Generalized Topological Operator (g-Tg-Operator) Theory in Generalized Topological Spaces (Tg-Spaces): Part IV. Generalized Derived (g-Tg-Derived) and Generalized Coderived (g-Tg-Coderived) Operators

In a recent paper ({Cf.} {KHODABOCUS_2023_4}), we have introduced the definitions and studied the essential properties of the generalized topological operators g-Derg, g-Codg: P(Ω) P(Ω) (g-Tg-derived and g-Tg-coderived operators) in a generalized topological space Tg = (Ω,Tg) (Tg-space). Mainly, we have shown that $({g-Derg},{g-Codg}): P(Ω)(Ω) P(Ω)(Ω)is a pair of both dual and monotone{g-Tg}-operators that is(∅,Ω)$, $(∪,∩)-preserving, and(⊆,⊇)-preserving relative to{g-T}g$-(open, closed) sets. We have also shown that $({g-Der}g,{g-Cod}g): P(Ω)(Ω) P(Ω)(Ω)is a pair of weaker and stronger{g-Tg}-operators. In this paper, we define by transfinite recursion on the class of successor ordinals theδᵗʰ$-iterates ${g-Der}g(δ)$, ${g-Cod}g(δ): P(Ω) P(Ω)({g-Tg(δ)}-derived and{g-Tg(δ)}-coderived operators) of{g-Der}g$, ${g-Cod}g: P(Ω) P(Ω)$, respectively, and study their basic properties in a $Tg$-space. Moreover, we establish the necessary and sufficient conditions for $({g-Der}g(δ),{g-Cod}g(δ)): P(Ω)(Ω) P(Ω)(Ω)$ to be a pair of ${g-Tg}$-derived and ${g-Tg}$-coderived operators in $Tg$. Finally, we diagram various relationships amongst $derg(δ)$, ${g-Der}g(δ)$, $codg(δ)$, ${g-Cod}g(δ): P(Ω) P(Ω)$ and present a nice application to support the overall study.