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Correction

Correction: Ali, M., et al. Study on the Development of Neutrosophic Triplet Ring and Neutrosophic Triplet Field. Mathematics 2018, 6, 46

1
Department of Mathematics, Süleyman Demirel University, 32260 Isparta, Turkey
2
Department of Mathematics, University of New Mexico, 705 Gurley Avenue, Gallup, NM 87301, USA
*
Author to whom correspondence should be addressed.
Mathematics 2019, 7(6), 565; https://doi.org/10.3390/math7060565
Submission received: 5 May 2019 / Revised: 19 June 2019 / Accepted: 19 June 2019 / Published: 24 June 2019
We have found the following errors in the article which was recently published in Mathematics [1]:
1. In Example 1, 3 gives rise to the neutrosophic triplet (3, 3, 3). However, 3 has two neutrals: neut(3) = {3, 5}, but 3 does not give rise to a neutrosophic triplet for neut(3)= 5, since anti(3) does not exist in 6 with respect to neut(3) = 5.
2. In Example 2, 10 is not a neutrosophic triplet group. 7 is the classical unitary element of the set 10 . Therefore 10 is a neutrosophic extended triplet group.
3. In classical ring theory, for any ring ( R , + , . ) , 0 is the additive identity element. However, in a neutrosophic triplet ring ( N , , # ) , 0 is an ordinary element and the element 0 is not used in definition. Also N may not have such an element. So, in Definition 8 and subsequent parts of the paper, when using the element 0, the element 0 should be defined.
4. In classical ring theory, for any ring ( R , + , . ) , n∙a is defined by a + + a and a n is defined by a a (n times). In neutrosophic triplet ring (NTR), we do not know the definition of a n . So before Definition 11, the element a n should be defined.
5. For the proof of Theorem 3, Theorem 1 was used. So Theorem 3 must satisfy the hypothesis of Theorem 1. Also according to definition of a n , Theorem 3 should be rewritten.
6. Proposition 1 and its proof are not true. The sentences “if a is not a zero divisor, so a is cancellable” and “if a is cancellable, a is not a zero divisor” are not true. These statements cannot be obtained from the given definitions and theorems.
7. The set P(X) in Example 3 is not neutrosophic triplet field. P(X) has identity elements X and for the operations   and   , respectively. Therefore P(X) is a neutrosophic extended triplet group.
8. The counterexamples given for Theorem 5 do not satisfy the distributive law since 1 # ( 1 2 ) ( 1 # 1 ) ( 1 # 2 ) .
9. In the proof of Theorem 6, the set N is not NTF since 5 # ( 5 5 ) ( 5 # 5 ) ( 5 # 5 ) .
10. The proof of Theorem 7(2) is not true. If c U , then f 1 ( c ) is a set. If f is not a function, f 1 ( c ) can be equal to an empty set. Then f 1 ( c ) f 1 ( d ) is not in f 1 ( U ) . We can prove it by the following:
Let a , b f 1 ( U ) . Then f ( a ) , f ( b ) U and f ( a ) f ( b ) = f ( a b ) U . Hence we get a b f 1 ( U ) . The proof of a # b f 1 ( U ) is similar. Also, since f ( a ) U and n e u t * ( f ( a ) ) = f ( n e u t * ( a ) ) U , we have n e u t * ( a ) f 1 ( U ) . The proof of n e u t # ( a ) f 1 ( U ) is similar.
11. The proof of Theorem 7(3) is not true. If i I   and   r N T R 2 , then f 1 ( i )   and   f 1 ( r ) is a set. If f is not a function, f 1 ( i )   and   f 1 ( r ) can be equal to an empty set. Then f 1 ( i ) f 1 ( r ) is not in f 1 ( I ) . We can prove it by the following:
Let a f 1 ( U ) and r N T R 1 . Then f ( a ) I   a n d   f ( r ) N T R 2 and f ( a ) f ( r ) = f ( a r ) I . Hence we get a r f 1 ( I ) . The remaining part of the proof is similar.
12. The proof of Theorem 7(4) should be proven as the following:
Let j f ( J )   and r N T R 2 . Since f is onto, then h J exists such that f ( h ) = j and s N T R 1 such that f ( s ) = r . Then h s J and we get f ( h s ) = f ( h ) f ( s ) = j r f ( J ) .

Reference

  1. Ali, M.; Smarandache, F.; Khan, M. Study on the Development of Neutrosophic Triplet Ring and Neutrosophic Triplet Field. Mathematics 2018, 6, 46. [Google Scholar] [CrossRef]

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MDPI and ACS Style

Çeven, Y.; Smarandache, F. Correction: Ali, M., et al. Study on the Development of Neutrosophic Triplet Ring and Neutrosophic Triplet Field. Mathematics 2018, 6, 46. Mathematics 2019, 7, 565. https://doi.org/10.3390/math7060565

AMA Style

Çeven Y, Smarandache F. Correction: Ali, M., et al. Study on the Development of Neutrosophic Triplet Ring and Neutrosophic Triplet Field. Mathematics 2018, 6, 46. Mathematics. 2019; 7(6):565. https://doi.org/10.3390/math7060565

Chicago/Turabian Style

Çeven, Yılmaz, and Florentin Smarandache. 2019. "Correction: Ali, M., et al. Study on the Development of Neutrosophic Triplet Ring and Neutrosophic Triplet Field. Mathematics 2018, 6, 46" Mathematics 7, no. 6: 565. https://doi.org/10.3390/math7060565

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