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Computer Modeling in Engineering & Sciences
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DOI: 10.32604/cmes.2021.016996

ARTICLE

Neutrosophic N-Structures Applied to Sheffer Stroke BL-Algebras

Tugce Katican1, Tahsin Oner1, Akbar Rezaei2,* and Florentin Smarandache3

1Department of Mathematics, Ege University, Izmir, 35100, Turkey
2Department of Mathematics, Payame Noor University, Tehran, 19395-4697, Iran
3Department of Mathematics and Science, University of New Mexico, Gallup, 87301, NM, USA
*Corresponding Author: Akbar Rezaei. Email: rezaei@pnu.ac.ir
Received: 18 April 2021; Accepted: 25 May 2021

Abstract: In this paper, we introduce a neutrosophic N-subalgebra, a (ultra) neutrosophic N-filter, level sets of these neutrosophic N-structures and their properties on a Sheffer stroke BL-algebra. By defining a quasi-subalgebra of a Sheffer stroke BL-algebra, it is proved that the level set of neutrosophic N-subalgebras on the algebraic structure is its quasi-subalgebra and vice versa. Then we show that the family of all neutrosophic N-subalgebras of a Sheffer stroke BL-algebra forms a complete distributive lattice. After that a (ultra) neutrosophic N-filter of a Sheffer stroke BL-algebra is described, we demonstrate that every neutrosophic N-filter of a Sheffer stroke BL-algebra is its neutrosophic N-subalgebra but the inverse is generally not true. Finally, it is presented that a level set of a (ultra) neutrosophic N-filter of a Sheffer stroke BL-algebra is also its (ultra) filter and the inverse is always true. Moreover, some features of neutrosophic N-structures on a Sheffer stroke BL-algebra are investigated.

Keywords: Sheffer stroke BL-algebra; (ultra) filter; neutrosophic 𝒩-subalgebra; (ultra) neutrosophic 𝒩-filter

1  Introduction

Fuzzy set theory, which has the truth (t) (membership) function and state positive meaning of information, is introduced by Zadeh [1] as a generalization the classical set theory. This led scientists to find negative meaning of information. Hence, intuitionistic fuzzy sets [2] which are fuzzy sets with the falsehood (f) (nonmembership) function were introduced by Atanassov. However, there exist uncertainty and vagueness in the language, as well as positive ana negative meaning of information. Thus, Smarandache defined neutrosophic sets which are intuitionistic fuzzy sets with the indeterminacy/neutrality (i) function [3,4]. Thereby, neutrosophic sets are determined on three components: (t,i,f):(truth, indeterminacy, falsehood) [5]. Since neutrosophy enables that information in language can be comprehensively examined at all points, many researchers applied neutrosophy to different theoretical areas such as BCK/BCI-algebras, BE-algebras, semigroups, metric spaces, Sheffer stroke Hilbert algebras and strong Sheffer stroke non-associative MV-algebras [6–15] so as to improve devices imitating human behaviours and thoughts, artificial intelligence and technological tools.

Sheffer stroke (or Sheffer operation) was originally introduced by Sheffer [16]. Since Sheffer stroke can be used by itself without any other logical operators to build a logical system which is easy to control, Sheffer stroke can be applied to many logical algebras such as Boolean algebras [17], ortholattices [18], Sheffer stroke Hilbert algebras [19]. On the other side, BL-algebras were introduced by Hájek as an axiom system of his Basic Logic (BL) for fuzzy propositional logic, and he widely studied many types of filters [20]. Moreover, Oner et al. [21] introduced BL-algebras with Sheffer operation and investigated some types of (fuzzy) filters.

We give fundamental definitions and notions about Sheffer stroke BL-algebras, N-functions and neutrosophic N-structures defined by these functions on a crispy set X. Then a neutrosophic N-subalgebra and a (τ,γ,ρ)-level set of a neutrosophic N-structure are presented on Sheffer stroke BL-algebras. By defining a quasi-subalgebra of a Sheffer stroke BL-algebra, it is proved that every (τ,γ,ρ)-level set of a neutrosophic N-subalgebra of the algebra is the quasi-subalgebra and the inverse is true. Also, we show that the family of all neutrosophic N-subalgebras of this algebraic structure forms a complete distributive lattice. Some properties of neutrosophic N-subalgebras of Sheffer stroke BL-algebras are examined. Indeed, we investigate the case which N-functions defining a neutrosophic N-subalgebra of a Sheffer stroke BL-algebra are constant. Moreover, we define a (ultra) neutrosophic N-filter of a Sheffer stroke BL-algebra by N-functions and analyze many features. It is demonstrated that (τ,γ,ρ)-level set of a neutrosophic N-filter of a Sheffer stroke BL-algebra is its filter but the inverse does not hold in general. In fact, we propound that (τ,γ,ρ)-level set of a (ultra) neutrosophic N-filter of a Sheffer stroke BL-algebra is its (ultra) filter and the inverse is true. Finally, new subsets of a Sheffer stroke BL-algebra are defined by the N-functions and special elements of the algebra. It is illustrated that these subsets are (ultra) filters of a Sheffer stroke BL-algebra for the (ultra) neutrosophic N-filter but the special conditions are necessary to prove the inverse.

2  Preliminaries

In this section, basic definitions and notions on Sheffer stroke BL-algebras and neutrosophic N-structures.

Definition 2.1. [18] Let H=⟨H,∣⟩ be a groupoid. The operation ∣ is said to be a Sheffer stroke (or Sheffer operation) if it satisfies the following conditions:

(S1) x∣y=y∣x,

(S2) (x∣x)∣(x∣y)=x,

(S3) x∣((y∣z)∣(y∣z))=((x∣y)∣(x∣y))∣z,

(S4) (x∣((x∣x)∣(y∣y)))∣(x∣((x∣x)∣(y∣y)))=x.

Definition 2.2. [21] A Sheffer stroke BL-algebra is an algebra (C,∨,∧,∣,0,1) of type (2, 2, 2, 0, 0) satisfying the following conditions:

(sBL −1) (C,∨,∧,0,1) is a bounded lattice,

(sBL −2) (C,∣) is a groupoid with the Sheffer stroke,

(sBL −3) c1∧c2=(c1∣(c1∣(c2∣c2)))∣(c1∣(c1∣(c2∣c2))),

(sBL −4) (c1∣(c2∣c2))∨(c2∣(c1∣c1))=1,

for all c1,c2∈C.

1=0∣0 is the greatest element and 0=1∣1 is the least element of C.

Proposition 2.1. [21] In any Sheffer stroke BL-algebra C, the following features hold, for all c1,c2,c3∈C:

(1) c1∣((c2∣(c3∣c3))∣(c2∣(c3∣c3)))=c2∣((c1∣(c3∣c3))∣(c1∣(c3∣c3))),

(2) c1∣(c1∣c1)=1,

(3) 1∣(c1∣c1)=c1,

(4) c1∣(1∣1)=1,

(5) (c1∣1)∣(c1∣1)=c1,

(6) (c1∣c2)∣(c1∣c2)≤c3⇔c1≤c2∣(c3∣c3)

(7) c1≤c2 iff c1∣(c2∣c2)=1,

(8) c1≤c2∣(c1∣c1),

(9) c1≤(c1∣c2)∣c2,

(10) (a) (c1∣(c1∣(c2∣c2)))∣(c1∣(c1∣(c2∣c2)))≤c1,

  (b) (c1∣(c1∣(c2∣c2)))∣(c1∣(c1∣(c2∣c2)))≤c2.

(11) If c1≤c2, then

  (i) c3∣(c1∣c1)≤c3∣(c2∣c2),

  (ii) (c1∣c3)∣(c1∣c3)≤(c2∣c3)∣(c2∣c3),

  (iii) c2∣(c3∣c3)≤c1∣(c3∣c3).

(12) c1∣(c2∣c2)≤(c3∣(c1∣c1))∣((c3∣(c2∣c2))∣(c3∣(c2∣c2))),

(13) c1∣(c2∣c2)≤(c2∣(c3∣c3))∣((c1∣(c3∣c3))∣(c1∣(c3∣c3))),

(14) ((c1∨c2)∣c3)∣((c1∨c2)∣c3)=((c1∣c3)∣(c1∣c3))∨((c2∣c3)∣(c2∣c3)),

(15) c1∨c2=((c1∣(c2∣c2))∣(c2∣c2))∧((c2∣(c1∣c1))∣(c1∣c1)).

Lemma 2.1. [21] Let C be a Sheffer stroke BL-algebra. Then

(c1∣(c2∣c2))∣(c2∣c2)=(c2∣(c1∣c1))∣(c1∣c1),

for all c1,c2∈C.

Corollary 2.1. [21] Let C be a Sheffer stroke BL-algebra. Then

c1∨c2=(c1∣(c2∣c2))∣(c2∣c2),

for all c1,c2∈C.

Lemma 2.2. [21] Let C be a Sheffer stroke BL-algebra. Then

c1∣((c2∣(c3∣c3))∣(c2∣(c3∣c3)))=(c1∣(c2∣c2))∣((c1∣(c3∣c3))∣(c1∣(c3∣c3))),

for all c1,c2,c3∈C.

Definition 2.3. [21] A filter of C is a nonempty subset P⊆C satisfying

(SF−1) ifc1,c2∈P, then (c1∣c2)∣(c1∣c2)∈P,(SF−2) if c1∈P and c1≤c2, then c2∈P.

Proposition 2.2. [21] Let P be a nonempty subset of C. Then P is a filter of C if and only if the following hold:

(SF−3)1∈P,(SF−4) c1∈P and c1∣(c2∣c2)∈P imply c2∈P.

Definition 2.4. [21] Let P be a filter of C. Then P is called an ultra filter of C if it satisfies c∈P or c∣c∈P, for all c∈C.

Lemma 2.3. [21] A filter P of C is an ultra filter of C if and only if c1∨c2∈P implies c1∈P or c2∈P, for all c1,c2∈C.

Definition 2.5. [8] F(X,[−1,0]) denotes the collection of functions from a set X to [−1,0] and an element of F(X,[−1,0]) is called a negative-valued function from X to [−1,0] (briefly, N-function on X). An N-structure refers to an ordered pair (X, f) of X and N-function f on X.

Definition 2.6. [12] A neutrosophic N-structure over a nonempty universe X is defined by

XN:=X(TN,IN,FN)={x(TN(x),IN(x),FN(x)):x∈X}

where TN, IN and FN are N-functions on X, called the negative truth membership function, the negative indeterminacy membership function and the negative falsity membership function, respectively.

Every neutrosophic N-structure XN over X satisfies the condition (∀x∈X)(−3≤TN(x)+IN(x)+FN(x)≤0).

Definition 2.7. [13] Let XN be a neutrosophic N-structure on a set X and τ,γ,ρ be any elements of [ −1, 0] such that −3≤τ+γ+ρ≤0. Consider the following sets:

TNτ:={x∈X:TN(x)≤τ},INγ:={x∈X:IN(x)≥γ}

and

FNρ:={x∈X:FN(x)≤ρ}.

The set

XN(τ,γ,ρ):={x∈X:TN(x)≤τ,IN(x)≥γ and TN(x)≤ρ}

is called the (τ,γ,ρ)-level set of XN. Moreover, XN(τ,γ,ρ)=TNτ∩INγ∩FNρ.

Consider sets

XNct:={x∈X:TN(x)≤TN(ct)},XNci:={x∈X:IN(x)≥IN(ci)}

and

XNcf:={x∈X:FN(x)≤FN(cf)},

for any ct,ci,cf∈X. Obviously, ct∈XNct,ci∈XNci and cf∈XNcf [13].

3  Neutrosophic N-Structures

In this section, neutrosophic N-subalgebras and neutrosophic N-filters on Sheffer stroke BL-algebras. Unless otherwise specified, C denotes a Sheffer stroke BL-algebra.

Definition 3.1. A neutrosophic N-structure CN on a Sheffer stroke BL-algebra C is called a neutrosophic N-subalgebra of C if the following condition is valid:

min{TN(c1),TN(c2)}≤TN(c1∣(c2∣c2)),max{IN(c1),IN(c2)}≥IN(c1∣(c2∣c2)) andmax{FN(c1),FN(c2)}≥FN(c1∣(c2∣c2)), (1)

for all c1,c2∈C.

Example 3.1. Consider a Sheffer stroke BL-algebra C where the set C={0,a,b,c,d,e,f,1} and the Sheffer operation ∣, the join operation ∨ and the meet operation ∧ on C has the Cayley tables in Tab. 1 [21]. Then a neutrosophic N-structure

CN={x(−0.08,−0.999,−0.26):x=d,1}∪{x(−0.92,−0.52,−0.0012):x∈C−{d,1}}

on C is a neutrosophic N-subalgebra of C.

Definition 3.2. Let CN be a neutrosophic N-structure on a Sheffer stroke BL-algebra C and τ,γ,ρ be any elements of [ −1, 0] such that −3≤τ+γ+ρ≤0. For the sets

TNτ:={c∈C:TN(c)≥τ},INγ:={c∈C:IN(c)≤γ}

and

FNρ:={c∈C:FN(c)≤ρ},

the set

CN(τ,γ,ρ):={c∈C:TN(c)≥τ,IN(c)≤γ and FN(c)≤ρ}

is called the (τ,γ,ρ)-level set of CN. Moreover, CN(τ,γ,ρ)=TNτ∩INγ∩FNρ.

Definition 3.3. A subset D of a Sheffer stroke BL-algebra C is called a quasi-subalgebra of C if c1∣(c2∣c2)∈D, for all c1,c2∈D. Obviously, C itself and {1} are quasi-subalgebras of C.

Example 3.2. Consider the Sheffer stroke BL-algebra C in Example 3.1. Then {0,a,f,1} is a quasi-subalgebra of C.

Theorem 3.1. Let CN be a neutrosophic N-structure on a Sheffer stroke BL-algebra C and τ,γ,ρ be any elements of [ −1, 0] such that −3≤τ+γ+ρ≤0. If CN is a neutrosophic N-subalgebra of C, then the nonempty level set CN(τ,γ,ρ) of CN is a quasi-subalgebra of C.

Proof. Let CN be a neutrosophic N-subalgebra of C and c1, c2 be any elements of CN(τ,γ,ρ), for τ,γ,ρ∈[−1,0] with −3≤τ+γ+ρ≤0. Then TN(c1),TN(c2)≥τ,IN(c1),IN(c2)≤γ and FN(c1),FN(c2)≤ρ. Since

τ≤min{TN(c1),TN(c2)}≤TN(c1∣(c2∣c2)),IN(c1∣(c2∣c2))≤max{IN(c1),IN(c2)}≤γ

and

FN(c1∣(c2∣c2))≤max{FN(c1),FN(c2)}≤ρ,

for all c1,c2∈C, we obtain that c1∣(c2∣c2)∈TNτ, c1∣(c2∣c2)∈INγ and c1∣(c2∣c2)∈FNρ, and so, c1∣(c2∣c2)∈TNτ∩INγ∩FNρ=CN(τ,γ,ρ). Hence, CN(τ,γ,ρ) is a quasi-subalgebra of C.

Theorem 3.2. Let CN be a neutrosophic N-structure on a Sheffer stroke BL-algebra C and TNτ,INγ and FNρ be quasi-subalgebras of C, for all τ,γ,ρ∈[−1,0] with −3≤τ+γ+ρ≤0. Then CN is a neutrosophic N-subalgebra of C.

Proof. Let CN be a neutrosophic N-structure on a Sheffer stroke BL-algebra C, and TNτ,INγ and FNρ be quasi-subalgebras of C, for all τ,γ,ρ∈[−1,0] with −3≤τ+γ+ρ≤0. Suppose that c1 and c2 be any elements of C such that w1=TN(c1∣(c2∣c2))<min{TN(c1),TN(c2)}=w2, c1 and c1. If τ1=12(w1+w2)∈[−1,0), γ1=12(t1+t2)∈[−1,0) and ρ1=12(r1+r2)∈[−1,0), then , and . Thus, c1,c2∈TNτ1, c1,c2∈INγ1 and c1,c2∈FNρ1 but c1∣(c2∣c2)∉TNτ1, c1∣(c2∣c2)∉INγ1 and c1∣(c2∣c2)∉FNρ1, which are contradictions. Hence, min{TN(c1),TN(c2)}≤TN(c1∣(c2∣c2)), IN(c1∣(c2∣c2))≤max{IN(c1),IN(c2)} and FN(c1∣(c2∣c2))≤max{FN(c1),FN(c2)}, for all c1,c2∈C. Thereby, CN is a neutrosophic N-subalgebra of C.

Theorem 3.3. Let {CNi:i∈N} be a family of all neutrosophic N-subalgebras of a Sheffer stroke BL-algebra C. Then {CNi:i∈N} forms a complete distributive lattice.

Proof. Let D be a nonempty subset of {CNi:i∈N}. Since CNi is a neutrosophic N-subalgebra of C, for all i∈N, it satisfies the condition (1). Then ⋂D satisfies the condition (1). Thus, ⋂D is a neutrosophic N-subalgebra of C. Let E be a family of all neutrosophic N-subalgebras of C containing ⋃{CNi:i∈N}. Thus, ⋂E is also a neutrosophic N-subalgebra of C. If ⋀i∈NCNi=⋂i∈NCNi and ⋁i∈NCNi=⋂E, then ({CNi:i∈N},⋁,⋀) forms a complete lattice. Also, it is distibutive by the definitions of ⋁ and ⋀.

Lemma 3.1. Let CN be a neutrosophic N-subalgebra of a Sheffer stroke BL-algebra C. Then TN(c)≤TN(1), IN(c)≥IN(1) and FN(c)≥FN(1), for all c∈C.

Proof. Let CN be a neutrosophic N-subalgebra of C. Then it follows from Poposition 2.1 (2) that

TN(c)=min{TN(c),TN(c)}≤TN(c∣(c∣c))=TN(1),IN(1)=IN(c∣(c∣c))≤max{IN(c),IN(c)}=IN(c)

and

FN(1)=FN(c∣(c∣c))≤max{FN(c),FN(c)}=FN(c),

for all c∈C.

The inverse of Lemma 3.1 is not true in general.

Example 3.3. Consider the Sheffer stroke BL-algebra C in Example 3.1. Then a neutrosophic N-structure

CN={x(−0.01,−0.1,−0.11):x=a,b,1}∪{x(−0.1,−0.01,−0.01):x∈C−{a,b,1}}

on C is not a neutrosophic N-subalgebra of C since max{FN(a),FN(b)}=−0.11<−0.01=FN(f)=FN(a∣(b∣b)).

Lemma 3.2. A neutrosophic N-subalgebra CN of a Sheffer stroke BL-algebra C satisfies TN(c1)≤TN(c1∣(c2∣c2)), IN(c1)≥IN(c1∣(c2∣c2)) and FN(c1)≥FN(c1∣(c2∣c2)), for all c1,c2∈C if and only if TN, IN and FN are constant.

Proof. Let CN be a a neutrosophic N-subalgebra of C such that TN(c1)≤TN(c1∣(c2∣c2)), IN(c1)≥IN(c1∣(c2∣c2)) and FN(c1)≥FN(c1∣(c2∣c2)), for all c1,c2∈C. Since TN(1)≤TN(1∣(c∣c))=TN(c), IN(1)≥IN(1∣(c∣c))=IN(c) and FN(1)≥FN(1∣(c∣c))=FN(c) from Proposition 2.1 (3), it is obtained from Lemma 3.1 that TN(c) = TN(1), IN(c) = IN(1) and FN(c) = FN(1), for all c∈C. Hence, TN, IN and FN are constant.

Conversely, it is obvious since TN, IN and FN are constant.

Definition 3.4. A neutrosophic N-structure CN on a Sheffer stroke BL-algebra C is called a neutrosophic N-filter of C if

1.    c1≤c2 implies TN(c1)≤TN(c2), IN(c2)≤IN(c1) and FN(c2)≤FN(c1),

2.    min{TN(c1),TN(c2)}≤TN((c1∣c2)∣(c1∣c2)), IN((c1∣c2)∣(c1∣c2))≤max{IN(c1),IN(c2)} and FN((c1∣c2)∣(c1∣c2))≤max{FN(c1),FN(c2)},

for all c1,c2∈C.

Example 3.4. Consider the Sheffer stroke BL-algebra C in Example 3.1. Then a neutrosophic N-structure

CN={x(−0.3,−1,−0.15):x=c,e,f,1}∪{x(−1,−0.7,0):x=0,a,b,d}

on C is a neutrosophic N-filter of C.

Theorem 3.4. Let CN be a a neutrosophic N-structure on a Sheffer stroke BL-algebra C. Then CN is a neutrosophic N-filter of C if and only if

min{TN(c1),TN(c1∣(c2∣c2))}≤TN(c2)≤TN(1),IN(1)≤IN(c2)≤max{IN(c1),IN(c1∣(c2∣c2))} andFN(1)≤FN(c2)≤max{FN(c1),FN(c1∣(c2∣c2))}, (2)

for all c1,c2∈C.

Proof. Let CN be a neutrosophic N-filter of C. Then it follows from (sBL-3) and Definition 3.4 that

min{TN(c1),TN(c1∣(c2∣c2))}≤TN((c1∣(c1∣(c2∣c2)))∣(c1∣(c1∣(c2∣c2))))=TN(c1∧c2)≤TN(c2)≤TN(1),IN(1)≤IN(c2)≤IN(c1∧c2)=IN((c1∣(c1∣(c2∣c2)))∣(c1∣(c1∣(c2∣c2))))≤max{IN(c1),IN(c1∣(c2∣c2))}

and

FN(1)≤FN(c2)≤FN(c1∧c2)=FN((c1∣(c1∣(c2∣c2)))∣(c1∣(c1∣(c2∣c2))))≤max{FN(c1),FN(c1∣(c2∣c2))},

for all c1,c2∈C.

Conversely, let CN be a a neutrosophic N-structure on C satisfying the condition (2). Assume that c1≤c2. Then c1∣(c2∣c2)=1 from Proposition 2.1 (7). Thus,

TN(c1)=min{TN(c1),TN(1)}=min{TN(c1),TN(c1∣(c2∣c2))}≤TN(c2),IN(c2)≤max{IN(c1),IN(c1∣(c2∣c2))}=max{IN(c1),IN(1)}=IN(c1)

and

FN(c2)≤max{FN(c1),FN(c1∣(c2∣c2))}=max{FN(c1),FN(1)}=FN(c1),

for all c1,c2∈C. Also, it follows from Proposition 2.1 (9), (S1) and (S2) that

min{TN(c1),TN(c2)}≤min{TN(c1),TN(c1∣(c1∣c2))}=min{TN(c1),TN(c1∣(((c1∣c2)∣(c1∣c2))∣((c1∣c2)∣(c1∣c2))))}≤TN((c1∣c2)∣(c1∣c2)), IN((c1∣c2)∣(c1∣c2))≤max{IN(c1),IN(c1∣(((c1∣c2)∣(c1∣c2))∣((c1∣c2)∣(c1∣c2))))}=max{IN(c1),IN(c1∣(c1∣c2))}≤max{IN(c1),IN(c2)}

and

FN((c1∣c2)∣(c1∣c2))≤max{FN(c1),FN(c1∣(((c1∣c2)∣(c1∣c2))∣((c1∣c2)∣(c1∣c2))))}=max{FN(c1),FN(c1∣(c1∣c2))}≤max{FN(c1),FN(c2)},

for all c1,c2∈C. Thus, CN is a neutrosophic N-filter of C.

Corollary 3.1. Let CN be a neutrosophic N-filter of a Sheffer stroke BL-algebra C. Then

1.    min{TN(c3),TN(c3∣(((c2∣(c1∣c1))∣(c1∣c1))∣((c2∣(c1∣c1))∣(c1∣c1))))}≤TN((c1∣(c2∣c2))∣(c2∣c2)), IN((c1∣(c2∣c2))∣(c2∣c2))≤max{IN(c3),IN(c3∣(((c2∣(c1∣c1))∣(c1∣c1))∣((c2∣(c1∣c1))∣(c1∣c1))))} and FN((c1∣(c2∣c2))∣(c2∣c2))≤max{FN(c3),FN(c3∣(((c2∣(c1∣c1))∣(c1∣c1))∣((c2∣(c1∣c1))∣(c1∣c1))))},

2.    min{TN(c3),TN(c3∣((c1∣(c2∣c2))∣∣(c1∣(c2∣c2))))}≤TN(c1∣(c2∣c2)), IN(c1∣(c2∣c2))≤max{IN(c3),IN(c3∣((c1∣(c2∣c2))∣(c1∣(c2∣c2))))} and FN(c1∣(c2∣c2))≤max{FN(c3),FN(c3∣((c1∣(c2∣c2))∣(c1∣(c2∣c2))))},

3.    min{TN(c1∣((c2∣(c3∣c3))∣∣(c2∣(c3∣c3)))),TN(c1∣(c2∣c2))}≤TN(c1∣(c3∣c3)), IN(c1∣(c3∣c3))≤max{IN(c1∣((c2∣(c3∣c3))∣(c2∣(c3∣c3)))),IN(c1∣(c2∣c2))} and FN(c1∣(c3∣c3))≤max{FN(c1∣((c2∣(c3∣c3))∣(c2∣(c3∣c3)))),FN(c1∣(c2∣c2))},

4.    TN(c1∣(c2∣c2))=TN(1), IN(c1∣(c2∣c2))=IN(1) and FN(c1∣(c2∣c2))=FN(1) imply TN(c1)≤TN(c2), IN(c2)≤IN(c1) and FN(c2)≤FN(c1),

for all c1,c2,c3∈C.

Proof. It is proved from Theorem 3.4, Lemma 2.1 and Lemma 2.2.

Lemma 3.3. Let CN be a neutrosophic N-structure on a Sheffer stroke BL-algebra C. Then CN is a neutrosophic N-filter of C if and only if

c1≤c2∣(c3∣c3) implies (min{TN(c1),TN(c2)}≤TN(c3),IN(c3)≤max{IN(c1),IN(c2)}  andFN(c3)≤max{FN(c1),FN(c2)},) (3)

for all c1,c2,c3∈C.

Proof. Let CN be a neutrosophic N-filter of C and c1≤c2∣(c3∣c3). Then it is obtained from Definition 3.4 (1) and Theorem 3.4 that

min{TN(c1),TN(c2)}≤min{TN(c2),TN(c2∣(c3∣c3))}≤TN(c3),IN(c3)≤max{IN(c2),IN(c2∣(c3∣c3))}≤max{IN(c1),IN(c2)}

and

FN(c3)≤max{FN(c2),FN(c2∣(c3∣c3))}≤max{FN(c1),FN(c2)},

for all c1,c2,c3∈C.

Conversely, let CN be a neutrosophic N-structure on C satisfying the condition (3). Since it is known from Proposition 2.1 (4) that c≤1=c∣(1∣1), for all c∈C, we get that TN(c)=min{TN(c),TN(c)}≤TN(1), {IN(1)≤max{IN(c),IN(c)}=IN(c)} and {FN(1)≤max{FN(c),FN(c)}=FN(c)}, for all c∈C. Suppose that c1≤c2. Since we have c1≤c2=1∣(c2∣c2) from Proposition 2.1 (3), it is obtained that TN(c1)=min{TN(c1),TN(1)}≤TN(c2), IN(c2)≤max{IN(c1),IN(1)}=IN(c1) and FN(c2)≤max{FN(c1),FN(1)}=FN(c1). Since c1≤(c1∣c2)∣c2=c2∣(((c1∣c2)∣(c1∣c2))∣((c1∣c2)∣(c1∣c2))) from Proposition 2.1 (9), (S1) and (S2), it follows that

min{TN(c1),TN(c2)}≤TN((c1∣c2)∣(c1∣c2)),IN((c1∣c2)∣(c1∣c2))≤max{IN(c1),IN(c2)}

and

FN((c1∣c2)∣(c1∣c2))≤max{FN(c1),FN(c2)},

for all c1,c2∈C. Thus, CN is a neutrosophic N-filter of C.

Lemma 3.4. Every neutrosophic N-filter of a Sheffer stroke BL-algebra C is a neutrosophic N-subalgebra of C.

Proof. Let CN be a neutrosophic N-filter of C. Since

((c1∣c2)∣(c1∣c2))∣((c1∣(c2∣c2))∣(c1∣(c2∣c2)))=c1∣((((c1∣c2)∣(c1∣c2))∣(c2∣c2))∣(((c1∣c2)∣(c1∣c2))∣(c2∣c2)))=c1∣((c1∣((c2∣(c2∣c2))∣(c2∣(c2∣c2))))∣(c1∣((c2∣(c2∣c2))∣(c2∣(c2∣c2)))))=c1∣((c1∣(1∣1))∣(c1∣(1∣1)))=c1∣(1∣1)=1

from Proposition 2.1 (1), (2), (4) and (S3), it follows from Proposition 2.1 (7) that (c1∣c2)∣(c1∣c2)≤c1∣(c2∣c2), for all c1,c2∈C. Then

min{TN(c1),TN(c2)}≤TN((c1∣c2)∣(c1∣c2))≤TN(c1∣(c2∣c2)),IN(c1∣(c2∣c2))≤IN((c1∣c2)∣(c1∣c2))≤max{IN(c1),IN(c2)}

and

FN(c1∣(c2∣c2))≤FN((c1∣c2)∣(c1∣c2))≤max{FN(c1),FN(c2)},

for all c1,c2∈C. Thereby, CN is a neutrosophic N-subalgebra of C.

The inverse of Lemma 3.4 is usually not true.

Example 3.5. Consider the Sheffer stroke BL-algebra C in Example 3.1. Then a neutrosophic N-structure

CN={0(−1,0,0),1(0,−1,−1)}∪{x(−0.5,−0.5,−0.5):x∈C−{0,1}}

on C is a neutrosophic N-subalgebra of C whereas it is not a neutrosophic N-filter of C since min{TN(a),TN(b)}=−0.5>−1=TN((a∣b)∣(a∣b)).

Definition 3.5. Let CN be a neutrosophic N-structure on a Sheffer stroke BL-algebra C. Then an ultra neutrosophic N-filter CN of C is a neutrosophic N-filter of C satisfying TN(c) = TN(1), IN(c) = IN(1), FN(c) = FN(1) or TN(c∣c)=TN(1), IN(c∣c)=IN(1), FN(c∣c)=FN(1), for all c∈C.

Example 3.6. Consider the Sheffer stroke BL-algebra C in Example 3.1. Then a neutrosophic N-structure

CN={x(−0.02,−0.77,−0.6):x=b,d,f,1}∪{x(−0.79,−0.05,−0.41):x=0,a,c,e}

on C is an ultra neutrosophic N-filter of C.

Remark 3.1. By Definition 3.5, every ultra neutrosophic N-filter of a Sheffer stroke BL-algebra C is a neutrosophic N-filter of C but the inverse does not generally hold.

Example 3.7. Consider the Sheffer stroke BL-algebra C in Example 3.1. Then a neutrosophic N-filter

CN={x(−0.18,−0.82,−0.57): x=e,1}∪{x(−1,−0.64,−0.43):x∈C−{e,1}}

of C is not ultra since TN(a)≠TN(1)≠TN(a∣a)=TN(f), IN(a)≠IN(1)≠IN(a∣a)=IN(f) and FN(a)≠FN(1)≠TFN(a∣a)=FN(f).

Lemma 3.5. Let CN be a neutrosophic N-filter of a Sheffer stroke BL-algebra C. Then CN is an ultra neutrosophic N-filter of C if and only if TN(c1)≠TN(1),TN(c2)≠TN(1), IN(c1)≠IN(1),IN(c2)≠IN(1) and FN(c1)≠FN(1),FN(c2)≠FN(1) imply TN(c1∣(c2∣c2))=TN(1)=TN(c2∣(c1∣c1)), IN(c1∣(c2∣c2))=IN(1)=IN(c2∣(c1∣c1)) and FN(c1∣(c2∣c2))=FN(1)=FN(c2∣(c1∣c1)), for all c1,c2∈C.

Proof. Let CN be an ultra neutrosophic N-filter of C, and TN(c1)≠TN(1),TN(c2)≠TN(1), IN(c1)≠IN(1),IN(c2)≠IN(1) and FN(c1)≠FN(1),FN(c2)≠FN(1), for any c1,c2∈C. Then TN(c1∣c1)=TN(1)=TN(c2∣c2), IN(c1∣c1)=IN(1)=IN(c2∣c2) and FN(c1∣c1)=FN(1)=FN(c2∣c2). Since

(c1∣c1)∣((c1∣(c2∣c2))∣(c1∣(c2∣c2)))=(c2∣c2)∣((c1∣(c1∣c1))∣(c1∣(c1∣c1)))=(c2∣c2)∣(1∣1)=1

and

(c2∣c2)∣((c2∣(c1∣c1))∣(c2∣(c1∣c1)))=(c1∣c1)∣((c2∣(c2∣c2))∣(c2∣(c2∣c2)))=(c1∣c1)∣(1∣1)=1

from (S1), (S3), Proposition 2.1 (2) and (4), it follows from Theorem 3.4 that

TN(1)=min{TN(1),TN(1)}=min{TN(c1∣c1),TN((c1∣c1)∣((c1∣(c2∣c2))∣(c1∣(c2∣c2))))}≤TN(c1∣(c2∣c2)),IN(c1∣(c2∣c2))≤max{IN(c1∣c1),IN((c1∣c1)∣((c1∣(c2∣c2))∣(c1∣(c2∣c2))))}=max{IN(1),IN(1)}=IN(1),FN(c1∣(c2∣c2))≤max{FN(c1∣c1),FN((c1∣c1)∣((c1∣(c2∣c2))∣(c1∣(c2∣c2))))}=max{FN(1),FN(1)}=FN(1),

and similarly, TN(1)≤TN(c2∣(c1∣c1)), IN(c2∣(c1∣c1))≤IN(1), FN(c2∣(c1∣c1))≤FN(1). Hence, we obtain from Theorem 3.4 that TN(c1∣(c2∣c2))=TN(1)=TN(c2∣(c1∣c1)), IN(c1∣(c2∣c2))=IN(1)=IN(c2∣(c1∣c1)) and FN(c1∣(c2∣c2))=FN(1)=FN(c2∣(c1∣c1)), for all c1,c2∈C.

Conversely, let CN be a neutrosophic N-filter of C such that TN(c1)≠TN(1),TN(c2)≠TN(1), IN(c1)≠IN(1),IN(c2)≠IN(1) and FN(c1)≠FN(1),FN(c2)≠FN(1) imply TN(c1∣(c2∣c2))=TN(1)=TN(c2∣(c1∣c1)), IN(c1∣(c2∣c2))=IN(1)=IN(c2∣(c1∣c1)) and FN(c1∣(c2∣c2))=FN(1)=FN(c2∣(c1∣c1)), for all c1,c2∈C. Assume that TN(c)≠TN(1)≠TN(0)=TN(1∣1), IN(c)≠IN(1)≠IN(0)=IN(1∣1) and FN(c)≠FN(1)≠FN(0)=FN(1∣1). Hence, TN(c∣c)=TN(1∣((c∣c)∣(c∣c)))=TN(c∣1)=TN(c∣((1∣1)∣(1∣1)))=TN(1), TN((1∣1)∣(c∣c))=TN(1), IN(c∣c)=IN(1∣((c∣c)∣(c∣c)))=IN(c∣1)=IN(c∣((1∣1)∣(1∣1)))=IN(1), IN((1∣1)∣(c∣c))=IN(1) and FN(c∣c)=FN(1∣((c∣c)∣(c∣c)))=FN(c∣1)=FN(c∣((1∣1)∣(1∣1)))=FN(1), FN((1∣1)∣(c∣c))=FN(1) from Proposition 2.1 (3), (4), (S1) and (S2). Suppose that TN(c∣c)≠TN(1)≠TN(0)=TN(1∣1), IN(c)≠IN(1)≠IN(0)=IN(1∣1) and FN(c)≠FN(1)≠FN(0)=FN(1∣1). Thus, TN(c)=TN(1∣(c∣c))=TN((c∣c)∣((1∣1)∣(1∣1)))=TN(1), TN((1∣1)∣((c∣c)∣(c∣c)))=TN(1), IN(c)=IN(1∣(c∣c))=IN((c∣c)∣((1∣1)∣(1∣1)))=IN(1), IN((1∣1)∣((c∣ c)∣(c∣c)))=IN(1) and FN(c)=FN(1∣(c∣c))=FN((c∣c)∣((1∣1)∣(1∣1)))=FN(1), FN((1∣1)∣((c∣c)∣(c∣c)))=FN(1) from Proposition 2.1 (3), (4), (S1) and (S2). Therefore, CN is an ultra neutrosophic N-filter of C.

Lemma 3.6. Let CN be a neutrosophic N-filter of a Sheffer stroke BL-algebra C. Then CN is an ultra neutrosophic N-filter of C if and only if TN(c1∨c2)≤TN(c1)∨TN(c2), IN(c1)∨IN(c2)≤IN(c1∨c2) and FN(c1)∨FN(c2)≤FN(c1∨c2), for all c1,c2∈C.

Proof. Let CN be an ultra neutrosophic N-filter of C. If TN(c1) = TN(1), IN(c1) = IN(1), FN(c1) = FN(1) or TN(c2) = TN(1), IN(c2) = IN(1), FN(c2) = FN(1), then the proof is completed from Theorem 3.4. Assume that TN(c1)≠TN(1)≠TN(c2),IN(c1)≠IN(1)≠IN(c2) and FN(c1)≠FN(1)≠FN(c2). Thus, we have from Lemma 3.5 that TN(c1∣(c2∣c2))=TN(1)=TN(c2∣(c1∣c1)), IN(c1∣(c2∣c2))=IN(1)=IN(c2∣(c1∣c1)) and FN(c1∣(c2∣c2))=FN(1)=FN(c2∣(c1∣c1)), for all c1,c2∈C. Since

TN(c1∨c2)=min{TN(1),TN(c1∨c2)}=min{TN(c1∣(c2∣c2)),TN((c1∣(c2∣c2))∣(c2∣c2))}≤TN(c2),IN(c2)≤max{IN(c1∣(c2∣c2)),IN((c1∣(c2∣c2))∣(c2∣c2))}=max{IN(1),IN(c1∨c2)}=IN(c1∨c2),FN(c2)≤max{FN(c1∣(c2∣c2)),FN((c1∣(c2∣c2))∣(c2∣c2))}=max{FN(1),IN(c1∨c2)}=FN(c1∨c2),

and similarly, TN(c1∨c2)=TN(c2∨c1)≤TN(c1), IN(c1)≤IN(c2∨c1)=IN(c1∨c2), FN(c1)≤FN(c2∨c1)=FN(c1∨c2) from Corollary 2.1 and Theorem 3.4, it follows that TN(c1∨c2)≤TN(c1)∨TN(c2), IN(c1)∨IN(c2)≤IN(c1∨c2) and FN(c1)∨FN(c2)≤FN(c1∨c2), for all c1,c2∈C.

Conversely, let CN be a neutrosophic N-filter of C satisfying that TN(c1∨c2)≤TN(c1)∨TN(c2), IN(c1)∨IN(c2)≤IN(c1∨c2) and FN(c1)∨FN(c2)≤FN(c1∨c2), for any c1,c2∈C. Since

TN(1)=TN(c∣(c∣c))=TN((c∣((c∣c)∣(c∣c)))∣((c∣c)∣(c∣c)))=TN(c∨(c∣c))≤TN(c)∨TN(c∣c),IN(c)∨IN(c∣c)≤IN(c∨(c∣c))=IN((c∣((c∣c)∣(c∣c)))∣((c∣c)∣(c∣c)))=IN(c∣(c∣c))=IN(1)

and

FN(c)∨FN(c∣c)≤FN(c∨(c∣c))=FN((c∣((c∣c)∣(c∣c)))∣((c∣c)∣(c∣c)))=FN(c∣(c∣c))=FN(1)

from Proposition 2.1 (2), (S1), (S2) and Corollary 2.1, it is obtained from Theorem 3.4 that TN(c)∨TN(c∣c)=TN(1), IN(c)∨IN(c∣c)=IN(1) and FN(c)∨FN(c∣c)=FN(1), and so, TN(c) = TN(1), IN(c) = IN(1), FN(c) = FN(1) or TN(c∣c)=TN(1), IN(c∣c)=IN(1), FN(c∣c)=FN(1), for all c∈C. Thus, CN is an ultra neutrosophic N-filter of C.

Theorem 3.5. Let CN be a neutrosophic N-structure on a Sheffer stroke BL-algebra C and τ,γ,ρ be any elements of [ −1, 0] with −3≤τ+γ+ρ≤0. If CN is a (ultra) neutrosophic N-filter of C, then the nonempty subset CN(τ,γ,ρ) is a (ultra) filter of C.

Proof. Let CN be a neutrosophic N-filter of C and CN(τ,γ,ρ)≠∅, for τ,γ,ρ∈[−1,0] with −3≤τ+γ+ρ≤0. Asumme that c1,c2∈CN(τ,γ,ρ). Since τ≤TN(c1),τ≤TN(c2), IN(c1)≤γ,IN(c2)≤γ, FN(c1)≤ρ and FN(c2)≤ρ, it follows that

τ≤min{TN(c1),TN(c2)}≤TN((c1∣c2)∣(c1∣c2)),IN((c1∣c2)∣(c1∣c2))≤max{IN(c1),IN(c2)}≤γ

and

FN((c1∣c2)∣(c1∣c2))≤max{FN(c1),fN(c2)}≤ρ.

Then (c1∣c2)∣(c1∣c2)∈TNτ,INγ,FNρ, and so, (c1∣c2)∣(c1∣c2)∈CN(τ,γ,ρ). Suppose that c1∈CN(τ,γ,ρ) and c1≤c2. Since τ≤TN(c1)≤TN(c2), IN(c2)≤IN(c1)≤γ and FN(c2)≤FN(c1)≤ρ, we have that c2∈TNτ,INγ,FNρ, and so, c2∈CN(τ,γ,ρ). Hence, CN(τ,γ,ρ) is a filter of C. Moreover, let CN be an ultra neutrosophic N-filter of C. Assume that c1∨c2∈CN(τ,γ,ρ). Since τ≤TN(c1∨c2), IN(c1∨c2)≤γ and FN(c1∨c2)≤ρ, it is obtained from Lemma 3.6 that τ≤TN(c1∨c2)≤TN(c1)∨TN(c2), IN(c1)∨IN(c2)≤IN(c1∨c2)≤γ and FN(c1)∨FN(c2)≤FN(c1∨c2)≤ρ, for all c1,c2∈C. Thus, τ≤TN(c1), IN(c1)≤γ, FN(c2)≤ρ or τ≤TN(c2), IN(c2)≤γ, FN(c2)≤ρ, and so, c1∈CN(τ,γ,ρ) or c2∈CN(τ,γ,ρ). By Lemma 2.3, CN(τ,γ,ρ) is an ultra filter of C.

Theorem 3.6. Let CN be a neutrosophic N-structure on a Sheffer stroke BL-algebra C, and TNτ,INγ and FNρ be (ultra) filters of C, for all τ,γ,ρ∈[−1,0] with −3≤τ+γ+ρ≤0. Then CN is a (ultra) neutrosophic N-filter of C.

Proof. Let CN be a neutrosophic N-structure on C, and TNτ,INγ and FNρ be filters of C, for all τ,γ,ρ∈[−1,0] with −3≤τ+γ+ρ≤0. Assume that

τ1=TN((c1∣c2)∣(c1∣c2))<min{TN(c1),TN(c2)}=τ2,γ1=max{IN(c1),IN(c2)}<IN((c1∣c2)∣(c1∣c2))=γ2

and

ρ1=max{FN(c1),fN(c2)}<FN((c1∣c2)∣(c1∣c2))=ρ2,

for some c1,c2∈C. If τ0=12(τ1+τ2), γ0=12(γ1+γ2), ρ0=12(ρ1+ρ2)∈[−1,0), then τ1<τ0<τ2, γ1<γ0<γ2 and ρ1<ρ0<ρ2. So, (c1∣c2)∣(c1∣c2)∉TNτ0,INγ0,FNρ0 when c1,c2∈TNτ0,INγ0,FNρ0, which contradict with (SF-1). Thus

min{TN(c1),TN(c2)}≤TN((c1∣c2)∣(c1∣c2)),IN((c1∣c2)∣(c1∣c2))≤max{IN(c1),IN(c2)}

and

FN((c1∣c2)∣(c1∣c2))≤max{FN(c1),fN(c2)},

for all c1,c2∈C. Let c1≤c2. Suppose that TN(c2) < TN(c1), IN(c1) < IN(c2) and FN(c1) < FN(c2), for some c1,c2∈C. If τ∗=12(TN(c1)+TN(c2)), γ∗=12(IN(c1)+IN(c2)), ρ∗=12(FN(c1)+FN(c2))∈[−1,0), then c2, c1 and c1 Hence, c1∈TNτ∗,INγ∗,FNρ∗ but c2∉TNτ∗,INγ∗,FNρ∗ which is a contradiction with (SF-2). Therefore, TN(c1)≤TN(c2), IN(c2)≤IN(c1) and FN(c2)≤FN(c1), for all c1,c2∈C. Thereby, CN is a neutrosophic N-filter of C.

Also, let TNτ,INγ and FNρ be ultra filters of C, for all τ,γ,ρ∈[−1,0] with −3≤τ+γ+ρ≤0, and TN(c1∨c2)=τ, IN(c1∨c2)=γ and FN(c1∨c2)=ρ. Since c1∨c2∈TNτ,INγ,FNρ, it follows from Lemma 2.3 that c1∈TNτ,INγ,FNρ or c2∈TNτ,INγ,FNρ. Thus, TN(c1∨c2)=τ≤TN(c1),TN(c2), IN(c1),IN(c2)≤γ=IN(c1∨c2) and FN(c1),FN(c2)≤ρ=FN(c1∨c2), and so, TN(c1∨c2)≤TN(c1)∨TN(c2), IN(c1)∨IN(c2)≤IN(c1∨c2) and FN(c1)∨FN(c2)≤FN(c1∨c2), for all c1,c2∈C. By Lemma 3.6, CN is an ultra neutrosophic N-filter of C.

Definition 3.6. Let C be a Sheffer stroke BL-algebra. Define

CNct:={c∈C:TN(ct)≤TN(c)},CNci:={c∈C:IN(c)≤IN(ci)}

and

CNcf:={c∈C:FN(c)≤FN(cf)},

for all ct,ci,cf∈C. It is obvious that ct∈CNct,ci∈CNci and cf∈CNcf.

Example 3.8. Consider the Sheffer stroke BL-algebra C in Example 3.1. Let ct = a, ci = b, cf=c∈C,

TN(x)={ −0.18if x=0, a, f, 1−0.29otherwise, IN(x)={ 0if x=d, e, f−1otherwise and FN(x)={ −0.55if x=0, 1−0.56if x=a, b, c−0.57if x=d, e, f.

Then

CNa={x∈C:TN(a)≤TN(x)}={x∈C:−0.18≤TN(x)}={0,a,f,1},CNxb={x∈C:IN(x)≤IN(b)}={x∈C:IN(x)≤−1}={0,a,b,c,1}

and

CNc={x∈C:FN(x)≤FN(c)}={x∈C:FN(x)≤−0.56}={a,b,c,d,e,f}.

Theorem 3.7. Let ct, ci and cf be any elements of a Sheffer stroke BL-algebra C. If CN is a (ultra) neutrosophic N-filter of C, then CNct,CNci and CNcf are (ultra) filters of C.

Proof. Let ct, ci and cf be any elements of C and CN be a neutrosophic N-filter of C. Assume that c1,c2∈CNct,CNci,CNcf. Since TN(ct)≤TN(c1),TN(ct)≤TN(c2), IN(c1)≤IN(ci),IN(c2)≤IN(ci) and FN(c1)≤FN(cf),FN(c2)≤FN(cf), we get that

TN(ct)≤min{TN(c1),TN(c2)}≤TN((c1∣c2)∣(c1∣c2)),IN((c1∣c2)∣(c1∣c2))≤max{IN(c1),IN(c2)}≤IN(ci)

and

FN((c1∣c2)∣(c1∣c2))≤max{FN(c1),FN(c2)}≤FN(cf).

Then (c1∣c2)∣(c1∣c2)∈CNct,CNci,CNcf. Suppose that c1∈CNct,CNci,CNcf and c1≤c2. Since TN(ct)≤TN(c1)≤TN(c2), IN(c2)≤IN(c1)≤IN(ci) and FN(c2)≤FN(c1)≤FN(cf), it is obtained that c2∈CNct,CNci,CNcf. Thus, CNct,CNci, CNcf are filters of C.

Let CN be an ultra neutrosophic N-filter of C and c1∨c2∈CNct,CNci, CNcf. Since

TN(ct)≤TN(c1∨c2)≤TN(c1)∨TN(c2),IN(c1)∨IN(c2)≤IN(c1∨c2)≤IN(ci)

and

FN(c1)∨FN(c2)≤FN(c1∨c2)≤FN(cf)

from Lemma 3.6, it follows that TN(ct)≤TN(c1), IN(c1)≤IN(ci), FN(c1)≤FN(cf) or TN(ct)≤TN(c2), IN(c2)≤IN(ci), FN(c2)≤FN(cf). Hence, c1∈CNct,CNci,CNcf or c2∈CNct,CNci,CNcf. Therefore, CNct,CNci and CNcf are ultra filters of C from Lemma 2.3.

Example 3.9. Consider the Sheffer stroke BL-algebra C in Example 3.1. For a neutrosophic N-filter

CN={x(−0.21,−0.41,−0.61):x=0,a,b,d}∪{x(−0.13,−0.53,−0.93):x=c,e,f,1}

of C, ct = b, ci = c and cf=f∈C, the subsets

CNb={x∈C:TN(b)≤TN(x)}={x∈C:−0.21≤TN(x)}=C,CNc={x∈C:IN(x)≤IN(c)}={x∈C:IN(x)≤−0.53}={c,e,f,1}

and

CNf={x∈C:FN(x)≤FN(f)}={x∈C:FN(x)≤−0.93}={c,e,f,1}

of C are filters of C. Also, CNb,CNc and CNf are ultra since CN is ultra.

The inverse of Theorem 3.7 does not hold in general.

Example 3.10. Consider the Sheffer stroke BL-algebra C in Example 3.1. Then

CNc={x∈C:TN(c)≤TN(x)}={x∈C:−0.11≤TN(x)}=C,CNd={x∈C:IN(x)≤IN(d)}={x∈C:IN(x)≤0}=C

and

CNe={x∈C:FN(x)≤FN(e)}={x∈C:FN(x)≤−0.12}=C

of C are filters of C but a neutrosophic N-structure

CN={x(−0.11,0,−0.12):x=0,c,d,e}∪{x(0,−1,−0.87):x=a,b,f,1}

is not a neutrosophic N-filter of C since TN(d) = −0.11 < 0 = TN(a) when a≤d.

Theorem 3.8. Let ct, ci and cf be any elements of a Sheffer stroke BL-algebra C and CN be a neutrosophic N-structure on C.

1.    If CNct,CNci and CNcf are filters of C, then

TN(c1)≤min{TN(c2∣(c3∣c3)),TN(c2)}⇒TN(c1)≤TN(c3),max{IN(c2∣(c3∣c3)),IN(c2)}≤IN(c1)⇒IN(c3)≤IN(c1) andmax{FN(c2∣(c3∣c3)),FN(c2)}≤FN(c1)⇒FN(c3)≤FN(c1),

for all c1,c2,c3∈C.

2.    If CN satisfies the condition (4) and

c1≤c2 implies TN(c1)≤TN(c2),IN(c2)≤IN(c1) and FN(c2)≤FN(c1),

for all c1,c2,c3∈C, then CNct,CNci and CNcf are filters of C, for all ct∈TN−1, ci∈IN−1 and cf∈FN−1.

Proof. Let CN be a neutrosophic N-structure on C.

1.    Assume that CNct,CNci and CNcf are filters of C, for all ct,ci,cf∈C, and c1, c2 and c3 are any elements of C such that TN(c1)≤min{TN(c2∣(c3∣c3)),TN(c2)}, max{IN(c2∣(c3∣c3)),IN(c2)}≤IN(c1) and max{FN(c2∣(c3∣c3)),FN(c2)}≤FN(c1). Since c2∣(c3∣c3),c2∈CNct,CNci,CNcf where ct = ci = cf = c1, we have from (SF-4) that c3∈CNct,CNci,CNcf where ct = ci = cf = c1. So, TN(c1)≤TN(c3), IN(c3)≤IN(c1) and FN(c3)≤FN(c1), for all c1,c2,c3∈C.

2.    Suppose that CN be a neutrosophic N-structure on C satisfying the conditions (4) and (5), for any ct∈TN−1,ci∈IN−1 and cf∈FN−1. Let c1,c2∈CNct,CNci,CNcf. Since c2≤(c2∣c1)∣c1=c1∣(((c1∣c2)∣(c1∣c2))∣((c1∣c2)∣(c1∣c2))) from Proposition 2.1 (9), (S1)–(S2), and TN(ct)≤TN(c1),TN(ct)≤TN(c2), IN(c1)≤IN(ci),IN(c2)≤IN(ci), FN(c1)≤FN(cf) and FN(c2)≤FN(cf), it follows from the condition (5) that

TN(ct)≤min{TN(c1),TN(c2)}≤min{TN(c1),TN(c1∣(((c1∣c2)∣(c1∣c2))∣((c1∣c2)∣(c1∣c2))))},max{IN(c1),IN(c1∣(((c1∣c2)∣(c1∣c2))∣((c1∣c2)∣(c1∣c2))))}≤max{IN(c1),IN(c2)}≤IN(ci)

and max{FN(c1),FN(c1∣(((c1∣c2)∣(c1∣c2))∣((c1∣c2)∣(c1∣c2))))}≤max{FN(c1),FN(c2)}≤FN(cf).

Thus, TN(ct)≤TN((c1∣c2)∣(c1∣c2)), IN((c1∣c2)∣(c1∣c2))≤IN(ci) and FN((c1∣c2)∣(c1∣c2))≤FN(cf) from the condition (4), and so, (c1∣c2)∣(c1∣c2)∈CNct,CNci,CNcf. Let c1≤c2 and c1∈CNct,CNci,CNcf. Since TN(ct)≤TN(c1)≤TN(c2), IN(c2)≤IN(c1)≤IN(ci) and FN(c2)≤FN(c1)≤FN(cf) from condition (5), it is obtained that c2∈CNct,CNci,CNcf. Thereby, CNct,CNci and CNcf are filters of C.

Example 3.11. Consider the Sheffer stroke BL-algebra C in Example 3.1. Let

TN(x)={ −0.07if x=1−0.77otherwise, IN(x)={ −0.63if x=e, 10otherwise, and FN(x)={ −0.84if x=a, d, e, 1−0.42otherwise.

Then the filters CNct=C,CNci={e.1} and CNcf={a,d,e,1} of C satisfy the condition (4), for the elements ct = a, ci = e and cf = d of C.

Also, let

CN={x(−0.91,−0.23,−0.001):x∈C−{1}}∪{1(−0.17,−0.86,−0.79)}

be a neutrosophic N-structure on C satisfying the conditions (4) and (5). Then the subsets

CNct={x∈C:TN(f)≤TN(x)}={x∈C:−0.91≤TN(x)}=C,CNci={x∈C:IN(x)≤IN(b)}={x∈A:IN(x)≤−0.23}=C

and

CNcf={x∈C:FN(x)≤FN(1)}={x∈C:FN(x)≤−0.79}={1}

of C are filters of C, where ct = f, ci = b and cf = 1 of C.

4  Conclusion

In the study, neutrosophic N-structures defined by N-functions on Sheffer stroke BL-algebras have been examined. By giving basic definitions and notions of Sheffer stroke BL-algebras and neutrosophic N-structures on a crispy set X, a neutrosophic N-subalgebra and a (τ,γ,ρ)-level set of a neutrosophic N-structure are defined on Sheffer stroke BL-algebras. We determine a quasi-subalgebra of a Sheffer stroke BL-algebra and prove that the (τ,γ,ρ)-level set of a neutrosophic N-subalgebra of a Sheffer stroke BL-algebra is its quasi-subalgebra and vice versa. Besides, it is stated that the family of all neutrosophic N-subalgebras of the algebra forms a complete distributive lattice. It is illustrated that every neutrosophic N-subalgebra of a Sheffer stroke BL-algebra satisfies TN(x)≤TN(1), IN(1)≤IN(x) and FN(1)≤FN(x), for all elements x of the algebra but the inverse does not generally hold. We interpret the case which N-functions defining a neutrosophic N-subalgebra of a Sheffer stroke BL-algebra are constant. Also, a (ultra) neutrosophic N-filter of a Sheffer stroke BL-algebra is described and some properties are analysed. Indeed, it is proved that every neutrosophic N-filter of a Sheffer stroke BL-algebra is the neutrosophic N-subalgebra but the inverse is not true in general, and that the (τ,γ,ρ)-level set of a (ultra) neutrosophic N-filter of a Sheffer stroke BL-algebra is its (ultra) filter and the inverse is always true. After that the subsets CNct,CNci and CNcf of a Sheffer stroke BL-algebra are described by means of N-functions and any elements ct, ci and cf of this algebraic structure, it is demonstrated that these subsets are (ultra) filters of a Sheffer stroke BL-algebra if CN is the (ultra) neutrosophic N-filter.

In future works, we wish to study on plithogenic structures and relationships between neutrosophic N-structures on some algebraic structures.

Acknowledgement: The authors are thankful to the referees for a careful reading of the paper and for valuable comments and suggestions.

Funding Statement: The authors received no specific funding for this study.

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

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