The stated WQO gives finite equivariant bases

Exact statement

Let $\mathcal A$ be an $\omega$ -well-structured, totally ordered data domain. For every Noetherian commutative coefficient ring $K$ , each embedding-equivariant ideal of $K[\mathcal A]$ has a finite equivariant basis.

StatusSupported by the published equivariant Hilbert-basis theorem

External reviewThe finite-basis theorem appeared at LICS 2024. No external review of this FPRD specialization is recorded.

Context

A monomial is a finite induced data substructure labelled by its positive exponents. Label-increasing embedding is monomial divisibility modulo the source embedding action.

Proof or evidence

The source's equivariant Noetherianity criterion transfers the displayed WQO to finite generation of every invariant ideal, including the transition ideal.

What the two structural hypotheses imply →

Verification notes

The coefficient-ring hypothesis and embedding action were restored. The record explicitly rejects the invalid inference from finite generation to a terminating basis algorithm or a per-instance certificate bound.

Limitations

This is an existence theorem, not by itself a decision procedure.
No complexity bound follows.

Open work

Keep existence of a finite basis separate from effective computation of one.