Finitely Generated Congruences in Semirings and Canonical Positive Models
Snehinh Sen
TL;DR
This work develops Congruence Noetherian (c-Noetherian) theory for semirings and connects it to classical Noetherian behavior via a Hilbert Basis Criterion, revealing when semialgebras over a semiring inherit c-Noetherianity and how this interacts with ring-theoretic Noetherian properties. It then analyzes positive models, focusing on the canonical positive model $S=R_{ egeq 0}$ associated to real-order subrings, proving that $S$ is flat over $b{N}$ and that its c-Noetherian or c-principal status aligns with $R$ being Noetherian or a PID, respectively. A central result shows that for real orders, every nontrivial quotient of $S$ is finite, and that congruence structure on $S$ can be described via $k$-ideals, linking semiring finiteness to classical ring-theoretic finiteness. The paper thus extends Noetherian phenomena to semiring contexts and validates the canonical positive model as a robust bridge between semiring and ring theory, with precise finiteness and flatness properties codified.
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 $\mathbb{N}$.