Computer Modeling in Engineering & Sciences

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

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

for all c1,c2∈C.

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

for all c1,c2∈C.

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

for all c1,c2,c3∈C.

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

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:

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

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:

and

The set

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

Consider sets

and

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

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

and

the set

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

and

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

and

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

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

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

and

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,

and

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

and

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),