A Short Introduction To The Symbolic Turiyam Vector Spaces and Complex Numbers
Mikail Bal, Gaziantep University, Turkey
Prem Kumar Singh1, Department of Computer science and Engineering, Gandhi institute of Technology and Management-Visakhapatnam 530045, India
Katy D. Ahmad2, Islamic university Of Gaza, Palestine
2Email: Katyon765@gmail.com
1Email: premsingh.csjm@gmail.com
Abstract:
This paper is dedicated to define for the first time the concept of Symbolic Turiyam vector space as a generalization of the corresponding neutrosophic one by using the algebra of symbolic Turiyam set. Also, we illustrate many examples to show and clarify the validity of this work.
Key words: Turiyam matrix, Turiyam Set, Turiyam Space, Turiyam Context.
Introduction
Recently, Turiyam set [91-93] is produced for dealing the data beyond three-way fuzzy space [1-10] for knowledge processing tasks. In this process the symbolic turiyam ring structures [78] is also introduced as a new algebraic generalization of the corresponding neutrosophic one [1-10]. In this process, a problem is arises for defining the Turiyam space for precise representation of Turiyam set. Some of the researchers tried to represent the three-way fuzzy space data sets using Neutrosophic set [11-20] and its algebraic structures [21-30] with an extensive properties [31-40]. The refined neutrosophic set provided a way to deal with many-valued data sets in three-way fuzzy space [41-50] and its topology [51-60] for multi-decision tasks [61-75]. The problem arises when the data set contains independent values for acceptation, rejection and uncertain parts as well as Turiyam [76-78]. It is observed in neutrosophic set which contains independent acceptation, rejection and uncertain values exists in intuitionistic fuzzy set [79-84]. It become more crucial in case every opinion considered as independent like acceptation, rejection, uncertain and None of the above condition [85-88]. One of the suitable examples is COVID 19 data which contain fourth dimensions uncertainty for the analysis. The people who got recovered can be considered as true regions(t), people who died due to COVID 19 can be considered as false regions(f), people who are still active can be considered as indeterminacy(i), people who got vaccinated can be considered as Turiya or Liberated state(l). The refusal degree means people who still did not come under these regions can be represented as 1-(t +i +f +l) [89-93]. To deal with these types of data set current paper introduce Turiyam space in this paper.
In the literature, mant authors have contributed to algebraic structures in neutrosophic systems such as neutrosophic rings, spaces, modules , and matrices [10-25, 31-45, 50-66].
In this work, we extend the previous efforts to the case of the symbolic Turiyam set, where we define the symbolic Turiyam vector space, the AH-turiyam subspace, AHS-Turiyam subspace, and AH-Turiyam linear transformation, which are considered as generalizations of the AH-substructures defined in [6-8, 27].
This work may be very useful in future studied to generalize classical algebraic structures and to study the rerlationships between Turiyam sets and classical sets.
Main Discussion
Definition
Let be a field and be a vector space over .
Let be the corresponding symbolicTuriyam field (STF). We define the symbolic Turiyam vector space (STVS) as follows:
= .
Example
Let be the 2-dimensional Eucledian vector space over the real field . The corresponding (STVS) over is:
= .
.
Theorem
Let be the symbolic Turiyam vector space (STVS) over the (STF) , hence is a module over in the ordinary algebraic meaning.
Definition:
Let = be a Symbolic Turiyam vector space. Let ; i=0..4 be subspaces of V. We define the corresponding Turiyam AH-subpace as the following:
= .
If , then is called Symbolic Turiyam AHS-subspace.
Example
Let be a vector space over .
, are two subspace of .
is an AH-subspace.
is an AH-subspace.
Definition
Let be linear transformation; . We define the corresponding AH-linear Turiyam transformation as follows:
.
If , then we get AHS-Turiyam linear transformation.
Example
Consider the following classical linear transformations:
.
.
Now, we are able to build AH-Turiyam linear transformation as follows:
.
Also, we can build an AHS-Turiyam linear transformation as follows:
,
.
Definition
We define the Turiyam complex number as follows:
.
The cingucate of is defined:
The set of all Turiyam complex numbers is defined by .
Remark
contains the neutrosophic field of complex numbers .
Example
Take , .
We have:
, .
Remark
, , .
Remark
Not all Turiyam complex numbers have inverses, for example is not invertible.
Conclusion
In this paper, we have defined the Symbolic Turiyam vector spaces and Turiyam Complex numbers. Also, we have illustrated many examples to clarify the validity of our definitions.
This work is considered as a part of a largest project to define and study the Turiyam algebraic structures.
Acknowledgements: Author thanks the editorial team for the valuable time.
Funding :Author declares that, there is no funding for this paper.
Conflicts of Interest: Author declares that, there is no conflict of interest.
Ethics approval: This article does not contain any studies with human or animals participants.
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