A New Algebraic Structure of Complex Pythagorean Fuzzy Subfield

The concept of complex Pythagorean fuzzy set
$(\mathbf {CPFS})$
is recent development in the field of fuzzy set
$(\mathbf {FS})$
theory. The significance of this concept lies in the fact that this theory assigned membership grades
$\psi $
and non-membership grades
$\hat {\psi }$
from unit circle in plane, i.e., in the form of a complex number with the condition
$(\psi)^{2}+ (\hat {\psi })^{2}\le 1$
instead from [0, 1] interval. This is an expressive technique for dealing with uncertain circumstances. The aim of this study is to proceed the classification of the unique framework of
$\mathbf {CPFS}$
in algebraic structure that is field theory and examine its numerous algebraic features. Also, we initiate the important examples and results of certain field. Furthermore, we illustrate that every complex Pythagorean fuzzy subfield (
$\mathbf {CPFSF}$
) generates two Pythagorean fuzzy subfields
$(\mathbf {PFSFs})$
. We also prove many useful algebraic aspects of this notion for a
$\mathbf {CPFSF}$
. Moreover, we demonstrate that intersection of two complex Pythagorean fuzzy subfields
$(\mathbf {CPFSFs})$
is also
$\mathbf {CPFSF}$
. Additionally, we discuss the novel idea of level subsets of
$\mathbf {CPFSFs}$
and demonstrate that level subset of
$\mathbf {CPFSF}$
form subfield. Additionally, we improve the application of this theory to show the concept of the direct product of two
$\mathbf {CPFSFs}$
is also a
$\mathbf {CPFSF}$
and produce several novel results on direct product of
$\mathbf {CPFSFs}$
. Finally, we explore the homomorphic images and inverse images of
$\mathbf {CPFSFs}$
.