Operational algebraic properties and subsemigroups of semigroups in view of $k$-folded $\mathcal{N}$-structures

The concept of $k$-folded $\mathcal{N}$-structures ( $k$-F $\mathcal{N}$Ss) is an essential concept to be considered for tackling intricate and tricky data. In this study, we want to broaden the notion of $k$-F $\mathcal{N}$S by providing a general algebraic structure for tackling $k$-folded $\mathcal{N}$-data by fusing the conception of semigroup and $k$-F $\mathcal{N}$S. First, we introduce and study some algebraic properties of $k$-F $\mathcal{N}$Ss, for instance, subset, characteristic function, union, intersection, complement and product of $k$-F $\mathcal{N}$Ss, and support them by illustrative examples. We also propose $k$-folded $\mathcal{N}$-subsemigroups ( $k$-F $\mathcal{N}$SBs) and $\tilde{\zeta }$- $k$-folded $\mathcal{N}$-subsemigroups ( $\tilde{\zeta }$- $k$-F $\mathcal{N}$SBs) in the structure of semigroups and explore some attributes of these concepts. Characterizations of subsemigroups are considered based on these concepts. Using the notion of $k$-folded $\mathcal{N}$-product, characterizations of $k$-F $\mathcal{N}$SBs are also discussed. Further, we obtain a necessary condition of a $k$-F $\mathcal{N}$SB to be a $k$-folded $\mathcal{N}$-idempotent. Finally, relations between $k$-folded $\mathcal{N}$-intersection and $k$-folded $\mathcal{N}$-product are displayed, and how the image and inverse image of a $k$-F $\mathcal{N}$SB become a $k$-F $\mathcal{N}$SB is studied.