Computer Modeling in Engineering & Sciences |

DOI: 10.32604/cmes.2021.016996

ARTICLE

Neutrosophic

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

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

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:

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,

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

Definition 2.1. [18] Let

(S1)

(S2)

(S3)

(S4)

Definition 2.2. [21] A Sheffer stroke BL-algebra is an algebra

(sBL −1)

(sBL −2)

(sBL −3)

(sBL −4)

for all

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

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10) (a)

(b)

(11) If

(i)

(ii)

(iii)

(12)

(13)

(14)

(15)

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

for all

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

for all

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

for all

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

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

Lemma 2.3. [21] A filter P of C is an ultra filter of C if and only if

Definition 2.5. [8]

Definition 2.6. [12] A neutrosophic

where TN, IN and FN are

Every neutrosophic

Definition 2.7. [13] Let XN be a neutrosophic

and

The set

is called the

Consider sets

and

for any

In this section, neutrosophic

Definition 3.1. A neutrosophic

for all

Example 3.1. Consider a Sheffer stroke BL-algebra C where the set

on C is a neutrosophic

Definition 3.2. Let CN be a neutrosophic

and

the set

is called the

Definition 3.3. A subset D of a Sheffer stroke BL-algebra C is called a quasi-subalgebra of C if

Example 3.2. Consider the Sheffer stroke BL-algebra C in Example 3.1. Then

Theorem 3.1. Let CN be a neutrosophic

Proof. Let CN be a neutrosophic

and

for all

Theorem 3.2. Let CN be a neutrosophic

Proof. Let CN be a neutrosophic

Theorem 3.3. Let

Proof. Let D be a nonempty subset of

Lemma 3.1. Let CN be a neutrosophic

Proof. Let CN be a neutrosophic

and

for all

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

on C is not a neutrosophic

Lemma 3.2. A neutrosophic

Proof. Let CN be a a neutrosophic

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

Definition 3.4. A neutrosophic

1.

2.

for all

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

on C is a neutrosophic

Theorem 3.4. Let CN be a a neutrosophic

for all

Proof. Let CN be a neutrosophic

and

for all

Conversely, let CN be a a neutrosophic

and

for all

and

for all

Corollary 3.1. Let CN be a neutrosophic

1.

2.

3.

4.

for all

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

Lemma 3.3. Let CN be a neutrosophic

for all

Proof. Let CN be a neutrosophic

and

for all

Conversely, let CN be a neutrosophic

and

for all

Lemma 3.4. Every neutrosophic

Proof. Let CN be a neutrosophic

from Proposition 2.1 (1), (2), (4) and (S3), it follows from Proposition 2.1 (7) that

and

for all

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

on C is a neutrosophic

Definition 3.5. Let CN be a neutrosophic

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

on C is an ultra neutrosophic

Remark 3.1. By Definition 3.5, every ultra neutrosophic

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

of C is not ultra since

Lemma 3.5. Let CN be a neutrosophic

Proof. Let CN be an ultra neutrosophic

and

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

and similarly,

Conversely, let CN be a neutrosophic

Lemma 3.6. Let CN be a neutrosophic

Proof. Let CN be an ultra neutrosophic

and similarly,

Conversely, let CN be a neutrosophic

and

from Proposition 2.1 (2), (S1), (S2) and Corollary 2.1, it is obtained from Theorem 3.4 that

Theorem 3.5. Let CN be a neutrosophic

Proof. Let CN be a neutrosophic

and

Then

Theorem 3.6. Let CN be a neutrosophic

Proof. Let CN be a neutrosophic

and

for some

and

for all

Also, let

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

and

for all

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

Then

and

Theorem 3.7. Let ct, ci and cf be any elements of a Sheffer stroke BL-algebra C. If CN is a (ultra) neutrosophic

Proof. Let ct, ci and cf be any elements of C and CN be a neutrosophic

and

Then

Let CN be an ultra neutrosophic

and

from Lemma 3.6, it follows that

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

of C, ct = b, ci = c and

and

of C are filters of C. Also,

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

and

of C are filters of C but a neutrosophic

is not a neutrosophic

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

1. If

for all

2. If CN satisfies the condition (4) and

for all

Proof. Let CN be a neutrosophic

1. Assume that

2. Suppose that CN be a neutrosophic

and

Thus,

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

Then the filters

Also, let

be a neutrosophic

and

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

In the study, neutrosophic

In future works, we wish to study on plithogenic structures and relationships between neutrosophic

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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