On Structural Properties of <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"> <a:mi>ξ</a:mi> </a:math>-Complex Fuzzy Sets and Their Applications

Complex fuzzy sets are the novel extension of Zadeh’s fuzzy sets. In this paper, we comprise the introduction to the concept of <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M2"> <a:mi>ξ</a:mi> </a:math> -complex fuzzy sets and proofs of their various set theoretical properties. We define the notion of <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" id="M3"> <c:mfenced open="(" close=")" separators="|"> <c:mrow> <c:mi>α</c:mi> <c:mo>,</c:mo> <c:mi>δ</c:mi> </c:mrow> </c:mfenced> </c:math> -cut sets of <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" id="M4"> <h:mi>ξ</h:mi> </h:math> -complex fuzzy sets and justify the representation of an <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" id="M5"> <j:mi>ξ</j:mi> </j:math> -complex fuzzy set as a union of nested intervals of these cut sets. We also apply this newly defined concept to a physical situation in which one may judge the performance of the participants in a given task. In addition, we innovate the phenomena of <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" id="M6"> <l:mi>ξ</l:mi> </l:math> -complex fuzzy subgroups and investigate some of their fundamental algebraic attributes. Moreover, we utilize this notion to define level subgroups of these groups and prove the necessary and sufficient condition under which an <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" id="M7"> <n:mi>ξ</n:mi> </n:math> -complex fuzzy set is <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" id="M8"> <p:mi>ξ</p:mi> </p:math> -complex fuzzy subgroup. Furthermore, we extend the idea of <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" id="M9"> <r:mi>ξ</r:mi> </r:math> -complex fuzzy normal subgroup to define the quotient group of a group <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" id="M10"> <t:mi>G</t:mi> </t:math> by this particular <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" id="M11"> <v:mi>ξ</v:mi> </v:math> -complex fuzzy normal subgroup and establish an isomorphism between this quotient group and a quotient group of <x:math xmlns:x="http://www.w3.org/1998/Math/MathML" id="M12"> <x:mi>G</x:mi> </x:math> by a specific normal subgroup <z:math xmlns:z="http://www.w3.org/1998/Math/MathML" id="M13"> <z:msub> <z:mrow> <z:mi>G</z:mi> </z:mrow> <z:mrow> <z:msup> <z:mrow> <z:mi>A</z:mi> </z:mrow> <z:mrow> <z:mi>ξ</z:mi> </z:mrow> </z:msup> </z:mrow> </z:msub> </z:math> .