Stabilizer

Definition

For each trace $t$, the stabilizer is an endomorphism ${\sigma }_t:H_t\to H_t$ that is monotone and idempotent. It sends each recognition to its stabilized form.

Plain-language meaning

The stabilizer models repeated checking or normalization. After enough iterations, recognitions stop changing: they are stable.

Minimal structure

• A map ${\sigma }_t$ on each fiber.
• ${\sigma }_t$ is monotone: if $a\le b$ then ${\sigma }_t(a)\le {\sigma }_t(b)$.
• ${\sigma }_t$ is idempotent: ${\sigma }_t({\sigma }_t(a))={\sigma }_t(a)$.

Entry conditions

Use stabilizers only when:

• Your recognitions can be normalized to a stable form.
• Normalization does not depend on history (idempotent).

Example

Let $H_t=\{\perp <m<\top \}$. Define ${\sigma }_t$ to send $m$ to $m$ and keep $\perp ,\top$ fixed. Then ${\sigma }_t$ is monotone and idempotent.