(FE1 )(y1 ∗ y2 ) ∪ α. We assume that Ψ−1 (y1 ) ̸= ∅ and Ψ−1 (y2 ) ̸= ∅. Then Ψ(FE1 )(y1 ) ∩ Ψ(FE1 )(y2 ) ∩ β =
∪
FE1 (x1 ) ∩
x1 ∈Ψ−1 (y1 )
=
∪
∪
FE1 (x2 ) ∩ β
x2 ∈Ψ−1 (y1 )
(FE1 (x1 ) ∩ FE1 (x2 ) ∩ β)
x1 ∈Ψ−1 (y1 ),x2 ∈Ψ−1 (y2 )
⊆
∪
x1 ∈Ψ−1 (y1 ),x2 ∈Ψ−1 (y2 )
FE1 (x1 ∗ x2 ) ∪ α
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=
FE1 (x) ∪ α
x∈Ψ−1 (y1 ∗y2 )
= Ψ(FE1 (y1 ∗ y2 ) ∪ α. This imply Ψ(FE1 )(y1 ) ∩ Ψ(FE1 )(y2 ) ⊆(α,β) Ψ(FE1 (y1 ∗ y2 ) hold. Hence, soft image Ψ(FE1 ) is an (α, β)-soft intersectional BCK/BCI-algebras over U. Let FE1 and FE2 be two (α, β)-soft intersectional BCK/BCI-algebras. We have to show that FE1 × FE2 = FE1 ×E2 is an (α, β)-soft intersectional BCK/BCI-algebras is defined by the operation: (x1 , y1 ) ∗ (x2 , y2 ) = (x1 ∗ x2 , y1 ∗ y2 ) for all x1 , y1 ∈ E1 and x2 , y2 ∈ E2 . Proposition 5.20: Let FE1 and FE2 be two (α, β)-soft intersectional BCK/BCIalgebras of E1 and E2 respectively. Then FE1 ∧ FE2 is an (α, β)-soft intersectional BCK/BCI-algebras of E1 × E2 over U . Then FE1 ∧E2 (x1 , y1 ) ∩ FE1 ∧E2 (x2 , y2 ) ⊆(α,β) FE1 ∧E2 ((x1 , y1 ) ∗ (x2 , y2 )) holds. Proof: Let FE1 and FE2 be two (α, β)-soft intersectional BCK/BCI-algebras of E1 and E2 over U , respectively. Then for all (x1 , y1 ), (x2 , y2 ) ∈ E1 × E2 FE1 ∧E2 (x1 , y1 ) ∩ FE1 ∧E2 (x2 , y2 ) ∩ β = {FE1 (x1 ) ∩ FE2 (y1 )} ∩ {FE1 (x2 ) ∩ FE2 (y2 )} ∩ β = {FE1 (x1 ) ∩ FE2 (y1 ) ∩ β} ∩ {FE1 (x2 ) ∩ FE2 (y2 ) ∩ β} = {FE1 (x1 ) ∩ FE1 (x2 ) ∩ β} ∩ {FE2 (y1 ) ∩ FE2 (y2 ) ∩ β} ⊆ {FE1 (x1 ∗ x2 ) ∪ α} ∩ {FE2 (y1 ∗ y2 ) ∪ α} = FE1 (x1 ∗ x2 ) ∩ FE2 (y1 ∗ y2 ) ∪ α = FE1 ∧E2 ((x1 , y1 ) ∗ (x2 , y2 )) ∪ α.
This imply FE1 ∧E2 (x1 , y1 ) ∩ FE1 ∧E2 (x2 , y2 ) ⊆(α,β) FE1 ∧E2 ((x1 , y1 ) ∗ (x2 , y2 )). Therefore, FE1 ∧E2 is an (α, β)-soft intersectional BCK/BCI-algebras over U .
Theorem 5.21: Let FE and GE be two (α, β)-soft intersectional BCK/BCI-algebras of E over U . Then FE ∩ GE is also an (α, β)-soft intersectional BCK/BCI-algebras over U . Then (FE ∩ GE )(x) ∩ (FE ∩ GE )(y) ⊆(α,β) (FE ∩ GE )(x ∗ y) holds. Proof: Let FE and GE be two (α, β)-soft intersectional BCK/BCI-algebras of E over U . Then (FE ∩ GE )(x) ∩ (FE ∩ GE )(y) ∩ β = (FE (x) ∩ GE (x)) ∩ (FE (y) ∩ GE (y)) ∩ β = {FE (x) ∩ FE (y) ∩ β} ∩ {GE (x) ∩ GE (y) ∩ β} ⊆ {FE (x ∗ y) ∪ α} ∩ {GE (x ∗ y) ∪ α} = (FE ∩ GE )(x ∗ y) ∪ α. Therefore, (FE ∩ GE )(x) ∩ (FE ∩ GE )(y) ⊆(α,β) (FE ∩ GE )(x ∗ y) hold. Hence, FE ∩ GE is an (α, β)-soft intersectional BCK/BCI-algebras over U .
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6 (α, β)-soft intersectional sets on BCK/BCI -ideals In this section, we introduced the concept of (α, β)-soft intersectional BCK/BCI-ideal of BCK/BCI-algebras and investigated some of its characterisation in this direction. Definition 6.1: Let E be a BCK/BCI-algebras. A soft set FA over U is called an (α, β)soft intersectional BCK/BCI-ideal if it satisfies the following conditions: (1) FA (x) ∩ β ⊆ FA (0) ∪ α (2) FA (x ∗ y) ∩ FA (y) ∩ β ⊆ FA (x) ∪ α for all x, y ∈ E. By using the definition of order relation in Section 3, the Definition 6.1 is equivalent to the following. Definition 6.2: Let E be a BCK/BCI-algebras and A ⊆ E. A soft set FA over U is called an (α, β)-soft intersectional BCK/BCI-ideal if it satisfies the conditions: e (α,β) FA (0) (1) FA (x)⊆ e (α,β) FA (x) for all x, y ∈ E. (2) FA (x ∗ y) ∩ FA (y)⊆ Example 6.3: Let U = N (set of natural numbers) be the initial universal set. Let E = N (set of natural numbers) and defined a binary operation ∗ on E such that x∗y =
x (x, y)
for all x, y ∈ X, where (x, y) is the greatest common divisor of x and y. Then (X; ∗, 1) is a BCK-algebra. For a subalgebra A = {1, 2, 3, 4, 5} of E, let α = 6N and β = 3N . Define a A-soft set FA over U by FA (1) = N , FA (2) = 2N , FA (3) = 4N , FA (4) = 8N and FA (5) = 12N . Then we can easily check that FA is an (α, β)-soft intersectional BCKideal of A over U . Example 6.4: Let U = Z (set of integers) be the universal set and E = {0, 1, a, b, c} be a BCI-algebra with the following Caley table: ∗ 0 1 a b c
0 0 1 a b c
1 a b c 0 c b a 0 c b a a 0 c b b a 0 c c b a 0
We define a soft set (GA , A) over U as follows GA (0) = z, GA (1) = ∅, GA (a) = 2z, GA (b) = 3z and GA (c) = 4z is not an (α, β)-soft intersectional ideal of BCI-algebra over U , where β = 5z and α = 15z, because GA (b ∗ c) ∩ GA (c) ∩ β = 20z * 3z = GA (b) ∪ α. Proposition 6.5: Let E be a BCK/BCI-algebras. If a A-soft set FA over U is an (α, β)soft intersectional BCK/BCI-ideal of E, then for any x, y, z ∈ E e (α,β) FA (x) (1) x ≤ y ⇒ FA (y)⊆ e (α,β) FA (x). (2) x ∗ y ≤ z ⇒ FA (y) ∩ FA (z)⊆
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Proof: (1) Let x, y ∈ A be such that x ≤ y. Then x ∗ y = 0 ∈ A, so from Definition 6.1, we have, FA (y) ∩ β = FA (0) ∩ FA (y) ∩ β = FA (x ∗ y) ∩ FA (y) ∩ β ⊆ FA (x) ∪ α. e (α,β) FA (x). Therefore, FA (y)⊆ (2) Let x, y, z ∈ X be such that x ∗ y ≤ z. Then (x ∗ y) ∗ z = 0, and so, we have FA (z) = FA (0) ∗ FA (z) = FA ((x ∗ y) ∗ z) ∩ FA (z) ⊆ FA (x ∗ y). It follows that FA (y) ∩ FA (z) ∩ β ⊆ FA (x ∗ y) ∩ FA (y) ∪ α ⊆ FA (x) ∪ α. e (α,β) FA (x). Hence, FA (y) ∩ FA (z)⊆
Corollary 6.6: Let E be a BCK/BCI-algebras. Then every (α, β)-soft intersectional BCK/BCI-algebras (F, E) satisfies the following conditions for all x, y, z ∈ E e (α,β) F(x) when x ≤ y (1) F(y)⊆ e (α,β) F(x) when x ∗ y ≤ z. (2) F(y) ∩ F(z)⊆ Corollary 6.7: Let E be a BCK/BCI-algebras. Let A be a subalgebra of E, then every (α, β)-soft intersectional BCK/BCI-ideal FA over U satisfies the condition: (....(x ∗ a1 ) ∗ ....) ∗ an = 0 ⇒
n ∩
e (α,β) FA (x) FA (ak )⊆
k=1
for all x, a1 , a2 , ....., an ∈ E. Proof: It is obvious by induction.
Proposition 6.8: Let E be a BCK/BCI-algebras and A be a subalgebras of E. Then for any A-soft set FA over U is an (α, β)-soft intersectional BCK/BCI-ideal, then the following are equivalent: for all x, y, z ∈ E e (α,β) FA (x ∗ y) (1) FA ((x ∗ y) ∗ y)⊆ e (α,β) FA ((x ∗ z) ∗ (y ∗ z)). (2) FA ((x ∗ y) ∗ z)⊆ Proof: Assume that (1) holds. Let x, y, z ∈ A such that ((x ∗ (y ∗ z)) ∗ z) ∗ z = ((x ∗ z) ∗ (y ∗ z)) ∗ z ≤ (x ∗ y) ∗ z. It follows from Proposition 6.5(1), and by (C1 ) and (C2 ) that FA ((x ∗ y) ∗ z) ∩ β ⊆ FA (((x ∗ (y ∗ z)) ∗ z) ∗ z) ∪ α ⊆ FA ((x ∗ (y ∗ z)) ∗ z) ∪ α = FA ((x ∗ z) ∗ (y ∗ z)) ∪ α. e (α,β) FA ((x ∗ z) ∗ (y ∗ z)). This implies, FA ((x ∗ y) ∗ z)⊆ Next, we assume that (2) holds. If replace z instead of y in (2), then by (C3 ) and (C6 ) FA ((x ∗ z) ∗ z) ∩ β ⊆ FA ((x ∗ z) ∗ (z ∗ z)) ∪ α = FA ((x ∗ z) ∗ 0) ∪ α = FA (x ∗ z) ∪ α. This implies (1) holds.
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Corollary 6.9: Let E be a BCK/BCI-algebras. For any (α, β)-soft intersectional BCK/BCI-ideal (F, E) over U , the following are equivalent: e (α,β) F(x ∗ y) (1) F((x ∗ y) ∗ y)⊆ e (2) F((x ∗ y) ∗ z)⊆(α,β) F((x ∗ z) ∗ (y ∗ z)). Theorem 6.10: Let E be a BCK-algebra. For a given subalgebra A of E, every A-soft set is an (α, β)-intersectional BCK/BCI-ideal over U is a A-soft (α, β)-intersectional BCK-algebra over U . Proof: Let FA be A-soft (α, β)-intersectional BCK/BCI-ideal over U . Then by Definition 6.1(2), (C8 ), (C3 ), (C5 ) and by Definition 6.1(1), we have FA (x ∗ y) ∪ α ⊇ FA ((x ∗ y) ∗ x) ∩ FA (x) ∩ β = FA ((x ∗ x) ∗ y) ∩ FA (x) ∩ β = FA (0 ∗ y) ∩ FA (x) ∩ β = FA (0) ∩ FA (x) ∩ β ⊇ FA (y) ∩ FA (x) ∩ β. Corollary 6.11: Let E be a BCK-algebra. Then every (α, β)-soft intersectional BCKideal over U is an (α, β)-soft intersectional BCK-algebra over U . The following example ensures the converse of the above Corollary 6.11 is not true in general. Example 6.12: Let U = N (set of natural numbers) be the universal set and E = {0, a, b, c, d} be a BCK-algebra with the following Caley table: ∗ 0 a b c d
0 a b c 0 0 0 0 a 0 0 0 b b 0 0 c c c 0 d c c a
d 0 0 0 0 0
Defined a soft set (FA , A) over U as follows FA (0) = N , FA (a) = 4N , FA (b) = 2N , FA (c) = 3N and FA (d) = 8N . Then FA is an (α, β)-soft intersectional BCK-algebra over U , but not an (α, β)-soft intersectional BCK-ideal over U , where β = 6N and α = 12N , since FA (d ∗ b) ∩ FA (b) ∩ β = 6N * 4N = FA (d) ∪ α. Theorem 6.13: Let E be a BCK/BCI-algebras. If a soft set (F, E) over U is an (α, β)-soft intersectional BCK/BCI-ideal over U , then t-support of (F, E) is an (α, β)intersectional BCK/BCI-ideal of E for all t ∈ E. Proof: It is obvious that 0 ∈ F (E t ). Let x, y ∈ E such that x ∗ y ∈ F(E t ) and y ∈ F (E t ). Then from Definition 5.13, we get F(t) ⊆ F (x ∗ y) ∩ β and F(t) ⊆ F (y) ∩ β, which indicate F(t) ⊆ F(x ∗ y) ∩ F(y) ∩ β ⊆ F(x) ∪ α. Therefore, (α, β)-t support F(E t ) of (F, E) is a BCK/BCI-ideal of E.
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Theorem 6.14: Let Ψ : E1 → E2 be a homomorphism of BCK/BCI-algebras and FE2 ∈ S(U ). If FE2 is an (α, β)-soft intersectional BCK/BCI-ideal over U , then soft pre-image Ψ−1 (FE2 ) of FE2 under Ψ is also an (α, β)-soft intersectional BCK/BCIideal over U . Then e (α,β) Ψ−1 (FY (0)) (1) Ψ−1 (FY )(x)⊆ −1 e (α,β) Ψ−1 (FY )(x1 ) holds. (2) Ψ (FY )(x1 ∗ x2 ) ∩ Ψ−1 (FY )(x2 )⊆ Proof: For all x ∈ E1 Ψ−1 (FE2 )(x)
∩
β = FE2 (Ψ(x)) ⊆ FE2 (Ψ(0))
∩ ∪
= Ψ−1 (FE2 (0))
β
α ∪
α.
e (α,β) Ψ−1 (FE2 (0)) holds. Therefore, Ψ−1 (FE2 )(x)⊆ For any x1 , x2 ∈ E1 , we have Ψ−1 (FE2 )(x1 ∗ x2 ) ∩ Ψ−1 (FE2 )(x2 )
∩
β = FE2 (Ψ(x1 ∗ x2 ))
∩
FE2 (Ψ(x2 )) ∪ ⊆ FE2 (Ψ(x1 ∗ x2 ) ∗ Ψ(x2 )) α ∪ = FE2 (Ψ(x1 )) α ∪ = Ψ−1 (FE2 )(x1 ) α.
e (α,β) Ψ−1 (FE2 )(x1 ) holds. Therefore, Ψ−1 (FE2 )(x1 ∗ x2 ) ∩ Ψ−1 (FE2 )(x2 )⊆ −1 Hence, Ψ (FE2 ) is an (α, β)-soft intersectional BCK/BCI-ideal over U .
Theorem 6.15: Let Ψ : E1 → E2 be a homomorphism of BCK/BCI-algebras and FE1 ∈ S(U ). If FE1 is an (α, β)-soft intersectional BCK/BCI-ideal over U and Ψ is injective, then the soft image of Ψ(FE1 ) under Ψ is also an (α, β)-soft intersectional BCK/BCI-ideal over U . Then the followings hold e (α,β) Ψ(FE1 )(0) (1) Ψ(FE1 )(y)⊆ e (α,β) Ψ(FE1 )(y1 ). (2) Ψ(FE1 )(y1 ∗ y2 )⊆ Proof: Let y1 , y2 ∈ E2 . If at least one of Ψ−1 (y1 ) and Ψ−1 (y2 ) is empty then Ψ(FE1 )(y1 ∗ y2 ) ∩ Ψ(FE1 )(y2 ) ∩ β ⊆ Ψ(FE1 )(y1 ) ∪ α. We assume that Ψ−1 (y1 ) ̸= ∅ and Ψ−1 (y2 ) ̸= ∅. Then Ψ(FE1 )(y)
∩
β=
∪ ∪
{FE1 (x)|Ψ(x) = y, x ∈ E1 }
∩
β
{FE1 (x) ∩ β|Ψ(x) = y, x ∈ E1 } ∪ ⊆ {FE1 (x) α|Ψ(x) = 0, x ∈ E1 } ∪ = Ψ(FE1 )(0) α.
=
∪
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e (α,β) Ψ(FE1 )(0) holds. Therefore, Ψ(FE1 )(y)⊆ (Ψ(FE1 )(y1 ∗ y2 )
∩
(Ψ(FE1 )(y2 )
∩
∪
{FE1 (x1 ∗ x2 )|Ψ(x1 ∗ x2 ) ∩ = y1 ∗ y2 , x1 ∗ x2 ∈ E1 } ∪ ∩ {FE1 (x2 )|Ψ(x2 ) = y2 , x2 ∈ X} β ∪ ∩ = {FE1 (x1 ∗ x2 ) FE1 (x2 ) ∩ β|Ψ(x1 )
β=
= y1 , Ψ(x2 ) = y2 , x1 , x2 ∈ E1 } ∪ ∪ ⊆ {FE1 (x1 ) α|Ψ(x1 ) = y1 , x1 ∈ E1 } ∪ = Ψ(FE1 )(y1 ) α. e (α,β) Ψ(FE1 )(y1 ) holds. Therefore, (Ψ(FE1 )(y1 ∗ y2 )⊆ Hence, Ψ(FE ) is an (α, β)-soft intersectional BCK/BCI-ideals over U . Proposition 6.16: Let FE1 and FE2 be two (α, β)-soft intersectional BCK/BCIalgebras of E1 and E2 respectively respectively over U . Then FE1 ∧E2 (x1 , y1 ) ∩ e (α,β) FE1 ∧E2 ((x1 , x2 ). FE1 ∧E2 (x2 , y2 )⊆ Proof: Let FE1 and FE2 be two (α, β)-soft intersectional BCK/BCI-ideals of E1 and E2 over U , respectively. Then for all (x1 , y1 ), (x2 , y2 ) ∈ E1 × E2 . We have FE1 ∧E2 (x1 , y1 ) ∩ FE1 ∧E2 (x2 , y2 ) ∩ β = {FE1 (x1 ∗ y1 ) ∩ FE2 (y1 )} ∩{FE1 (x2 ∗ y2 ) ∩ FE2 (y2 )} ∩ β = {FE1 (x1 ∗ y1 ) ∩ FE2 (y1 ) ∩ β} ∩{FE1 (x2 ∗ y2 ) ∩ FE2 (y2 ) ∩ β} ⊆ {FE1 (x1 ) ∪ α} ∩ {FE2 (x2 ) ∪ α} = FE1 (x1 ) ∩ FE2 (x2 ) ∪ α = FE1 ∧E2 ((x1 , x2 ) ∪ α. e (α,β) FE1 ∧E2 ((x1 , x2 ). Therefore, FE1 ∧E2 (x1 , y1 ) ∩ FE1 ∧E2 (x2 , y2 )⊆ Theorem 6.17: Let FE and GE be two (α, β)-soft intersectional BCK/BCI-algebras of E over U . Then FE ∩ GE is also an (α, β)-soft intersectional BCK/BCI-algebras over U . Then e (α,β) (FE ∩ GE )(0) (1) FE ∩ GE )(x)⊆ e (α,β) (FE ∩ GE )(x) hold for all x, y ∈ E. (2) (FE ∩ GE )(x ∗ y) ∩ (FE ∩ GE )(y)⊆ Proof: Let FE and GE be two (α, β)-soft intersectional BCK/BCI-algebras of E over U . Then (FE ∩ GE )(x) ∩ β = (FE (x) ∩ GE (x)) ∩ β = {FE (x) ∩ β} ∩ {GE (x) ∩ β} ⊆ {FE (0) ∪ α} ∩ {GE (0) ∪ α} = (FE ∩ GE )(0) ∪ α.
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C. Jana and M. Pal (FE ∩ GE )(x ∗ y) ∩ (FE ∩ GE )(y) ∩ β = (FE (x ∗ y) ∩ GE (x ∗ y)) ∩(FE (y) ∩ GE (y)) ∩ β = {FE (x ∗ y) ∩ FE (y) ∩ β} ∩{GE (x ∗ y) ∩ GE (y) ∩ β} ⊆ {FE (x) ∪ α} ∩ {GE (x) ∪ α} = (FE ∩ GE )(x) ∪ α.
This implies that e (α,β) (FE ∩ GE )(0) (FE ∩ GE )(x)⊆ and e (α,β) (FE ∩ GE )(x). (FE ∩ GE )(x ∗ y) ∩ (FE ∩ GE )(y)⊆ Hence, FE ∩ GE is an (α, β)-soft intersectional BCK/BCI-ideal over U .
7 Conclusion The soft set theory is an important mathematical tool for dealing with uncertainties and have huge applications in real life situations. As a generalisation of intersectional soft BCK/BCI-algebras and ideals. In this paper, we introduced the notions of (α, β)-soft intersectional BCK/BCI-algebras and also studied (α, β)-soft intersectional BCK/BCI-ideals. We investigated some of their characterisations in details. We hope that this work will give a deep impact on the upcoming research in this field and other soft algebraic study to open up a new horizons of interest and innovations. It is our hope that this work will serve as a foundation for further study of BCK/BCI-algebras. This results can be applied other algebraic structure such as hemiring, topology and other mathematical branches. One may be applied this concept to study some application fields like decision-making, knowledge base system, data analysis, etc.
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