unitary semi-ring on finite nexuses

This paper introduces and investigates a novel algebraic framework for structures composed of finite addresses, which we term nexuses. a nexus n is defined as a set of finite sequences (or addresses), and we define its order, n(n), as the maximum integer value appearing among its constituent addresses. within this context, we introduce two custom binary operations, ⊕n and n, parameterized by a natural number n. the central construction of this work is the generation of a semi-ring hnin from a given finite-order nexus n, where n = n(n). this generated structure is shown to itself be a nexus, thereby preserving the core combinatorial properties of the original set while enriching it with a coherent algebraic architecture. notably, the semi-ring hnin is unitary, with the multiplicative identity given by the unit address (1).