Abstract
The set $\underline{SET}(n)$, generated by an arbitrary natural number $n$, was defined in [3]. There, and in [5,6], some arithmetic functions and arithmetic operators of a modal and topological types are defined over the elements of $\underline{SET}(n)$. Here, over the elements of $\underline{SET}(n)$ new arithmetic functions are defined and some of their basic properties are studied. Two standard modal topological structures over $\underline{SET}(n)$ are described. Perspectives for future research are discussed.
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