Monoids All Polygons Over Which Are $\omega$-Stable: Proof of the Mustafin – Poizat Conjecture

A monoid $S$ is called an $\omega$-stabilizer (superstabilizer, or stabilizer) if every $S$-polygon has an $\omega$-stable (superstable, or stable) theory. It is proved that every $\omega$-stabilizer is a regular monoid. This confirms the Mustafin – Poizat conjecture and allows us to end up the description of $\omega$-stabilizers.