Radical-Ideal Functors, a Support Bifibration, and Quantale Completion for Commutative Semirings

We organize ordinary, subtractive ($k$-), and strong ideal theory of commutative semirings into a functorial framework.

Radical extension is left adjoint to contraction and yields coherent-frame-valued functors naturally represented by the open-set frames of the corresponding prime spectra.

The comparison from ordinary to $k$-radical ideals is a natural nucleus whose components are surjective and, under coherent Stone duality, correspond to dense sublocale embeddings.

Ordinary, $k$-, and strong prime spectra form nested natural spectral functors, while universal support objects recover the spectra, radical frames, and complemented idempotents.

Finite supports assemble into a Grothendieck bifibration with a canonical bicartesian section.

For complete idealic semirings, $k$-ideal completion realizes a subtractive form of ideal quantale completion.

We compute the induced monad, identify its restriction to frames with the classical ideal-lattice monad, and prove that its Eilenberg--Moore category is equivalent to the category of integral commutative quantales.

Applications include a Stone-spectrum criterion for positive cones of $f$-rings and density criteria for $k$-prime spectra of $r$-semirings.