Algebraic properties of Indigenous semirings

TL;DR

This paper introduces Indigenous semirings, explores their algebraic and topological properties, and characterizes elements like units and idempotents, revealing their structure as examples of information algebras.

Contribution

It is the first to define Indigenous semirings, analyze their graph-theoretic and topological properties, and study their algebraic structure including ideals and elements.

Findings

Indigenous semirings are examples of information algebras.

The Zariski topology on Indigenous semirings is the Sierpiński space.

Characterization of units and idempotents in formal power series over these semirings.

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 Sierpi\'{n}ski 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.