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 $ \widetilde{\zeta} $-$ k $-folded $ \mathcal{N} $-subsemigroups ($ \widetilde{\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.