GENERALIZED TOPOLOGICAL OPERATOR ($\\operatorname{\\mathfrak{g}-\\mathfrak{T}_{\\mathfrak{g}}}$-OPERATOR) THEORY IN GENERALIZED TOPOLOGICAL SPACES ($\\mathcal{T}_{\\mathfrak{g}}$-SPACES): PART IV. GENERALIZED DERIVED ($\\operatorname{\\mathfrak{g}-\\mathfrak{T}_{\\mathfrak{g}}}$-DERIVED) AND GENERALIZED CODERIVED ($\\operatorname{\\mathfrak{g}-\\mathfrak{T}_{\\mathfrak{g}}}$-CODERIVED) OPERATORS

Authors : Mohammad Irshad Khodabocus, Noor-ul-hacq Sookıa, Radhakhrishna Dinesh Somanah

Pages : 128-165

Doi:10.33773/jum.1393185

View : 23 | Download : 13

Publication Date : 2024-07-31

Article Type : Research

Abstract :In a recent paper (\\textsc{Cf.} \\cite{KHODABOCUS_2023_4}), we have introduced the definitions and studied the essential properties of the generalized topological operators $\\operatorname{\\mathfrak{g}-Der}_{\\mathfrak{g}}$, $\\operatorname{\\mathfrak{g}-Cod}_{\\mathfrak{g}}: \\mathcal{P}\\left(\\Omega\\right) \\longrightarrow \\mathcal{P}\\left(\\Omega\\right)$ (\\textit{$\\operatorname{\\mathfrak{g}-\\mathfrak{T}}_{\\mathfrak{g}}$-derived} and \\textit{$\\operatorname{\\mathfrak{g}-\\mathfrak{T}}_{\\mathfrak{g}}$-coderived operators}) in a generalized topological space $\\mathfrak{T}_{\\mathfrak{g}} = \\left(\\Omega,\\mathcal{T}_{\\mathfrak{g}}\\right)$ (\\textit{$\\mathcal{T}_{\\mathfrak{g}}$-space}). Mainly, we have shown that $\\left(\\operatorname{\\mathfrak{g}-Der_{\\mathfrak{g}}},\\operatorname{\\mathfrak{g}-Cod_{\\mathfrak{g}}}\\right): \\mathcal{P}\\left(\\Omega\\right)\\times\\mathcal{P}\\left(\\Omega\\right) \\longrightarrow \\mathcal{P}\\left(\\Omega\\right)\\times\\mathcal{P}\\left(\\Omega\\right)$ is a pair of both \\textit{dual and monotone $\\operatorname{\\mathfrak{g}-\\mathfrak{T}_{\\mathfrak{g}}}$-operators} that is \\textit{$\\left(\\emptyset,\\Omega\\right)$, $\\left(\\cup,\\cap\\right)$-preserving}, and \\textit{$\\left(\\subseteq,\\supseteq\\right)$-preserving} relative to $\\operatorname{\\mathfrak{g}-\\mathfrak{T}}_{\\mathfrak{g}}$-(open, closed) sets. We have also shown that $\\left(\\operatorname{\\mathfrak{g}-Der}_{\\mathfrak{g}},\\operatorname{\\mathfrak{g}-Cod}_{\\mathfrak{g}}\\right): \\mathcal{P}\\left(\\Omega\\right)\\times\\mathcal{P}\\left(\\Omega\\right) \\longrightarrow \\mathcal{P}\\left(\\Omega\\right)\\times\\mathcal{P}\\left(\\Omega\\right)$ is a pair of \\textit{weaker} and \\textit{stronger $\\operatorname{\\mathfrak{g}-\\mathfrak{T}_{\\mathfrak{g}}}$-operators}. In this paper, we define by transfinite recursion on the class of successor ordinals the $\\delta^{\\operatorname{th}}$-iterates $\\operatorname{\\mathfrak{g}-Der}_{\\mathfrak{g}}^{\\left(\\delta\\right)}$, $\\operatorname{\\mathfrak{g}-Cod}_{\\mathfrak{g}}^{\\left(\\delta\\right)}: \\mathcal{P}\\left(\\Omega\\right) \\longrightarrow \\mathcal{P}\\left(\\Omega\\right)$ (\\textit{$\\operatorname{\\mathfrak{g}-\\mathfrak{T}_{\\mathfrak{g}}^{\\left(\\delta\\right)}}$-derived} and \\textit{$\\operatorname{\\mathfrak{g}-\\mathfrak{T}_{\\mathfrak{g}}^{\\left(\\delta\\right)}}$-coderived operators}) of $\\operatorname{\\mathfrak{g}-Der}_{\\mathfrak{g}}$, $\\operatorname{\\mathfrak{g}-Cod}_{\\mathfrak{g}}: \\mathcal{P}\\left(\\Omega\\right) \\longrightarrow \\mathcal{P}\\left(\\Omega\\right)$, respectively, and study their basic properties in a $\\mathcal{T}_{\\mathfrak{g}}$-space. Moreover, we establish the necessary and sufficient conditions for $\\bigl(\\operatorname{\\mathfrak{g}-Der}_{\\mathfrak{g}}^{\\left(\\delta\\right)},\\operatorname{\\mathfrak{g}-Cod}_{\\mathfrak{g}}^{\\left(\\delta\\right)}\\bigr): \\mathcal{P}\\left(\\Omega\\right)\\times\\mathcal{P}\\left(\\Omega\\right) \\longrightarrow \\mathcal{P}\\left(\\Omega\\right)\\times\\mathcal{P}\\left(\\Omega\\right)$ to be a pair of $\\operatorname{\\mathfrak{g}-\\mathfrak{T}_{\\mathfrak{g}}}$-derived and $\\operatorname{\\mathfrak{g}-\\mathfrak{T}_{\\mathfrak{g}}}$-coderived operators in $\\mathfrak{T}_{\\mathfrak{g}}$. Finally, we diagram various relationships amongst $\\operatorname{der}_{\\mathfrak{g}}^{\\left(\\delta\\right)}$, $\\operatorname{\\mathfrak{g}-Der}_{\\mathfrak{g}}^{\\left(\\delta\\right)}$, $\\operatorname{cod}_{\\mathfrak{g}}^{\\left(\\delta\\right)}$, $\\operatorname{\\mathfrak{g}-Cod}_{\\mathfrak{g}}^{\\left(\\delta\\right)}: \\mathcal{P}\\left(\\Omega\\right) \\longrightarrow \\mathcal{P}\\left(\\Omega\\right)$ and present a nice application to support the overall study.
Keywords : Generalized topological space ($\\mathcal{T}_{\\mathfrak{g}}$-space), generalized sets ($\\operatorname{\\mathfrak{g}-\\mathfrak{T}_{\\mathfrak{g}}}$-sets), $\\delta^{\\operatorname{th}}$-order generalized derived operator ($\\operatorname{\\mathfrak{g}-\\m