@article{Al-Masarwah2023, 
author = {Anas Al-Masarwah and Mohammed Alqahtani},
title = {Operational algebraic properties and subsemigroups of semigroups in view of    k-folded        N  -structures},
year = {2023},
journal = {AIMS Mathematics},
volume = {8},
number = {9},
pages = {22081-22096},
keywords = {semigroups, N-structure, k-folded N-structure, k-folded N-subsemigroup, k-folded N-product, k-folded N-idempotent},
url = {https://www.sciopen.com/article/10.3934/math.20231125},
doi = {10.3934/math.20231125},
abstract = {The concept of    k-folded        N  -structures (   k-F       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       N  S by providing a general algebraic structure for tackling    k-folded        N  -data by fusing the conception of semigroup and    k-F       N  S. First, we introduce and study some algebraic properties of    k-F       N  Ss, for instance, subset, characteristic function, union, intersection, complement and product of    k-F       N  Ss, and support them by illustrative examples. We also propose    k-folded        N  -subsemigroups (   k-F       N  SBs) and              ζ      ~      -   k-folded        N  -subsemigroups (             ζ      ~      -   k-F       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        N  -product, characterizations of    k-F       N  SBs are also discussed. Further, we obtain a necessary condition of a    k-F       N  SB to be a    k-folded        N  -idempotent. Finally, relations between    k-folded        N  -intersection and    k-folded        N  -product are displayed, and how the image and inverse image of a    k-F       N  SB become a    k-F       N  SB is studied.}
}