Unitary semi-ring on some nexuses

Abstract

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 (addresses) satisfying a closure property. We define its order, n(N), as the maximum integer value appearing among its constituent addresses. Two custom binary operations, ⊕ and ⊙n, parameterized by a natural number n, are introduced. The central construction is the generation of a semi-ring ⟨N⟩n from a given finite-order nexus N, where n = n(N). This generated structure is shown to be itself a nexus, thereby preserving the core combinatorial properties of the original set while enriching it with a coherent algebraic architecture. Notably, ⟨N⟩n is unitary, with multiplicative identity given by the unit address (1).