Glossary

5.2 Tropical oriented matroids

4 min read•august 20, 2024

Tropical oriented matroids extend classical oriented matroids to the tropical semiring. They provide a framework for studying tropical linear algebra and convexity, using covectors and axioms to represent tropical structures.

These matroids have various representations, including and chirotopes. They exhibit duality properties, enable the formulation of tropical linear programming problems, and find applications in tropical convexity and .

Tropical oriented matroids

  • Tropical oriented matroids generalize the concept of classical oriented matroids to the tropical semiring (R∪{∞},min⁡,+)(\mathbb{R} \cup \{\infty\}, \min, +)
  • Provide a combinatorial framework for studying tropical linear algebra and tropical convexity
  • Enable the application of matroid-theoretic techniques to solve problems in tropical geometry and optimization

Definition of tropical oriented matroids

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  • Defined as a collection of covectors satisfying certain axioms
  • Covectors represent the tropical analogues of signed vectors in classical oriented matroids
  • Each covector is a function from the ground set EE to the set {+,−,0}\{+, -, 0\}, indicating the sign of each element in the tropical setting

Axioms of tropical oriented matroids

  • Composition axiom: the composition of two covectors is again a covector
  • Vector elimination axiom: for any two covectors and an element e∈Ee \in E, there exists a third covector that agrees with the first on ee and with the second on E∖{e}E \setminus \{e\}
  • Zero covector axiom: the zero covector, which maps all elements to 00, is always present

Tropical Grassmann-Plücker relations

  • Tropical analogue of the classical Grassmann-Plücker relations
  • Provide a set of equations that characterize the Plücker coordinates of a tropical linear space
  • Enable the study of tropical Grassmannians and their combinatorial properties

Tropical pseudohyperplanes

  • Tropical analogue of classical hyperplanes
  • Defined as the set of points where a tropical linear form attains its minimum
  • Partition the tropical projective space into convex regions called sectors

Representations of tropical oriented matroids

Tropical pseudohyperplane arrangements

  • Collection of tropical pseudohyperplanes in a tropical projective space
  • Represent the underlying of a
  • Encode the covectors of the matroid through the sectors and their intersections

Tropical polytopes

  • Tropical analogue of classical polytopes
  • Defined as the intersection of finitely many tropical halfspaces
  • Can be represented by a arrangement, with each halfspace corresponding to a pseudohyperplane

Tropical chirotopes

  • Alternate representation of tropical oriented matroids
  • Defined as a function from ordered subsets of the ground set to {+,−,0}\{+, -, 0\}
  • Satisfy certain axioms that ensure compatibility with the covector axioms
  • Provide a more compact encoding of the combinatorial structure of the matroid

Duality in tropical oriented matroids

Dual tropical oriented matroids

  • Every tropical has a unique
  • Obtained by interchanging the roles of covectors and cocircuits (minimal non-zero covectors)
  • Duality preserves the combinatorial structure and axioms of the matroid

Tropical Farkas lemma

  • Tropical analogue of the classical Farkas lemma in linear programming
  • States that a tropical linear inequality system has a solution if and only if a certain dual inequality system has no solution
  • Provides a fundamental tool for studying feasibility and duality in tropical linear programming

Tropical linear programming duality

  • Tropical linear programming problems can be formulated using tropical oriented matroids
  • Primal and dual problems can be defined in terms of the covectors and cocircuits of the matroid
  • Strong duality holds, meaning that the optimal values of the primal and dual problems coincide

Tropical oriented matroid subdivisions

Tropical oriented matroid polytope subdivisions

  • Subdivision of a induced by a tropical oriented matroid
  • Obtained by intersecting the polytope with the tropical pseudohyperplanes of the matroid
  • Encodes the combinatorial structure of the matroid within the polytope

Tropical oriented matroid fan subdivisions

  • Subdivision of a tropical fan induced by a tropical oriented matroid
  • Obtained by intersecting the fan with the tropical pseudohyperplanes of the matroid
  • Provides a way to study the combinatorial properties of the matroid in the context of tropical fans

Tropical oriented matroid Bergman fans

  • Special case of
  • Arise from the Bergman fan of a matroid, which encodes its lattice of flats
  • Offer insights into the connection between tropical geometry and matroid theory

Applications of tropical oriented matroids

Tropical linear programming

  • Tropical oriented matroids provide a natural framework for formulating and solving tropical linear programming problems
  • Primal and dual problems can be defined using the covectors and cocircuits of the matroid
  • Duality results and the can be applied to study feasibility and optimality conditions

Tropical convexity

  • Tropical oriented matroids offer a combinatorial approach to studying tropical convex sets
  • Tropical polytopes and tropical halfspaces can be represented using tropical pseudohyperplane arrangements
  • Matroidal properties can be used to characterize the structure and properties of tropical convex sets

Tropical combinatorial optimization problems

  • Many classical combinatorial optimization problems have tropical analogues that can be formulated using tropical oriented matroids
  • Examples include tropical spanning tree problem, tropical shortest path problem, and tropical network flow problem
  • Matroidal techniques can be applied to develop efficient algorithms and gain insights into the structure of these problems

Key Terms to Review (28)

Algebraic Geometry: Algebraic geometry is a branch of mathematics that studies the geometric properties and relationships of solutions to polynomial equations. It connects algebra, specifically the theory of polynomials, with geometric concepts, allowing for the exploration of shapes and structures defined by these equations in various dimensions and fields.
Bernd Sturmfels: Bernd Sturmfels is a prominent mathematician known for his contributions to algebraic geometry, combinatorial geometry, and tropical geometry. His work has been influential in developing new mathematical theories and methods, particularly in understanding the connections between algebraic varieties and combinatorial structures.
Closure Operator: A closure operator is a mathematical tool that assigns to every subset of a given set a closed set in a way that satisfies specific properties. It plays a crucial role in the study of algebraic structures, particularly in understanding how certain properties can be preserved when subsets are taken into account. Closure operators help to formalize concepts of closure in various mathematical contexts, including tropical geometry and matroid theory.
Combinatorial Optimization: Combinatorial optimization is a field of optimization that focuses on finding the best solution from a finite set of discrete possibilities. It often deals with problems involving the arrangement, selection, and combination of elements to optimize certain criteria, like cost or efficiency. This concept is crucial in understanding structures and properties related to tropical geometry, as it intersects with various mathematical constructs and models.
Combinatorial Structure: A combinatorial structure refers to the arrangement and organization of discrete elements, often defined through specific relationships and rules. This concept plays a significant role in understanding how these elements interact within mathematical contexts, particularly in areas like matroid theory and intersection theory, where arrangements influence properties and outcomes.
Dual Matroid: A dual matroid is a concept in matroid theory where it reflects the relationship between two matroids defined on the same ground set. Essentially, for every independent set in one matroid, there corresponds a dependent set in its dual, showcasing a deep combinatorial connection. This duality provides insights into the structure and properties of matroids, particularly in tropical geometry where oriented matroids offer a framework to understand the geometric aspects of these mathematical objects.
Dual tropical oriented matroid: A dual tropical oriented matroid is a combinatorial structure that captures the duality relationship between tropical oriented matroids and their corresponding geometric objects. This concept emphasizes how the duality in the classical matroid theory is reflected in the tropical setting, where certain properties like independence and circuits maintain their significance. In this context, one can understand how geometric configurations relate through their respective dual structures, allowing for insights into both tropical geometry and combinatorial topology.
Giorgio Ottaviani: Giorgio Ottaviani is an influential mathematician known for his significant contributions to the field of algebraic geometry, particularly in tropical geometry. His work focuses on tropical polynomial functions and their applications, exploring the interplay between algebraic and combinatorial structures in mathematics.
Independence Axioms: Independence axioms are foundational principles that define the independence of subsets in a matroid, a structure that generalizes the concept of linear independence in vector spaces. These axioms provide a way to characterize independent sets, ensuring that the notion of independence is consistent and can be applied in various contexts, such as tropical geometry. Understanding these axioms is essential for analyzing the properties of tropical oriented matroids, where the independence relations can reflect combinatorial structures derived from tropical algebra.
Linear representation: Linear representation refers to a way of expressing objects, such as points or vectors, in terms of linear combinations of basis elements. This concept is fundamental in various mathematical areas, including geometry and algebra, as it allows for the analysis and manipulation of these objects in a structured manner. In the context of tropical oriented matroids, linear representations help to understand the relationships between points and their arrangements in a tropical geometric setting.
Oriented Matroid: An oriented matroid is a combinatorial structure that encodes the directional information of vectors in a vector space, generalizing the concept of linear independence. It incorporates the notion of orientation, which can be thought of as a way to assign directions to the hyperplanes associated with a set of vectors, allowing for a deeper understanding of their geometric arrangements. This structure is essential in the study of tropical geometry, where it provides insight into the relationships between points, lines, and higher-dimensional objects within the tropical setting.
Rota's Basis Conjecture: Rota's Basis Conjecture posits that for any finite set of vectors in a vector space, if you have a certain number of bases formed from these vectors, then you can always find a subset of those bases that can be combined to form a new basis. This idea is tied closely to the study of oriented matroids, where the structure and arrangement of vectors play a crucial role in understanding their geometric and combinatorial properties.
Tropical Chirotope: A tropical chirotope is a combinatorial structure that generalizes the concept of oriented matroids in the context of tropical geometry. It encodes information about the orientations and relationships of points in a tropical setting, allowing for the study of arrangements and properties of tropical polytopes and their intersections.
Tropical Curve: A tropical curve is a piecewise-linear object that emerges in tropical geometry, characterized by its vertices and edges formed from the tropicalization of algebraic curves. These curves provide a way to study the geometric properties of algebraic varieties in a new, combinatorial framework, linking them to polyhedral geometry and combinatorial structures.
Tropical duality: Tropical duality is a principle in tropical geometry that relates the combinatorial structures of objects in tropical spaces to their geometric counterparts. It offers a way to connect the concepts of Plücker coordinates and oriented matroids, revealing an interplay between algebraic and combinatorial properties in tropical settings.
Tropical Farkas Lemma: The Tropical Farkas Lemma is a principle in tropical geometry that provides necessary and sufficient conditions for the existence of tropical solutions to systems of inequalities. This lemma draws parallels to classical Farkas Lemma in linear algebra, enabling the understanding of tropical linear inequalities and their feasible solutions, which are essential in various areas like discrete convexity and oriented matroids.
Tropical grassmann-plücker relations: Tropical grassmann-plücker relations are identities that describe the relationships between tropical Plücker coordinates, which represent the points in tropical projective spaces. These relations connect the geometry of tropical varieties to classical algebraic geometry, enabling the study of linear subspaces in a tropical setting. Understanding these relations helps to establish the connection between tropical Plücker vectors and oriented matroids, providing insight into their combinatorial structures and geometric interpretations.
Tropical Grassmannian: The tropical Grassmannian is a combinatorial object that generalizes the classical Grassmannian to tropical geometry, capturing the essence of linear subspaces in a tropical setting. It arises naturally in various contexts, including the study of tropical polytopes and as a tool for understanding tropical varieties through their Plücker coordinates. This framework also connects deeply with concepts like tropical discriminants and Schubert calculus, providing insights into how different geometrical structures can be analyzed through the lens of tropical algebra.
Tropical Linear Programming Duality: Tropical linear programming duality is a framework that extends classical linear programming concepts into the tropical setting, where the usual addition is replaced by taking minimum and multiplication by addition. This duality principle allows one to relate primal and dual problems in tropical linear programming, providing insights into optimal solutions and feasibility, similar to its classical counterpart.
Tropical oriented matroid: A tropical oriented matroid is a combinatorial structure that extends the notion of oriented matroids to the setting of tropical geometry. It encapsulates the relationships and dependencies among points in a tropical space, using a tropical version of linear dependence that is based on valuations and piecewise-linear functions. This concept is crucial for understanding the properties of tropical hyperplane arrangements and their associated combinatorial structures.
Tropical oriented matroid Bergman fans: Tropical oriented matroid Bergman fans are combinatorial structures that arise in the study of tropical geometry, connecting concepts from matroid theory and toric varieties. They can be seen as tropical analogs of classical Bergman fans, encoding the information of oriented matroids through their polyhedral geometry and tropicalization process.
Tropical oriented matroid fan subdivisions: Tropical oriented matroid fan subdivisions refer to a specific type of subdivision of a fan associated with tropical oriented matroids, which represent combinatorial structures derived from tropical geometry. These subdivisions capture the relationships and dependencies among points in a tropical setting, allowing for a visualization of how various combinatorial properties interact. They play a crucial role in understanding the geometry and combinatorics of tropical varieties, particularly in the way they can be decomposed into simpler pieces.
Tropical oriented matroid polytope subdivisions: Tropical oriented matroid polytope subdivisions refer to the divisions of polytopes that arise from the combinatorial structures of oriented matroids within tropical geometry. These subdivisions reflect how the tropical structure impacts the underlying geometry, allowing for a more flexible understanding of properties such as convexity and intersection. This concept ties together various aspects of matroid theory and tropical geometry, showcasing how combinatorial properties influence geometric configurations.
Tropical Polytope: A tropical polytope is a geometric object defined in tropical geometry, which is a piecewise-linear analogue of classical polytopes. It is formed by taking the convex hull of a set of points in tropical space, where the operations of addition and multiplication are replaced by minimum and addition, respectively, allowing for a new way to study combinatorial structures and optimization problems.
Tropical product: The tropical product is an operation in tropical mathematics that replaces traditional multiplication with the minimum (or maximum) operation and addition. This operation forms the backbone of various calculations in tropical geometry, enabling the exploration of properties such as matrix operations, rank, and the structure of oriented matroids, where the concepts of addition and multiplication are fundamentally redefined.
Tropical Pseudohyperplane: A tropical pseudohyperplane is a geometric object in tropical geometry, representing the tropical analogue of a hyperplane in classical algebraic geometry. In this framework, it is defined as a piecewise linear function that partitions the tropical space into regions where the function takes on constant values. This concept is crucial for understanding the structure of tropical oriented matroids, where these pseudohyperplanes help define the arrangement of points and their relationships in a tropical setting.
Tropical pseudohyperplane arrangements: Tropical pseudohyperplane arrangements are collections of tropical hyperplanes in a tropical space, which is a combinatorial analogue of classical hyperplane arrangements in algebraic geometry. They play a vital role in the study of tropical oriented matroids by providing a geometric framework to understand the relationships and intersections between these tropical hyperplanes, leading to insights into the underlying combinatorial structure.
Tropical sum: The tropical sum is a fundamental operation in tropical mathematics, defined as the minimum of two elements, reflecting the tropical algebra's unique approach to addition. This operation plays a crucial role in understanding concepts like tropical rank and oriented matroids, as it transforms traditional arithmetic into a framework suited for studying geometric structures in tropical geometry.
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