A New $\omega$-Stable Plane

TL;DR

This paper constructs a new $oldsymbol{ ext{omega}}$-stable plane with unique properties, including non-representability over any field, and analyzes its model-theoretic features such as forking, closure, and Morley rank.

Contribution

It introduces a novel $oldsymbol{ ext{omega}}$-stable plane that is not one-based and lacks algebraic representation, expanding understanding of stable structures.

Findings

Constructed a non-one-based $oldsymbol{ ext{omega}}$-stable plane.

Characterized forking and closure properties in the structure.

Showed the structure has Morley rank $oldsymbol{ ext{omega}}$ and fails weak elimination of imaginaries.

Abstract

We use a variation on Mason's $\alpha$-function as a pre-dimension function to construct a not one-based $\omega$-stable plane $P$ (i.e. a simple rank $3$ matroid) which does not admit an algebraic representation (in the sense of matroid theory) over any field. Furthermore, we characterize forking in $Th(P)$, we prove that algebraic closure and intrinsic closure coincide in $Th(P)$, and we show that $Th(P)$ fails weak elimination of imaginaries, and has Morley rank $\omega$.