search for




 

Int-Soft Filters in Hoops
International Journal of Fuzzy Logic and Intelligent Systems 2019;19(3):213-222
Published online September 25, 2019
© 2019 Korean Institute of Intelligent Systems.

R. A. Borzooei1, M. Sabetkish1, E. H. Roh2, and M. Aaly Kologani3

1Department of Mathematics, Shahid Beheshti University, Tehran, Iran
2Department of Mathematics Education, Chinju National University of Education, Jinju, Korea
3Hatef Higher Education Institute, Zahedan, Iran
Correspondence to: E. H. Roh (ehroh9988@gmail.com)
Received May 20, 2019; Revised September 17, 2019; Accepted September 20, 2019.
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract

In this paper, we introduce the notion of int-soft filters in hoops and related properties are investigated. Characterizations of int-soft filters are discussed and we introduce a congruence relation by using int-soft filter and we show that the quotient is a hoop. Also, we introduce the notions of int-soft n-fold (positive) implicative and int-soft n-fold fantastic filters of hoops, we study some equivalence definitions of them and the relation between them, we prove that every int-soft n-fold positive implicative filter is an int-soft fantastic filter and int-soft implicative filter. Also, we investigate about the quotient structure that is made by them.

Keywords : Hoop, n-fold implicative (positive implicative, fantastic) filter, Soft set, Int-soft filter, γ-inclusive set.
1. Introduction

Non-classical logic has become a considerable formal tool for computer science and artificial intelligence to deal with fuzzy information and uncertainty information. Many-valued logic, a great extension and development of classical logic, has always been a crucial direction in non-classical logic. In order to research the many-valued logical system whose propositional value is given in a lattice, Bosbach [1, 2] proposed the concept of hoops, and discussed their some properties. Hoops are naturally ordered commutative residuated integral monoids. In the last years, hoops theory was enriched with deep structure theorems [13]. Many of these results have a strong impact with fuzzy logic. Particularly, from the structure theorem of finite basic hoops [3, Corollary 2.10], one obtains an elegant short proof of the completeness theorem for propositional basic logic [3, Theorem 3.8], introduced by Hajek [4]. The algebraic structures corresponding to Hajek’s propositional (fuzzy) basic logic, BL-algebras, are particular cases of hoops. The main example of BL-algebras in interval [0, 1] endowed with the structure induced by a t-norm. MV-algebras, product algebras and Gödel algebras are the most known classes of BL-algebras. Recent investigations are concerned with non-commutative generalizations for these structures. During these years, many researchers study on hoops in different way, and got some results on hoops [5]. Borzooei and Kologani [6] investigated different kind of filters on hoops such as (positive)implicative and fantastic filters, and studied the relation between them and investigated congruence relation that is made by them and studied about the quotients structure.

Molodtsov [7] introduced the concept of soft set as a new mathematical tool for dealing with uncertainties that is free from the difficulties that have troubled the usual theoretical approaches. Molodtsov pointed out several directions for the applications of soft sets. At present, works on the soft set theory are progressing rapidly. Maji et al. [8] described the application of soft set theory to a decision making problem. Maji et al. [9] also studied several operations on the theory of soft sets. Chen et al. [10] presented a new definition of soft set parametrization reduction, and compared this definition to the related concept of attributes reduction in rough set theory. The algebraic structure of set theories dealing with uncertainties has been studied by some authors. Feng [8] considered the application of soft rough approximations in multicriteria group decision-making problems. Aktas and Cagman [12] studied the basic concepts of soft set theory, and compared soft sets to fuzzy and rough sets, providing examples to clarify their differences. They also discussed the notion of soft groups. After than, many algebraic properties of soft sets are studied [1316]. In this paper, we introduce the notion of int-soft filters in hoops and related properties are investigated. Characterizations of int-soft filters are discussed and we introduce a congruence relation by using int-soft filter and we show that the quotient is a hoop. Also, we introduce the notions of int-soft n-fold (positive) implicative and int-soft n-fold fantastic filters of hoops, we study some equivalence definitions of them and the relation between them, we prove that every int-soft n-fold positive implicative filter is an int-soft fantastic filter and int-soft implicative filter. Also, we investigate about the quotient structure that is made by them.

2. Preliminaries

In this section, recollect some definitions an results which will be used in this paper

Definition 2.1 ( [1, 2])

A hoop is an algebra (H,⊙,→, 1) of type (2,2,0) such that, for all x, y, zH,

(HP1) (H,⊙,→, 1) is a commutative monoid.

(HP2) xx = 1.

(HP3) (xy) → z = x → (yz).

(HP4) x ⊙ (xy) = y ⊙ (yx).

On hoop H, we define xy if and only if xy = 1. It is easy to see that ≤ is a partial order relation on H. A hoop H is called bounded if there is an element 0 ∈ H such that, for all xH, 0 ≤ x. Let H be a bounded hoop. For all xH, we define a negation ′ on H by x′ = x → 0. If, for all xH, x″ = x, then the bounded hoop H is said to have the double negation property, (DNP), for short.

The following proposition provides some properties of hoop.

Proposition 2.2 ( [1, 2])

Let H be a hoop. Then, for all x, y, zH, the following properties hold,

(i) xyz if and only if xyz.

(ii) xy if and only if xy = 1.

(iii) xyx, y.

(iv) xyx.

(v) x ≤ (xy) → y.

(vi) x → 1 = 1 and 1 → x = x.

(vii) If xy, then zxzy, yzxz and xzyz.

(viii) xy ≤ (yz) → (xz).

Definition 2.3 ( [3, 17])

Let H be a hoop and ∅︀ ≠ FH such that 1 ∈ F. Then, for any x, y, zH, and n ∈ ℕ, F is called (i) a filter of H, if x, xyF, then yF.

(ii) an n-fold implicative filter of H, if xn → (yz) ∈ F and xnyF, then xnzF.

(iii) an n-fold positive implicative filter of H, if x → ((ynz) → y) ∈ F and xF, then yF.

(iv) an n-fold fantastic filter of H, if zF and z → (yx) ∈ F, then ((xny) → y) → xF.

Definition 2.4 ( [7])

Let U be an initial universe set, E be a set of parameters, ℘(U) be the power set of U and ∅︀ ≠ AE. Then a soft set (α,A) over U is defined to be the set of ordered pairs (α,A) = {(x, α(x)) | xA, α(x) ∈ ℘(U)}, where α : A → ℘(U). For a soft set (α,A) over U and a subset γ of U, the γ-inclusive set of (α,A), denoted by iH(α, γ), is defined to be the set

iH(α,γ):={xA|γα(x)}.
Notation

From now on, in this paper (H,⊙,→, 1) or simply H is a hoop, unless otherwise state.

3. Int-Soft Filters

In this section, we introduce the notion of int-soft filter of hoops and find some equivalence definitions of that. Also, we define a congruence relation by int-soft filter of H and show that the quotient, that is made by it, is a hoop.

Definition 3.1

A soft set (α,H) is called an int-soft filter of H over U if, for all γ ∈ ℘(U), γ-inclusive set ∅︀ ≠ iH(α, γ) of (α,H) is a filter of H.

Example 3.2

Let H = {0, a, b, c, 1}. Then, for any x, yH, define the operations ⊙ and → on H as follows:

0abc1011111ab1b11baa111c0ab1110abc1   0abc1000000a0a0aab00bbbc0abcc10abc1

Then by routine calculations, we can see that (H,⊙,→, 0, 1) is a bounded hoop. Let (λ,H) be a soft set over U = ℤ in H such that α : HP(U) and α(1) = ℤ, α(a) = 2ℤ, α(b) = 4ℤ, α(0) = 4ℤ and α(c) = ℤ. Then (α,H) is an int-soft filter of H over U.

Theorem 3.3

Let (α,H) be a soft set of H. Then, for any x, y, zH, the following statements are equivalent:

(i) (α,H) is an int-soft filter of H over U,

(ii) α(x) ⊆ α(1) and α(x) ∩ α(xy) ⊆ α(y),

(iii) α(x) ⊆ α(1) and α(xy) ∩ α(yz) ⊆ α(xz),

(iv) α(x) ⊆ α(1) and α(xy) ∩ α(xz) ⊆ α(yz),

(v) if xyz, then α(x) ∩ α(y) ⊆ α(z).

Proof

(i)⇒(ii) Suppose (α,H) is an int-soft filter of H over U. Then, for any x, yH, α(xy) ∩ α(x) = γ. Thus, γα(x) and γα(xy), and so xiH(α, γ) and xyiH(α, γ). Since iH(α, γ) is a filter of H and x⊙(xy) ∈ iH(α, γ), by Proposition 2.2(v), x ⊙ (xy) ≤ y, and so yiH(α, γ). Hence, γ = α(x) ∩ α(xy) ⊆ α(y). Now, let xH and α(x) = γx. Then xiH(α, γ). Since iH(α, γ) is a filter of H, 1 ∈ iH(α, γ) and γx = α(x) ⊆ α(1). Hence, α(x) ⊆ α(1).

(ii) ⇒ (iii) It is clear that, for any xH, α(x) ⊆ α(1). By Proposition 2.2(viii), for any x, y, zH, xy ≤ (yz) → (xz), and so

(xy)[(yz)(xz)]=1.

By (ii), we have

α(xy)α(yz)=α(xy)α(1)α(yz)=α(xy)α((xy)[(yz)(xz)])α(yz)α((yz)(xz)])α(yz)α(xz).

Hence, α(xy) ∩ α(yz) ⊆ α(xz).

(iii) ⇒ (iv) Let x, y, zH. Since yzyz, we have yz → (yz). Then by Proposition 2.2(vii), xyx → (z → (yz)), and so xy ≤ (xz) → (yz). Thus, (xy) → ((xz) → (yz)) = 1. Hence, by (iii), we get that

α(1(xy))α((xy)((xz)(yz)))α(xz)α((xz)(yz))α(xz)=α(1(xz))α((xz)(yz))α(1(yz))=α(yz).

Hence, α(xy) ∩ α(yz) ⊆ α(xz).

(iv) ⇒ (v) Let x, y, zH such that xyz. It is clear that x → (yz) = 1. Then by (iv), we have,

α(x)α(y)α(x(yz))α(y)α(yz)=α(1y)α(yz)α(1z)=α(z).

(v) ⇒ (i) Let xiH(α, γ). Then γα(x). Since, for any xH, x → (x → 1) = 1, we have α(x) = α(x) ∩ α(x) ⊆ α(1). Hence, for any xiH(α, γ), α(x) ⊆ α(1). Now, suppose for any x, yH, x, xyiH(α, γ). Then γα(x) ∩ α(xy). Since (x ⊙ (xy)) → y = x → ((xy) → y) = 1 by (v), we have α(x) ∩ α(xy) ⊆ α(y) and so γα(y). Hence, yiH(α, γ). Therefore, (α,H) is an int-soft filter of H.

Corollary 3.4

Let H be a bounded hoop and (α,H) be an int-soft filter of H. Then, for any x, y, zH, the following sentences hold:

(i) If xy, then α(x) ⊆ α(y).

(ii) α(xy) ⊆ α(x) ∩ α(y).

(iii) α(x′) ⊆ α(xy).

(iv) α((xy) → y) ∩ α(x) ⊆ α(y).

Proof

(i) Let x, yH such that xy = 1. Since (α,H) is an int-soft filter of H, we have α(x) ∩ α(xy) ⊆ α(y), and so α(x) ⊆ α(y). By (i) and Proposition 2.2, the proof of other cases is clear.

Proposition 3.5

Let (α,H) be a soft set of H and for any xH, x2 = x. Then (α,H) is an int-soft filter of H over U if and only if α(x) ⊆ α(1) and

α((xy)z)α(xy)α(xz).
Proof

Let x, yH. Since ((1 ⊙ x) → y) = xy by assumption, we have α((1⊙x) → y)∩α(1 → x) ⊆ α(1 → y) and so α(x) ∩ α(xy) ⊆ α(y). Hence, (α,H) is an int-soft filter of H over U.

Conversely, let x, y, zH. Then by Proposition 2.1(viii),

(xy)(y(xz))x(xz)

Since, for any xH, x2 = x, we have (xy) ⊙ (y → (xz)) ≤ xz. Moreover, (α,H) is an int-soft filter of H, then α(x) ⊆ α(1) and by Corollary 3.4(ii), we have α((xy) → z) ∩ α(xy) ⊆ α(xz)

Theorem 3.6

Let x, yH and (α,H) be an int-soft filter of H. Then we define a binary relation ≡ on H by x(α,H)y if and only if α(xy) ∩ α(yx) = α(1). Then ≡(α,H) is a congruence relation on H.

Proof

By routine calculation it is clear that ≡(α,H) is reflective and symmetric. Now, we prove that ≡(α,H) is transitive. Let x, y, zH such that x(α,H)y and y(α,H)z. Then α(xy) ∩ α(yx) = α(1) and α(yz) ∩ α(zy) = α(1) By Proposition 2.2(viii), (xy) ⊙ (yz) ≤ (xy). Since (α,H) is an int-soft filter of H, by Corollary 3.4(ii), we have α(xy) ∩ α(yz) ⊆ α(xz) By the similar way, α(zy) ∩ α(yx) ⊆ α(zx) Thus,

α(1)α(xy)α(yx)α(yz)α(zy)α(zx)α(xz),

and so α(xz) ∩ α(zx) = α(1). Hence, x(α,H)z. Therefore, ≡(α,H) is an equivalence relation. Suppose x(α,H)y. We prove that for any zH, xz(α,H)yz. Since x(α,H)y, we have α(xy) ∩ α(yx) = α(1). By Proposition 2.2(viii), xy ≤ (yz) → (xz) and (α,H) is an int-soft filter of H, then by Corollary 3.4(i),α(xy) ⊆ α((yz) → (xz)) By the similar way, α(yx) ⊆ α((xz) → (yz)). Then

α(1)=α(xy)α(yx)vα((yz)(xz))α((xz)(yz)).

Hence, xz(α,H)yz. By repeating this method, it is easy to see that if x(α,H)y, then zx(α,H)zy. Now, let x(α,H)y. By Proposition 2.2(vii), xy ≤ (xz) → (yz). Since (α,H) is an int-soft filter of H, by Corollary 3.4(i), α(xy) ⊆ α((xz) → (yz)) and α(yx) ⊆ α((yz) → (xz)). Then it is clear that α(1) = α((xz) → (yz))∩α((yz) → (xz)). Hence, xz(α,H)yz. Therefore, ≡(α,H) is a congrunce relation on H.

Notation

Let define [x] = {yH | x(α,H)y} and H(α,H)={[x]xH}.

Theorem 3.7

Let H be a hoop and (α,H) be an int-soft filter on H. Then H(α,H), ⊗, ⇝, [1]) is a hoop, where, for any [x], [y]H(α,H), we define: [x] ⊗ [y] = [xy] and [x] ⇝ [y] = [xy] and [1] = {xH | α(x) = α(1)}.

Also, we can define an order on H(α,H) by [x] ≤ [y] if and only if α(xy) = α(1). Then (H(α,H), ≤) is a poset.

4. Int-Soft n-Fold Implicative Filters

In this section, we introduce the notion of int-soft n-fold implicative filter of hoop and investigate some properties and equivalence definitions of it. Also, we study the quotient that is made by int-soft n-fold implicative filters of hoop and we show that every filter of the quotient structure is an implicative filter.

Definition 4.1

A soft set (α,H) is an int-soft n-fold implicative filter of H over U if, for all γ ∈ ℘(U), an γ-inclusive set ∅︀ ≠ iH(α, γ) of (α,H) is an n-fold implicative filter of H.

The set of all int-soft n-fold implicative filters of H over U is denoted by INI(ℋ).

Example 4.2

According to Example 3.2, (α,H) is an int soft n-fold implicative filter of H over U.

Theorem 4.3

Every (α,H) ∈ INI(ℋ) is an int-soft filter of H.

Proof

Let (α,H) ∈ INI(ℋ). Then iH(α, γ) ≠ ∅︀. Thus, there exists xiH(α, γ). Since iH(α, γ) is an n-fold implicative filter of H, 1 ∈ iH(α, γ). Then it is clear that α(x) ⊆ α(1). Now, let x, yH and n ∈ ℕ such that x, xyiH(α, γ). Suppose γ = α(x) ∩ α(xy). Since iH(α, γ) is an n-fold implicative filter and 1n → (xy), 1nxiH(α, γ), we have 1nyiH(α, γ). So, yiH(α, γ). Thus, iH(α, γ) is a filter of H. Hence, (α,H) is an int-soft filter of H.

In the following example we show that the converse of Theorem 4.3 may not be true, in general.

Example 4.4

Let H = {0, a, b, 1}. Then, for any x, yH, define the operations ⊙ and → on H as follows:

0ab101111aa111b0a1110ab1         0ab100000a0aaab0abb10ab1

Then (H,⊙,→, 0, 1) is a bounded hoop. Let (α,H) be a soft set over U = ℤ in H such that α : HP(U) and α(1) = ℤ, α(a) = 4ℤ, α(b) = 2ℤ and α(0) = 4ℤ. Then (α,H) is an int-soft filter of H over U but it is not an int-soft n-fold implicative filter of H. Because ℤ = α(1) = α(an → (a → 0)) ∩ α(ana) ⊈ α(an → 0) = 4ℤ.

Theorem 4.5

Let (α,H) be an int-soft filter of H. Then, for any x, y, zH and n ∈ ℕ, the following statements are equivalent:

(i) (α,H) ∈ INI(ℋ),

(ii) α(xn → (yz)) ∩ α(xny) ⊆ α(xnz),

(iii) α(xnx2n) = α(1),

(iv) α(x2ny) ⊆ α(xny).

Proof

(i) ⇒ (ii) Since (α,H) ∈ INI(ℋ), by Theorem 4.3, it is clear that α(x) ⊆ α(1). Now, suppose x, y, zH and n ∈ ℕ. Let γ = α(xn → (yz)) ∩ α(xny). Then xn → (yz), xnyiH(α, γ). Since (α,H) ∈ INI(ℋ), iH(α, γ) is an n-fold implicative filter of H, then xnziH(α, γ). Thus γα(xnz). Hence,

α(xn(yz))α(xny)α(xnz).

(ii) ⇒ (iii) Let xH and n ∈ ℕ. Since x2nx2n = 1, we have α(1) = α(x2nx2n) = α(xn → (xnx2n)). Then by (ii),

α(1)=α(xn(xnx2n))α(xnxn)α(xnx2n).

Also, for any xH, α(x) ⊆ α(1). Then α(xnx2n) ⊆ α(1). Hence, α(xnx2n) = α(1).

(iii)⇒(iv) Let x, yH and n ∈ ℕ. By Proposition 2.2(viii), we have (xnx2n) ⊙ (x2ny) ≤ xny. Since (α,H) be an int-soft filter of H, by Corollary 3.4(ii) and (i), we have α(xnx2n) ∩ α(x2ny) ⊆ α(xny). By (iii), α(xnx2n) = α(1), then α(x2ny) ⊆ α(xny).

(iv) ⇒(i) Let x, y, zH and n ∈ ℕ such that xn → (yz), xnyiH(α, γ). Then by Proposition 2.2(viii),

(xny)(y(xnz))xn(xnz)=x2nz.

Since (α,H) be an int-soft filter of H, by Corollary 3.4(ii) and (i), we have α(xny) ∩ α(y → (xnz)) ⊆ α(x2nz).

Then by (iv), it is clear that

α(xny)α(y(xnz))α(x2nz)α(xnz).

Hence, xnziH(α, γ). Therefore, (α,H) ∈ INI(ℋ).

Corollary 4.6

If (α,H) is an int-soft filter of H such that, for any xH, x2 = x, then (α,H) ∈ INI(ℋ).

Proof

Let xH. Since x2 = x, we have xn = x2n, then xnx2n = 1, and so α(xnx2n) = α(1). Thus by Theorem 4.5, (α,H) ∈ INI(ℋ).

Theorem 4.7

Let (α,H) be an int-soft filter of H. Then (α,H) ∈ INI(ℋ) if and only if for any x, y, zH and n ∈ ℕ, α(xn → (yz)) ⊆ α((xny) → (xnz)).

Proof

Suppose (α,H) ∈ INI(ℋ). Then by Proposition 2.2(viii) and (vii), for any x, y, zH and n ∈ ℕ, we have:

xn(yz)xn((xny)(xnz))=xn(xn((xny)z))=x2n((xny)z).

Also, by Proposition 2.2(viii),

x2n((xny)z)(xnx2n)(xn((xny)z)).

Since (α,H) ∈ INI(ℋ), by Theorem 4.3, Corollary 3.4(i) and Theorem 4.5(iii), we get that

α(xn(yz))α(x2n((xny)z)α((xnx2n)((xn((xny)+z)))α(xnx2n)α((xny)(xnz)).

Conversely, let xH and n ∈ ℕ. Since xn → (xnx2n) = 1, by assumption, we have

α(1)=α(xn(xnx2n))α((xnxn)(xnx2n))=α(xnx2n).

Then α(xnx2n) = α(1). Hence, by Theorem 4.5, (α,H) ∈ INI(ℋ).

Notation

Let (α,H) and (β,H) be two int-soft filters of H over U. If for any xH, α(x) ⊆ β(x), then we denoted it by αβ.

Theorem 4.8

Let (α,H) and (β,H) be two int-soft filters of H over U such that αβ and α(1) = β(1). If (α,H) ∈ INI(ℋ), then (β,H) ∈ INI(ℋ).

Proof

By Theorem 4.5, for any xH and n ∈ ℕ, we have, β(1) = α(1) ⊆ α(xnx2n) ⊆ β(xnx2n). Then β(xnx2n) = β(1). Hence, by Theorem 4.5, (β,H) ∈ INI(ℋ).

Theorem 4.9

If (α,H) ∈ INI(ℋ), then it is an int-soft (n+1)-fold implicative filter of H.

Proof

Let (α,H) ∈ INI(ℋ). Then by Theorem 4.7, for any x, yH, we have,

α(x2(n+1)y)=α(xn+1(xn+1y))α((xn+1xn+1)(xn+1y))α(xn+1y).

Then by Theorem 4.5, (α,H) is an int-soft (n + 1)-fold implicative filter of H.

Theorem 4.10

Let (α,H) ∈ INI(ℋ). Then every filter of H is an implicative filter.

Proof

Let (α,H) ∈ INI(ℋ). Then by Theorem 4.3, (α,H) is an int-soft filter of H. Thus, by Theorem 3.7, H is well-define. Moreover, for any xH, it is clear that x2nxn = 1, and so α(x2nxn) = α(1). Also, by Theorem 4.5, α(xnx2n) = α(1). Thus, [xn] = [x2n], and so [x] = [x2]. By [?, Theorem 4.15], since class [1] is an int-soft filter of H and for any xH, [x2] = [x], then [1] is an implicative filter of H. Hence, by [?, Theorem 4.10], since, for any filter G of H(α,H)

and [1] ⊆ G, we get that G is an int-soft implicative filter of H(α,H).

Theorem 4.11

Let (α,H) be an int-soft filter of H. Then (α,H) is an int-soft implicative filter of H if and only if H(α,H) is a Brouwerian semilattice.

Proof

(⇒) Let (α,H) be an int-soft filter of H. Then by Theorem 3.7, H(α,H) is well-defined. Since H(α,H) is a hoop, then it is clear that H(α,H) is an ∧-semilattice. Now, it is enough to prove that for all x, y, zH, [x](α,H) ∧ [y](α,H) ≤ [z](α,H) if and only if [x](α,H) ≤ [y](α,H) ∈ [z](α,H). Let [x](α,H) ∧ [y](α,H) ≤ [z](α,H). Then by Proposition 2.2 (iii),

[x](α,H)[y](α,H)[x](α,H)[y](α,H)[z](α,H).

Thus, [x](α,H) ⊗ [y](α,H) ≤ [z](α,H). Since H(α,H) is a hoop, then by Proposition 2.2(i), [x](α,H) ≤ [y](α,H) ⇝ [z](α,H).

Conversely, suppose [x](α,H) ≤ [y](α,H) ⇝ [z](α,H). Then by Theorem 3.7, α(x → (yz)) = α(1), and so α(x → (yz)) ≥ α(1). Since (α,H) is an int-soft implicative filter of H, then by Theorem 4.7,

α(1)=α(x(yz))α((xy)(xz)).

Thus, [xy](α,H) ≤ [xz](α,H) and so

[x](α,H)[y](α,H)[x](α,H)[z](α,H).

Since H(α,H) is a hoop, then

[x](α,H)[y](α,H)[x](α,H)([x](α,H)[y](α,H))[z](α,H).

Therefore, H(α,H) is a Brouwerian semilattice.

(⇒) Since (α,H) is an int-soft filter of H, then it is clear that α(x) ⊆ α(1), for all xH. By assumption, H(α,H) is a Brouwerian semilattice, define [x](α,H)⊗[y](α,H) = [x](α,H)∧ [y](α,H), for all x, yH. Since [x](α,H) ≤ [x](α,H), then we have [x](α,H) ≤ [x](α,H) ∧ [x](α,H) = [x](α,H) ⊗ [x](α,H) = [x2](α,H). So [x](α,H) ≤ [x2](α,H). Then by Theorem 3.7, α(xx2) = α(1). Hence, by Theorem 4.3(iii), (α,H) is an int-soft implicative filter of H.

5. Int-Soft n-Fold Positive Implicative Filters

In this section, we introduce the notion of int-soft n-fold positive implicative filter and int-soft n-fold fantastic filter of hoop and investigate some properties and equivalence definitions of them. And we prove that every int-soft n-fold positive implicative filter is an int-soft fantastic filter.

Definition 5.1

A soft set (α,H) is an int-soft n-fold positive implicative filter of H over U if, for any γ ∈ ℘(U), γ-inclusive set ∅︀ ≠ iH(α, γ) of (α,H) is an n-fold positive implicative filter of H. The set of all int-soft n-fold positive implicative filters of H is denoted by INPI(ℋ).

Example 5.2

According to Example 3.2, it is clear that (α,H) is an int-soft n-fold positive implicative filter of H over U.

Theorem 5.3

Every (α,H) ∈ INPI(ℋ) is an int-soft filter of H.

Proof

Let (α,H) ∈ INPI(ℋ). Since iH(α, γ) ≠ ∅︀, we have xiH(α, γ) and we suppose γ = α(x). Since (α,H) ∈ INPI(ℋ), we have iH(α, γ) is an n-fold positive implicative filter of H, then 1 ∈ iH(α, γ) and so α(x) = γα(1). Now, let x, y, zH such that x, xyiH(α, γ). Suppose γ = α(x) ∩ α(xy). Since iH(α, γ) is an n-fold positive implicative filter of H, for any n ∈ ℕ, we have x, x → ((yn → 1) → y) ∈ iH(α, γ), then yiH(α, γ), and so

γ=α(x)α(xy)=α(x)α(x((yn1)y))α(y).

Hence, (α,H) is an int-soft filter of H.

Example 5.4

According to Example 3.2, H is a bounded hoop. Let define α(1) = ℤ, α(a) = 2ℤ, α(b) = 4ℤ, α(c) = 2ℤ and α(0) = 4ℤ. Then by routine calculations, we can see that (α,H) is an int-soft n-fold positive implicative filter but it is not an int-soft filter of H over U. Because

=α(1)=α(1((cn0)c))α(1)α(c)=2.

Theorem 5.5

Let (α,H) be an int-soft filter of H over U. Then, for any x, y, zH and n ∈ ℕ, the following conditions are equivalent:

(i) (α,H) ∈ INPI(ℋ),

(ii) α(x) ⊆ α(1) and α(x → ((ynz) → y)) ∩ α(x) ⊆ α(y),

(iii) α((xny) → x) ⊆ α(x),

(iv) α((xn → 0) → x) ⊆ α(x).

Proof

(i) ⇒ (ii) Let (α,H) ∈ INPI(ℋ), x, y, zH and n ∈ ℕ. Define γ = α(x) ∩ α(x → ((ynz) → y)). Then x → ((ynz) → y), xiH(α, γ). Moreover, since iH(α, γ) is an n-fold positive implicative filter of H, yiH(α, γ). Then, γ = α(x) ∩ α(x → ((ynz) → y)) ⊆ α(y).

(ii) ⇒ (iii) By (ii), we have α(1) ∩ α(1 → ((xny) → x)) ⊆ α(x).

(iii)⇒ (iv) It is enough let y = 0 in (iii).

(iv) ⇒ (i) Since (α,H) is an int-soft filter of H, for any x, y, zH and n ∈ ℕ,

α(x((ynz)y))α(x)α((ynz)y)).

Moreover, since 0 ≤ z, by Proposition 2.2(vii), yn → 0 ≤ ynz and so (ynz) → y ≤ (yn → 0) → y. Then ((ynz) → y) → ((yn → 0) → y) = 1 and by Corollary 3.4, α((ynz) → y)) ⊆ α((yn → 0) → y) ⊆ α(y). Hence, by (iv), (α,H) ∈ INPI(ℋ).

Theorem 5.6

Let (α,H) ∈ INPI(ℋ). Then (α,H) ∈ INI(ℋ).

Proof

Let (α,H) ∈ INPI(ℋ). Then by Definition 5.1, iH(α, γ) is an n-fold positive implicative filter of H, thus, by [?, Theorem 4.8], every n-fold positive implicative filter of H is an n-fold implicative filter of H, so iH(α, γ) is an n-fold implicative filter of H. Hence, (α,H) is an int-soft n-fold implicative filter of H.

Theorem 5.7

Let (α,H) ∈ INPI(ℋ). Then (α,H) is an int-soft (n+1)-fold positive implicative filter of H over U.

Proof

Let xH and n ∈ ℕ. Since xn+1xn, by Proposition 2.2(vii), xn → 0 ≤ xn+1 → 0, and so (xn+1 → 0) → x ≤ (xn → 0) → x. Since (α,H) ∈ INPI(ℋ), by Theorem 5.3 and Corollary 3.4, we have

α((xn+10)x))α((xn0)x)α(x).

Hence, by Theorem 5.5, (α,H) is an int-soft (n+1)-fold positive implicative filter of H.

Definition 5.8

A soft set (α,H) is called an int-soft n-fold fantastic filter of H over U if, for all γ ∈ ℘(U), γ-inclusive set ∅︀ ≠ iH(α, γ) of (α,H) is a n-fold fantastic filter of H.

The set of all int-soft n-fold fantastic filters of H is denoted by INFF(ℋ).

Example 5.9

According to Example 3.2, it is clear that (α,H) is an int-soft n-fold fantastic filter of H over U.

Theorem 5.10

Let (α,H) ∈ INFF(ℋ). Then (α,H) is an int-soft n-fold filter of H.

Proof

Let (α,H) ∈ INFF(ℋ). Since iH(α, γ) ≠ ∅︀, there is xiH(α, γ). Define γ = α(x). Since iH(α, γ) is an n-fold fantastic filter of H, 1 ∈ iH(α, γ) and so α(x) = γα(1). Now, let x, yH such that x, xyiH(α, γ). Then γ = α(x) ∩ α(xy). Since iH(α, γ) is an n-fold fantastic filter of H, ((yn → 1) → 1) → yiH(α, γ). Thus,

α(x)α(xy)=α(x)α(x(1y))α(((yn1))y)=α(y).

Hence, iH(α, γ) is an int-soft n-fold filter of H.

Example 5.11

According to Example 5.4, (α,H) is an int-soft filter of H over U but it is not an int-soft n-fold fantastic filter of H. Because,

=α(1)=α(1)α(1(0c))α(((cn0)0)c)=2.

Theorem 5.12

Let (α,H) be a soft set of H. Then, for any x, y, zH, the following statements are equivalent:

(i) (α,H) ∈ INFF(ℋ),

(ii) α(z) ∩ α(z → (yx)) ⊆ α(((xny) → y) → x), α(x) ⊆ α(1),

(iii) α(yx) ⊆ α(((xny) → y) → x), α(x) ⊆ α(1).

Proof

(i) ⇒ (ii) Suppose (α,H) ∈ INFF(ℋ), x, y, zH and γ ∈ ℘(U) be a γ-inclusive set. Let γf = α(z) ∩ α(z → (yx)). Then γfα(z) and γfα(z → (yx)), and so ziH(α, γ) and z → (yx) ∈ iH(α, γ). Since iH(α, γ) is an n-fold fantastic filter of H, we have ((xny) → y) → xiH(α, γ), thus γfα(((xny) → y) → x). Hence,

α(z)α(z(yx))α(((xny)y)x).

Let α(x) = γ. Then xiH(α, γ). Since iH(α, γ) is an int-soft n-fold fantastic filter, we have 1 ∈ iH(α, γ) and so γα(1). Hence, α(x) ⊆ α(1).

(ii) ⇒ (iii) It is enough to let z = 1.

(iii)⇒(i) Let xiH(α, γ). Then γα(x), by assumption α(x) ⊆ α(1). Thus γα(1), and so 1 ∈ iH(α, γ). Now, suppose x, yH such that yxiH(α, γ), then γα(yx). Since α(yx) ⊆ α(((xny) → y) → x), we get that γα(((xny) → y) → x). Hence, ((xny) → y) → xiH(α, γ). Hence, iH(α, γ) is an n-fold fantastic filter of H. Therefore, (α,H) ∈ INFF(ℋ).

Theorem 5.13

Every int-soft n-fold fantastic filter of H is an int-soft (n+1)-fold fantastic filter of H.

Proof

Let (α,H) ∈ INFF(ℋ), xH and n ∈ ℕ. By Proposition 2.2(iii) and (vii), xn+1xn, then xnyxn+1y. By repeating this method, we have (xn+1y) → y ≤ (xny) → y and so

((xny)y)x((xn+1y)y)x.

Since (α,H) is an int-soft n-fold fantastic filter of H, by Theorem 5.10, (α,H) is an int-soft n-fold filter of H and by Theorem 5.12,

α(yx)α(((xny)y)x)α(((xn+1y)y)x).

Hence, by Theorem 5.12, (α,H) is an int-soft (n+1)-fold fantastic filter of H.

Theorem 5.14

Let (α,H) ∈ INPI(ℋ). Then (α,H) ∈ INFF(ℋ).

Proof

Let x, yH and n ∈ ℕ. Then by Proposition 2.2(viii), we have ((xny) → y) ⊙ (yx) ≤ (xny) → x and so yx ≤ [(xny) → y] → [(xny) → x] Also, by Proposition 2.2(v), xn ≤ ((xny) → y) → x. Then by Proposition 2.2(viii), ((xny) → y) → x) → yxny and so xny) → (((xny) → y) → x) ≤ (((xny) → y) → x) → y) → (((xny) → y) → x). Hence, yx ≤ (((xny) → y) → x) → y) → (((xny) → y) → x). Since (α,H) ∈ INPI(ℋ), by Theorem 5.3, (α,H) is an int-soft filter of H. Then by Corollary 3.4(i), α(yx) ⊆ α((((xny) → y) → x) → y) → (((xny) → y) → x)) Moreover, since (α,H) ∈ INPI(ℋ), by Theorem 5.5(iii), we get that

α(yx)α((((xny)y)x)y)(((xny)y)x))α(((xny)y)x).

Hence, α(yx) ⊆ α(((xny) → y) → x). Therefore, by Theorem 5.12, (α,H) ∈ INFF(ℋ).

6. Conclusions

In this paper, the notion of int-soft filter in hoops were introduced and related properties investigated. Characterizations of int-soft filters discussed and a congruence relation was introduced by using int-soft filter and was proved the quotient is a hoop. Also, the notions of int-soft n-fold (positive) implicative and int-soft n-fold fantastic filters of hoops were introduced, some equivalence definitions of them and relation between them studied. Moreover, it was proved that every int-soft n-fold positive implicative filter is an int-soft fantastic filter and int-soft implicative filter. Also, the quotient structure that is made by them was investigated and proved that (α,H) is an int-soft implicative filter of H if and only if H(α,H) is a Brouwerian semilattice.

In the future works, int-soft of other filters such as prime, maximal, nodal and etc of hoops and the quotient structures can be study.

References
  1. Bosbach, B (1969). Komplementäre halbgruppen. axiomatik und arithmetic. Fundamenta Mathematicae. 64, 257-287.
    CrossRef
  2. Bosbach, B (1970). Komplementäre halbgruppen kongruenzen und quotienten. Fundamenta Mathematicae. 69, 1-14.
    CrossRef
  3. Agliano, P, Ferreirim, IM, and Montagna, F (2007). Basic hoops: an algebraic study of continuous t-norms. Studia Logica. 87, 73-98. https://doi.org/10.1007/s11225-007-9078-1
    CrossRef
  4. Hajek, P (1998). Metamathematics of Fuzzy Logic. Berlin: Springer
    CrossRef
  5. Namdar, A, Borzooei, RA, Borumand Saeid, A, and Aaly Kologani, M (2017). Some results in hoop algebras. Journal of Intelligent & Fuzzy Systems. 32, 1805-1813. https://doi.org/10.3233/JIFS-152553
    CrossRef
  6. Borzooei, RA, and Kologani, A (2014). Filter theory of hoop-algebras. Journal of Advanced Research in Pure Mathematics. 6, 72-86.
    CrossRef
  7. Molodtsov, D (1999). Soft set theory: first results. Computers & Mathematics with Applications. 37, 19-31. https://doi.org/10.1016/S0898-1221(99)00056-5
    CrossRef
  8. Maji, PK, Roy, AR, and Biswas, R (2002). An application of soft sets in a decision making problem. Computers & Mathematics with Applications. 44, 1077-1083. https://doi.org/10.1016/S0898-1221(02)00216-X
    CrossRef
  9. Maji, PK, Biswas, R, and Roy, A (2003). Soft set theory. Computers & Mathematics with Applications. 45, 555-562. https://doi.org/10.1016/S0898-1221(03)00016-6
    CrossRef
  10. Chen, D, Tsang, ECC, Yeung, DS, and Wang, X (2005). The parameterization reduction of soft sets and its applications. Computers & Mathematics with Applications. 49, 757-763. https://doi.org/10.1016/j.camwa.2004.10.036
    CrossRef
  11. Feng, F (2011). Soft rough sets applied to multicriteria group decision making. Annals of Fuzzy Mathematics and Informatics. 2, 69-80.
  12. Aktas, H, and Cagman, N (2007). Soft sets and soft groups. Information Sciences. 177, 2726-2735. https://doi.org/10.1016/j.ins.2006.12.008
    CrossRef
  13. Borzooei, RA, Mobini, M, and Ebrahimi, MM (2015). The category of soft sets. Journal of Intelligent & Fuzzy Systems. 28, 157-167. https://doi.org/10.3233/IFS-141286
    CrossRef
  14. Jun, YB (2008). Soft Bck/Bci-algebras. Computers & Mathematics with Applications. 56, 1408-1413. https://doi.org/10.1016/j.camwa.2008.02.035
    CrossRef
  15. Jun, YB, Lee, KJ, and Park, CH (2010). Fuzzy soft set theory applied to BCK/BCI-algebras. Computers & Mathematics with Applications. 59, 3180-3192. https://doi.org/10.1016/j.camwa.2010.03.004
    CrossRef
  16. Maji, PK, Biswas, P, and Roy, AR (2001). A fuzzy soft sets. Journal of Fuzzy Mathematics. 9, 589-602.
  17. Luo, C, Xin, X, and He, P (2017). n-fold (positive) implicative filters of hoops. Italian Journal of Pure and Applied Mathematics. 38, 631-642.
Biographies

R. A. Borzooei is a full professor at the Shahid Beheshti University. He is currently an Managing Editor and fonder of the Iranian Journal of Fuzzy Systems, and was an editorial board 5 journals. He published more than 220 publications in journals on logical algebras, algebraic hyperstructuers, and fuzzy graph theory.

E-mail: borzooei@sub.ac.ir


M. Sabetkish is a student at department of mathematics at Shahid Beheshti University, Iran. He has been working on research related to the soft and fuzzy logical algebras. He published more than 2 publications in journals on soft and fuzzy logical algebras.

E-mail: m.sabetkish@sub.ac.ir


E. H. Roh is a full professor at the Department of Mathematics Education, Chinju National University of Education. He is currently an Editor of 4 journals. He published more than 230 publications in peer-reviewed journals or conferences.

E-mail: ehroh9988@gmail.com


M. Aaly Kologani is a assistant professor at Hatef Higher Education Institute. He published more than 20 publications in peer-reviewed journals on logical algebras, algebraic hyperstructuers.

E-mail: mona4011@gmail.com