Invariant universality for projective planes

TL;DR

This paper investigates the complexity of bi-embeddability among countable projective planes, establishing their invariant universality and introducing a new Borel reducibility concept linked to category theory.

Contribution

It demonstrates the invariant universality of bi-embeddability relations for countable projective planes and introduces a novel Borel reducibility framework preserving stabilizers.

Findings

Bi-embeddability on countable projective planes is invariantly universal.

Established a connection between Borel reducibility and category-theoretic embeddings.

Proved the relations are complete analytic, indicating maximal complexity.

Abstract

We continue the work of [1, 2, 3] by analyzing the equivalence relation of bi-embeddability on various classes of countable planes, most notably the class of countable non-Desarguesian projective planes. We use constructions of the second author to show that these equivalence relations are invariantly universal, in the sense of [3], and thus in particular complete analytic. We also introduce a new kind of Borel reducibility relation for standard Borel G-spaces, which requires the preservation of stabilizers, and explain its connection with the notion of full embeddings commonly considered in category theory.