$\mathbb{P_{CIF}}$-$Z$-Open Sets in Cubic Intuitionistic Fuzzy Topology with Application to IoT-Enabled Smart Greenhouse Monitoring

$\mathbb{P_{CIF}}$-$Z$-Open Sets and IoT Greenhouse Application

Abstract

This paper introduces and investigates the concept of $Z$-open sets in cubic intuitionistic fuzzy topological spaces under $\mathbb{P_{CIF}}$-order. We establish fundamental properties of $\mathbb{P_{CIF}}$-$Z$-open sets and their relationships with other generalized open sets including $\delta$-open, preopen, and $\delta$-semiopen sets. Several characterizations of $\mathbb{P_{CIF}}$-$Z$-open sets are provided through interior and closure operators. We prove that the family of $\mathbb{P_{CIF}}$-$Z$-open sets forms a topology and investigate properties of $\mathbb{P_{CIF}}$-$Z$-interior and $\mathbb{P_{CIF}}$-$Z$-closure operators. The results demonstrate that $\mathbb{P_{CIF}}$-$Z$-open sets provide a unifying framework that generalizes several existing notions of openness in cubic intuitionistic fuzzy topological spaces, with significant implications for modeling hierarchical uncertainty in decision-making systems.