On $j$-fuzzy $\gamma \mathcal{I}$-open sets with some applications

In this paper, we first introduced and studied a new class of fuzzy open sets, called $j$-fuzzy $\gamma \mathcal{I}$-open ( $j$-F $\gamma \mathcal{I}$-open) sets on fuzzy ideal topological spaces ( $\mathcal{F}\mathcal{I}\mathcal{T}\mathcal{S}s$). The class of $j$-F $\gamma \mathcal{I}$-open sets is contained in the class of $j$-fuzzy strong $\beta$- $\mathcal{I}$-open ( $j$-FS $\beta \mathcal{I}$-open) sets and contains all $j$-fuzzy pre- $\mathcal{I}$-open ( $j$-FP $\mathcal{I}$-open) sets and $j$-fuzzy semi- $\mathcal{I}$-open ( $j$-FS $\mathcal{I}$-open) sets. We also defined and investigated the closure and interior operators with respect to the classes of $j$-F $\gamma \mathcal{I}$-closed sets and $j$-F $\gamma \mathcal{I}$-open sets. However, we explored and discussed novel types of fuzzy $\mathcal{I}$-separation axioms using $j$-F $\gamma \mathcal{I}$-closed sets, called $j$-F $\gamma \mathcal{I}$-regular spaces and $j$-F $\gamma \mathcal{I}$-normal spaces. Thereafter, we displayed and investigated the concept of fuzzy $\gamma \mathcal{I}$-continuity (F $\gamma \mathcal{I}$-continuity) using $j$-F $\gamma \mathcal{I}$-open sets. Also, we presented and characterized the concepts of fuzzy weak $\gamma \mathcal{I}$-continuity (FW $\gamma \mathcal{I}$-continuity) and fuzzy almost $\gamma \mathcal{I}$-continuity (FA $\gamma \mathcal{I}$-continuity), which are weaker forms of F $\gamma \mathcal{I}$-continuity. Moreover, we showed that F $\gamma$ $\mathcal{I}$-continuity $⟹$ FA $\gamma$ $\mathcal{I}$-continuity $⟹$ FW $\gamma$ $\mathcal{I}$-continuity, but the converse may not be true. Finally, we defined and studied some new fuzzy $\gamma \mathcal{I}$-mappings via $j$-F $\gamma \mathcal{I}$-open sets and $j$-F $\gamma \mathcal{I}$-closed sets, called F $\gamma \mathcal{I}$-open mappings, F $\gamma \mathcal{I}$-closed mappings, F $\gamma \mathcal{I}$-irresolute mappings, F $\gamma \mathcal{I}$-irresolute open mappings, and F $\gamma \mathcal{I}$-irresolute closed mappings. The relationships between these classes of mappings were discussed with the help of some examples.