Some Results on Neutrosophic Triplet Group and Their Applications
Abstract
:1. Introduction
1.1. Generalized Group
- (i)
- for all .
- (ii)
- For each there is a unique such that (existence and uniqueness of identity element).
- (iii)
- For each , there is such that (existence of inverse element).
- (i)
- For each , there is a unique .
- (ii)
- and if .
- (iii)
- If X is commutative, then X is a group.
1.2. Neutrosophic Triplet Group
2. Main Results
- 1.
- X is a generalized group if it satisfies the left (or right) cancellation law or X is a left (or right) quasigroup.
- 2.
- X is a generalized group if and only if each element has a unique .
- 3.
- Whenever X has the cancellation laws (or is a quasigroup), then X is a loop and group.
- Let x have at least two neutral elements, say . Then . Therefore, X is a generalized group. Similarly, X is a generalized group if it is has the right cancellation law or if it is a right quasigroup.
- This follows by definition.
- This is straightforward because every associative quasigroup is a loop and group.
2.1. Algebraic Properties of Neutrosophic Triplet Group
- (i)
- H is a neutrosophic triplet subgroup of X.
- (ii)
- For all , .
- (iii)
- For all , , and .
- (i)⇒ (ii)
- If H is an NTSG of X and , then . Therefore, by closure property, .
- (ii)⇒ (iii)
- If , and , then we have , , and , i.e., .
- (iii)⇒ (i)
- , so H is associative since X is associative. Obviously, for any , . Let , then . Therefore, . Thus, H is an NTSG of X.
2.2. Neutrosophic Triplet Group Homomorphism
- 1.
- for all .
- 2.
- for all .
- 3.
- If H is a neutrosophic triplet subgroup of X, then is a neutrosophic triplet subgroup of Y.
- 4.
- If K is a neutrosophic triplet subgroup of Y, then is a neutrosophic triplet subgroup of X.
- 5.
- If X is a normal neutrosophic triplet group and the set with the product
- Place in to obtain . Additionally, place in to obtain . Thus, for all .
- Place in to obtain . Additionally, place in to obtain . Thus, for all .
- If H is an NTSG of G, then . We shall prove that is an NTSG of Y by Lemma 2.Since for , . Let . Then and . Thus, . Therefore, is an NTSG of Y.
- If K is a neutrosophic triplet subgroup of Y, then . We shall prove that is an NTSG of Y by Lemma 2.Let . Then such that and . Thus, . Therefore, is an NTSG of X.
- Given the neutrosophic triplet group X and the set with the product . is a groupoid.For let . Then . Additionally, let . Then .Thus, and similarly, .On the other hand, , and similarly, .Therefore, is a neutrosophic triplet group. ☐
- 1.
- .
- 2.
- .
- 3.
- is a normal neutrosophic triplet group.
- 4.
- .
- 5.
- for all .
- 6.
- If X is finite, for all where is the index of in , i.e., the number of distinct left cosets of in .
- 7.
- .
- 8.
- If X is finite, .
- . Let , then . We shall use Lemma 2..Thus, is a neutrosophic triplet subgroup of X. For the a-normality, let , then . Therefore, for all . Therefore, .
- . . Therefore, . Let . Then . , and . Therefore, .. Thus, is a neutrosophic triplet subgroup of X.. Therefore, , and . Thus, . Therefore, .
- Let . Then . Therefore, . Thus, is a normal NTG.
- For all , . Therefore, based on Point 3 and Theorem 3, .
- Define a relation ≍ on as follows: if for all . . Therefore, ≍ is reflexive.. Therefore, ≍ is symmetric.. Therefore, ≍ is transitive and ≍ is an equivalence relation.The equivalence class . Therefore, .Thus, for all .
- If X is finite, then for all . Thus, for all where is the index of in , i.e., the number of distinct left cosets of in .
- Define a relation ∼ on X: if . ∼ is an equivalence relation on X, so and, therefore, .
- Hence, based on Point 7, if X is finite, then
- 1.
- f is a monomorphism if and only if for all ;
- 2.
- the factor set is a neutrosophic triplet group (neutrosophic triplet factor group) under the operation defined by
- Let and let . If , this implies thatSimilarly,Conversely, if f is mono, then . Let . Then . Therefore, for all .
- Let .
- Groupoid:
- Based on the multiplication , the factor set is a groupoid.
- Semigroup:
- .
- Neutrality:
- Let . Then and similarly, .
- Opposite:
- Let . Then . Similarly, .
is an NTG. ☐
2.3. Construction of Bol Algebraic Structures
3. Applications in Management and Sports
3.1. One-Way Management and Division of Labor
3.2. Two-Way Management Division of Labor
3.3. Sports
Author Contributions
Conflicts of Interest
References
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Jaíyéolá, T.G.; Smarandache, F. Some Results on Neutrosophic Triplet Group and Their Applications. Symmetry 2018, 10, 202. https://doi.org/10.3390/sym10060202
Jaíyéolá TG, Smarandache F. Some Results on Neutrosophic Triplet Group and Their Applications. Symmetry. 2018; 10(6):202. https://doi.org/10.3390/sym10060202
Chicago/Turabian StyleJaíyéolá, Tèmítópé Gbóláhàn, and Florentin Smarandache. 2018. "Some Results on Neutrosophic Triplet Group and Their Applications" Symmetry 10, no. 6: 202. https://doi.org/10.3390/sym10060202