Abstract
This paper introduces the concept of the direct product of sets that involve Fermatean neutrosophic elements in structures called INK-algebras. It defines terms like the direct product of Fermatean neutrosophic INK-ideals in INK-algebras and Fermatean neutrosophic sets (FNSs), Fermatean neutrosophic INK-ideals (FNINK-Is), and Fermatean neutrosophic closed INK-ideals (FNCINK-Is). The proof of theorems illustrating the relationships between these ideas is included in the paper. It also defines the INK-sub algebra embedded in an INK-algebra and gives a theorem elucidating the connection between the direct product of Fermatean neutrosophic INK-ideals and the images of these sub-algebras. In essence, the paper investigates and establishes connections between different mathematical ideas concerning INK-algebras and FNSs.
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