Algebraic properties of Indigenous semirings
Hussein Behzadipour, Henk Koppelaar, Peyman Nasehpour
TL;DR
The paper introduces Indigenous semirings as a concrete family within information algebras, built from Indigenous presemirings $I_k$ and extended to Indigenous semirings $S_k$. It develops graph-theoretic and topological invariants via the Indigenous graphs $\mathrm{IG}_k$ and the Zariski topology, showing the topology is the Sierpiński space in key cases; it then performs a thorough ideal-theoretic and polynomial/power-series analysis, yielding a local, totally ordered information algebra structure for $S_k$, a precise description of prime and radical ideals, and a complete characterization of units and idempotents in $S_k$, $S_k[X]$, and $S_k[[X]]$, including irreducibility criteria for quadratics. Localization results and monoid-semirings are shown to preserve the information-algebra property, connecting Indigenous numeration concepts to broader algebraic and topological frameworks. Collectively, the work provides a robust algebraic foundation for Indigenous numeration models with rich interactions among semiring theory, ideal theory, graph invariants, and power-series algebra.
Abstract
In this paper, we introduce Indigenous semirings and show that they are examples of information algebras. We also attribute a graph to them and discuss their diameters, girths, and clique numbers. On the other hand, we prove that the Zariski topology of any Indigenous semiring is the Sierpiński space. Next, we investigate their algebraic properties (including ideal theory). In the last section, we characterize units and idempotent elements of formal power series over Indigenous semirings.