Finitely Generated Congruences in Semirings and Canonical Positive Models

TL;DR

This paper explores Congruence Noetherian semirings, where all congruences are finitely generated, and demonstrates that canonical positive models of a real order are both Congruence Noetherian and flat over .

Contribution

It introduces the concept of Congruence Noetherian semirings and analyzes the properties of canonical positive models in this context.

Findings

Canonical positive models are Congruence Noetherian.

Canonical positive models are flat over .

Not all finitely generated models are necessarily Congruence Noetherian.

Abstract

In this paper, we inspect a relatively unexplored notion of finite generation in semirings, namely semirings in which all congruences are finitely generated. Such semirings are dubbed Congruence Noetherian. After developing sufficient background and examples, we focus on the canonical positive models of a real order and show that this obvious choice, though not finitely generated as an $\mathbb{N}$-module, is both Congruence Noetherian and flat over $\mathbb{N}$.