Linkage Axioms for Generic Tropical Oriented Matroids

TL;DR

The paper addresses the unification of the axiomatics for generic tropical oriented matroids and their related combinatorial objects. It synthesizes Ardila–Develin theory with matching fields, tope arrangements, ensembles, and trianguloids under a common linkage framework, and provides cryptomorphic correspondences among these models. Key contributions include corrected proofs of equivalences, new linkage covector constructions for trees, and extended axioms that reveal multiple equivalent TOM presentations. The work clarifies how fine mixed subdivisions of $n\Delta^{d-1}$ and triangulations of $\Delta^{n-1}\times\Delta^{d-1}$ arise from the same combinatorial data, enabling cross-application of results across tropical geometry, polyhedral combinatorics, and matrix minor theory.

Abstract

We present a comprehensive overview of Ardila and Develin's (generic) tropical oriented matroids, as well as many related objects and their axiomatics. Moreover, we use a unifying framework that elucidates the connections between these objects, including several new cryptomorphisms and properties.

Figures (11)

• Figure 1: A fine mixed subdivision of $4\Delta^2$ and the corresponding collection of trees. The vertices of $4\Delta^2$ are labeled with the corresponding index of $[\bar{d}]$ and the unit simplices in the subdivision are labeled with the corresponding unique index of $[n]$ for which the label is $\Delta^2$.
• Figure 2: A tropical pseudo-hyperplane arrangement obtained from the fine mixed subdivision in \ref{['fig:FMSexample']}.
• Figure 3: Three examples of types in the generic tropical oriented matroid and their corresponding cells/faces in the corresponding fine mixed subdivision in \ref{['fig:FMSexample']}. The one corresponding to a lattice point is a tope and the one corresponding to a rhombus is a tree-type.
• Figure 4: A linkage covector $\mathbb{T}_{\{2, 3, \underline{1}, \underline{2}\}}$ of a pointed matching field (all edges/vertices), an extended left linkage covector $\tilde{\mathbb{T}}_{\{2, 3\}, \{\bar{3}\}}$ (black edges/vertices only), and a left linkage covector $\mathbb{T}_{\{2, 3\}, \{\bar{3}\}}$ (solid edges only) of the corresponding matching left-semi-ensemble.
• Figure 5: An extended tope arrangement obtained from the fine mixed subdivision in \ref{['fig:FMSexample']}.

Theorems & Definitions (87)

• Definition 2.1: CayleyTri, Section 1.2
• Remark 2.2
• Definition 2.3
• Definition 2.4
• Lemma 2.5: Permutohedra, Lemma 12.6 and Trianguloids, Lemma 6.1
• Theorem 2.6: FlagArr, Proposition 7.2
• Lemma 2.7
• proof
• Remark 2.8
• Theorem 2.9: CayleyZono, Theorem 3.1