Maximal-Hull $z$-Ideals, Congruence Closures, and Coherent Frames of Commutative Semirings

We develop a spectral theory of $z$-ideals for commutative semirings.

The lattice $\mathsf{ZId}(S)$ of $z$-ideals is a \emph{coherent frame} for every commutative semiring $S$ -- unconditionally, without cancellativity, subtractivity, or Noetherian hypothesis -- so the prime spectrum $\mathsf{Spec}_z(S)$ is spectral.

Under an explicit finite-type hypothesis on the canonical congruence-generated closure~$g$, the lattice $\mathsf{Id}_{g}(S)$ of $g$-closed ideals is likewise a coherent frame, and $\mathsf{Spec}_g(S)$ is spectral and homeomorphic to the space of prime $g$-congruences.

These frame results are accompanied by a regularity criterion: a semiring with all multiplicative idempotents complemented is von Neumann regular if and only if every principal ideal is a $z$-ideal, extending Mason's classical theorem from rings.

Separating the maximal-ideal-hull $z$-closure from the maximal-congruence-hull $g$-closure -- operations that coincide in rings but diverge in semirings -- is a central theme, confirmed by explicit computations in $\mathbb{N}$ and power-set semirings.

Both constructions carry a complete functorial formulation.