Finitely Generated Congruences in Semirings and Canonical Positive Models

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 $\mathbb{N}$.