A Fraïssé theory for partial orders of a fixed finite dimension

TL;DR

The paper generalizes Fraïssé theory to finite-dimensional partial orders by adjoining $n$ linear realizers to form the class $\mathcal{PO}_{n,<_1,\ldots,<_n}$ and its Fraïssé limit $\mathbf{D}_{n,<_1,\ldots,<n}$. It provides a finite axiomatization $\mathsf{DPO}_{n,<_1,\ldots,<n}$ for this limit, proves the Ramsey property for the class, and deduces extreme amenability of $\mathrm{Aut}(\mathbf{D}_{n,<_1,\ldots,<n})$. The universal minimal flow of $\mathrm{Aut}(\mathbf{D}_n)$ is shown to be the extended logic action on the finite set of realizers $\mathcal{R}_n$ (with $|\mathcal{R}_n|=n!$), and a decomposition $\mathrm{Aut}(\mathbf{D}_n)=\mathrm{Aut}(\mathbf{D}_{n,<_1,\ldots,<n})\rtimes S_n$ provides an alternative route to the same flow. These results connect model theory, combinatorics, and topological dynamics for finite-dimensional posets and yield explicit descriptions of automorphism groups and their dynamics.

Abstract

For each $n\geq 2$, we show that the class of all finite $n$-dimensional partial orders, when expanded with $n$ linear orders which realize the partial order, forms a Fraïssé class and identify its Fraïssé limit $(D_n,<,<_1,\ldots,<_n)$. We give a finite axiomatization of this limit which specifies it uniquely up to isomorphism among countable structures. We then show that the aforementioned class of finite structures satisfies the Ramsey property and conclude, by the Kechris-Pestov-Todorčević correspondence, that the automorphism group of its Fraïssé limit is extremely amenable. Finally, we identify the universal minimal flow of the automorphism group of the reduct $(D_n,<)$.

Figures (9)

• Figure 1: A Hasse diagram for $(C_3,<)$.
• Figure 2: The amalgamation property (AP).
• Figure 3: The failure of the AP in $\mathcal{PO}_2$.
• Figure 4: $\mathbf{D}_2$ is not ultrahomogeneous.
• Figure 5: $(\mathbb{Q}^2,<,<_{\mathrm{lex}_1},<_{\mathrm{lex}_2})$ is not ultrahomogeneous.

Theorems & Definitions (47)

• Lemma 1.1: Szpilrajn zbMATH02566488
• Example 1.2
• Theorem 1.3: Theorem 10.4.2 in MR0150753
• Proposition 1.4
• proof
• Theorem 2.1: Fraïssé MR0069239, Theorem 7.1.2 in MR1221741
• Theorem 2.2: Fraïssé MR0069239, Theorem 7.1.7 in MR1221741
• Proposition 2.3
• proof
• Proposition 2.4