Realization spaces of matroids over hyperfields
We study realization spaces of matroids over hyperfields (in the sense of Baker and Bowler ). More precisely, given a matroid and a hyperfield we determine the space of all -matroids over . This can be seen as the matroid stratum of the hyperfield Grassmannians in the sense of Anderson and Davis .
For an algebraically determined class of hyperfields we give different descriptions of these realization spaces (e.g., in terms of Tutte groups or cross-ratios), allowing for explicit computations. When the hyperfield at hand is topological, the realization spaces have a natural topology. In this case, our models carry the correct homeomorphism type.
As applications of our methods we obtain a theorem on the existence of phased matroids that are not realizable over , as well as a result on the diffeomorphism type of complex hyperplane arrangements whose underlying matroid is uniform.
Matroids over hyperfields were introduced by Baker and Bowler in . They unify several flavors of matroid theory, including oriented matroids , valuated matroids  and phased matroids . Accordingly, they have applications to different areas of mathematics such as tropical geometry, Berkovich theory and classical algebraic geometry [4, §1].
A matroid over a hyperfield can be defined as a class of Grassmann-Plücker functions on a (finite) ground set with values in a hyperfield. Hyperfields are field-like objects where addition is allowed to be multivalued. An ordinary matroid corresponds to a matroid over the Krasner hyperfield, a matroid over the sign hyperfield corresponds to an oriented matroid and a matroid over the tropical hyperfield is a valuated matroid. (See Section 1.2 for precise definitions and examples.) Matroids over hyperfields admit the following “functorial” property: given a morphism of hyperfields and an -Matroid , there is an induced -Matroid . The Krasner hyperfield is the final object in the category of hyperfields and, accordingly, we can define the underlying matroid of any -Matroid as the push-forward with respect to the unique map . In this paper we study the following question:
What is the space of all matroids over a hyperfield
with a given underlying matroid?
In the case of oriented matroids, the corresponding space is a discrete and finite set that has been studied by Gel’fand, Rybnikov and Stone , who provide four different characterizations of it, up to a canonical operation on oriented matroids called reorientation (compare [7, §3.1 and Remark 3.2.3], [20, p. 121]).
In general, these spaces are not finite; moreover, many hyperfields of interest carry a topological structure which induce a topology on our realization spaces. Our aim is then to model the homeomorphism type of such realization spaces.
Generalizing the notion of reorientation to the context of hyperfields, we introduce the notion of rescaling class of a matroid over a hyperfield and we give several descriptions of (the homeomorphism type of) the space of rescaling classes of matroids over a fixed hyperfield and with a prescribed underlying matroid. Although our work is inspired by , we will see that working in the generality of hyperfields and accounting for topology introduces many new challenges. The reward is, then, a better structural understanding as well as a wider array of applications, of which we will outline a sample.
Readers not familiar with multivalued operations might find it unnatural to work with hyperfields. We briefly comment on the origin and some applications of these objects (for basic definitions and examples we refer to Section 1.2). The idea of multivalued algebraic objects goes back at least to 1934, when Marty introduced the notion of hypergroups . In particular, in 1956 Krasner introduced hyperrings in order to develop some technical tools in the study of approximations of valued fields . Ever since their first appearance, algebro-geometric properties of hyperrings have been investigated [10, 34]. In , Connes and Consani showed that Connes’ adèle class space of a global field has a hyperring structure, they investigated the connection between “vectorspaces” over the Krasner hyperfield and finite projective geometries, and they began the study of multivalued algebraic geometry on hyperrings. For a good overview and the connection to tropical geometry we refer to . In  and , Jaiung Jun further developed the theory of algebraic geometry over hyperrings by introducing integral hyperring schemes and used hyperrings in order to generalize the classical notion of valuations.
Matroids over hyperfields
Baker and Bowler presented several equivalent (or, in matroid theory parlance, “cryptomorphic”) descriptions of matroids over hyperfields – such as via circuits, dual pairs and Grassmann-Plücker functions – as well as a duality theory which depends on the choice of an involution of the hyperfield at hand. A special feature of this theory is the distinction of two notions of -matroids, namely strong and weak -matroids. Anderson contributed vector axioms in the strong case . For more details on definitions and examples on matroids over hyperfields we refer to Section 1.2. Note that the follow-up paper of Baker and Bowler  extends this theory to even more general algebraic structures, a line of research further advanced in the very recent preprint of Pendavingh .
When the hyperfield is a classical field, the space we aim at describing is known as the matroid stratum of the corresponding Grassmannian, or the realization space of the given matroid over the field at hand, going back to . In general, our spaces are related to the hyperfield realization spaces appearing in Anderson and Davis’ work on hyperfield Grassmannians (see  and Remark 2.17.(2)), where the notion of a topological hyperfield has been introduced.
In the special case of the sign hyperfield we recover the results of . Moreover, specializing to the tropical hyperfield our work amounts to describing the space of projective equivalence classes of valuated matroids  with prescribed underlying matroid. This is a quotient of the matroid’s Dressian, see [29, §4.4], hence the corresponding specialization of our results fits into the line of research studying the structure of Dressians, see [22, 23]. We do not pursue it here, but we mention as a sample the question of whether one of our descriptions could improve on the upper bound on the dimension of the Dressian of uniform matroids given in [25, Theorem 31].
Since our goal is to obtain descriptions for the space of all hyperfield matroids with a given underlying matroid, we first verify that the different equivalent definitions of matroids over hyperfields give rise to natural bijections (resp. homeomorphisms) between the corresponding spaces, allowing us to properly define “the” (topological) space of rescaling classes.
The space of hyperfield projective classes of a matroid , defined in terms of circuits and cocircuits of ;
Other than in the oriented matroid case (compare [20, Theorem 1]) for matroids over hyperfields the space 1 needs not be in bijection with the space of rescaling classes. In Section 3.2 we characterize algebraically those hyperfields for which this one-to-one correspondence holds. We name the corresponding class of hyperfields WAM hyperfields and show that the class of non-WAM hyperfields is non-empty.
The space of -cross-ratios, described geometrically as a subset of ;
A space of reduced cross-ratios, which affords easier geometric considerations, obtained by studying a new presentation of the inner Tutte group that eliminates redundant information.
Geometric and algebraic properties of such spaces can be used to tackle specific problems. For example, working with 4, we derive an explicit characterization of rescaling classes as solution of systems of equations. This allows in Proposition 4.11 to give upper bounds on the number of weak matroids over finite hyperfields with underlying matroid in terms of circuits of .
As a final structural result, with Theorem 5.1 we prove that if is a sub-hyperfield of then the space of -rescaling classes over a fixed matroid embed into that of -rescaling classes.
Our methods and results allow us to use topological and geometric techniques in order to obtain the following applications.
There exist phased matroids that are neither realizable over nor arising from the “complexification” of an oriented matroid (Theorem 6.2).
The diffeomorphism type of the complement manifold of any two arrangements of hyperplanes in complex space with uniform underlying matroid is determined by the underlying matroid itself (Corollary 6.6).
In Section 1 we recall the basics of matroids over hyperfields by giving the relevant definitions, examples and results. Then we introduce rescaling classes of matroids over hyperfields in Section 2. In Section 3 we introduce projective classes of matroids over hyperfields, WAM hyperfields and how these particular hyperfields are connected to the relation between projective classes and rescaling classes.
In Section 4 we give our characterizations of projective classes based on different variations of the Tutte group. (The definitions of those Tutte groups as well as other technical ingredients of the proofs are given in an Appendix.) Section 5 proves that the sub-hyperfield relation induces an embedding of the corresponding rescaling classes on a common matroid. Finally, in Section 6 we derive the stated applications to phased matroids and hyperplane arrangements. The paper is rounded up by a series of appendices that contain some technical computations and proofs which would otherwise have cluttered the main expository part.
Remark on the ArXiv history
The roots of this paper lie in the study of phasing spaces of matroid by the first and third author. That paper appeared as an earlier version of this ArXiv entry, and is now encompassed and superseded by the present work, which adopts the wider point of view of matroids over hyperfields.
We thank Laura Anderson, Christopher Eppolito, Jaiung Jun and Thomas Zavslasky for the warm hospitality and the very productive discussions during a visit at SUNY Binghamton. We also thank Alex Fink, Ivan Martino and Rudi Pendavingh for the opportunity to discuss an earlier version of our work. Moreover, we thank Richard Randell for feedback about Section 6.2, Peter Michor for advice regarding Lemma C.3, Alberto Cavallo for pointing out . We are also grateful to Michael Joswig and Benjamin Schröter for sharing their expertise on Dressians in friendly discussions during the 2018 special semester on tropical geometry at the Institute Mittag-Leffler.
All authors have been supported by the Swiss National Science Foundation Professorship grant PP00P2_150552/1.
1. Basics on matroids over hyperfields
In this introductory section we recall some basic definitions and results about matroids and matroids over hyperfields. For a thorough treatment of matroid theory we point to Oxley’s book , for basics on hyperfields we refer to Viro  while the foundations of matroids over hyperfields are laid in the preprint by Baker and Bowler .
A matroid is a pair , where is a finite set and is a collection of subsets of satisfying the following two conditions:
For all and , there exists such that
The set is called the ground set of . The members of are the bases of . The collection of subsets of elements of are the independent sets of , denoted by . A subset of that is not in is called dependent. Minimal inclusion dependent sets are called circuits and the family of circuits of will be denoted by . If no confusion arises, we write , and for the collections of independent sets, bases and circuits of .
The rank of a subset is defined by
and we define the rank of the matroid as . A subset of is spanning if .
Remark 1.1 (Cryptomorphisms).
Our definition in terms of bases can be replaced by a set of requirements for any of the set systems described by an italicized word above. This availability of different reformulations is a distinctive feature of matroid theory. The rules allowing to switch between these reformulations are called “cryptomorphisms”.
Remark 1.2 (Duality).
The family of complements of spanning sets of a matroid is the collection of independent sets of a matroid called dual to . The rank function of is linked to that of by .
Circuits and bases of are called cocircuits and cobases of . We write and for the families of cocircuits and cobases of . Again, if no confusion arises we write and for the families of cocircuits and cobases of .
Remark 1.3 (Representability).
A matroid is called representable if its ground set maps into a vector space so that a subset of is independent if and only if the corresponding vectors are linearly independent.
Example 1.4 (The Fano matroid).
The Fano matroid is defined on the ground set by the circuit set
It is representable over if and only if the characteristic of is two [32, Proposition 6.4.8].
Remark 1.5 (Minors).
Given a subset of the ground set of the matroid , the collection of all subsets of that are independent in satisfies the independence axioms. Thus, it is the set of independent sets of a matroid, called restriction of to and denoted by . The contraction of in is the matroid . A minor of is any matroid that can be obtained from through a sequence of restrictions and contractions.
We will often consider matroids “without minors of Fano or dual-Fano type”. By this we mean matroids for which neither the Fano matroid (see Example 1.4) nor its dual arise as minors.
Remark 1.7 (Connectedness).
Given matroids and with ground sets and and independent sets and , the direct sum of and is the matroid with ground set and independent sets
We say that is disconnected if there exists a proper non-empty subset of the ground set such that . We call connected otherwise. A connected component of is a maximal inclusion subset of such that is connected. From [32, Corollary 4.2.13] there is a unique (up to permutations) decomposition of as direct sum of connected matroids, allowing us to properly define the number of connected components of .
Given a set , a hyperoperation on is a map from to the collection of non-empty subsets of . If and are non-empty subsets of , we set
and we say that is commutative if for all , . We call associative if for all , , .
A commutative hypergroup is a tuple , where is a commutative and associative hyperoperation on such that
for all ;
For each there is a unique element of (denoted by and called the hyperinverse of ) such that ;
if and only if .
Given a commutative monoid , an element and a non-empty subset of we define
A commutative hyperring is a tuple such that
is a commutative hypergroup;
is a commutative monoid;
for all (Absorption rule);
for all , , (Distributive law).
A hyperfield is a commutative hyperring such that and all non-zero elements of have an inverse with respect to .
When no confusion arises, we denote a hyperfield by its underlying set and we write for the set of its non-zero elements. We will often denote by .
A sub-hyperfield of a hyperfield is a subset that itself is a hyperfield with respect to the operations induced by .
A hyperfield homomorphism is a map such that:
for any , ;
for any , .
An involution of the hyperfield is a hyperfield homomorphism such that . According to this definition the identity map of is an involution.
In the following statement we summarize some elementary algebraic properties of hyperfields that will be widely used in our work.
For a hyperfield with an involution the following properties hold:
for all ;
Since is an abelian group, we already know that
Thus, it suffices to see that and this follows immediately from
Similarly, to prove that it is enough to consider the relation below
The following notion of a topological hyperfield has been recently introduced by Anderson and Davis .
A topological hyperfield is a hyperfield with a topology on such that is open, the multiplication map is continuous, and the multiplicative inverse map is continuous.
A homomorphism of topological hyperfields is a hyperfield homomorphism that is continuous with respect to the given topology. Accordingly, when talking about topological hyperfields we consider only continuous involutions. In particular, every involution of a topological hyperfield is a homeomorphism.
The Krasner hyperfield defined on the set with the usual multiplication rule and hyperaddition law given by:
if or ;
The involution is the identity.
The hyperfield of signs defined on the set with the usual multiplication rule and hyperaddition law given by setting:
The involution is the identity.
The phase hyperfield defined on the set , where is the complex unit circle, with usual multiplication rule and hyperaddition law given by setting:
The involution is complex conjugation. Note that the name of this hyperfield is used differently in .
The tropical hyperfield defined on the set with multiplication rule defined by (and as absorbing element) and hyperaddition law given by setting:
The involution is the identity.
The triangle hyperfield defined on the set with usual multiplication rule and hyperaddition law defined by setting:
The involution is the identity.
1.3. Matroids over hyperfields
Throughout this work, we always assume that a hyperfield , an involution of and a finite ground set are given.
A hyperfield vector is any . The support of a hyperfield vector is the set
Definition 1.13 (Orthogonal hyperfield vectors).
Two hyperfield vectors and are orthogonal with respect to — denoted by — if
Two sets of hyperfield vectors are orthogonal with respect to — written — if for all and .
As explained in  there exist two different kinds of matroids over a hyperfield , that are called weak and strong . We now provide definitions and we recall cryptomorphisms for both cases.
Definition 1.14 (Grassmann–Plücker functions; [4, Definition 3.6]).
A rank weak Grassmann–Plücker function on with values in is a non-zero alternating function such that its support is the set of bases of a matroid and
for any two subsets and of with .
A rank strong Grassmann–Plücker function on with values in is a non-zero alternating function such that
for any two subsets and of .
We say that two weak (resp. strong) Grassmann–Plücker functions and are equivalent if for some .
Remark 1.15 (Matroids over the Krasner hyperfield).
In the case of being the Krasner hyperfield, strong and weak -matroids are the same and correspond to ordinary matroids, see 1.22.
Both axiom systems of Definition 1.14 ensure that the support of any weak (resp. strong) Grassmann–Plücker function is the set of bases of a matroid on which we call . We then call a subset if it is an independent set of the matroid . Accordingly, a is a maximal set.
In order to state weak (resp. strong) axioms of weak (resp. strong) we need at first to recall the notion of modular pair (resp. modular elimination structure).
As suggested by Baker and Bowler in [4, Section 1.2], the modular elimination can be interpreted in the following sense. If and are hyperfield vectors that are “sufficiently close” and there exists an index such that , then it is possible to “eliminate” i by (hyper-) summing and , i.e. there is with and for all .
To be more precise, given a family we say that , form a modular pair if , is a modular pair in the lattice of unions of supports of elements of . More generally (compare [4, Definition 3.7]), assume that we have a subset of , an indexed family with , and with for all but . We say that and give a modular elimination structure if the height of in the lattice of unions of supports of elements of is exactly .
Definition 1.16 (Weak ; [4, Definition 3.4]).
A set is the set of weak of a weak on if:
For all and all , ;
For all such that for some ;
[Weak modular elimination] For any modular pair , and for any with , there exists such that and for all .
Definition 1.17 (Strong ; [4, Definition 3.7]).
If we take we immediately notice that (C4)’ implies 4. Therefore, a strong on is also a weak on .
If is the set of weak (resp. strong) of a weak (resp. strong) on , the set is the set of circuits of a matroid . The rank of is defined to be the rank of the matroid . The following theorem asserts that Definition 1.14 and Definition 1.16 encode equivalent data.
Theorem 1.18 ([4, Theorem 3.13, Theorem 3.17]).
Given a set and a hyperfield , there exists a bijection between the set of all equivalence classes of rank weak (resp. strong) Grassmann–Plücker functions on a with values in and the set of all sets of weak (resp. strong) of a rank weak (resp. strong) on , determined as follows. For a weak (resp. strong) Grassmann–Plücker function and the corresponding set of weak (resp. strong) :
The set of all supports of elements of is the set of minimal non-empty sets;
The weak (resp. strong) are determined by the rule
for all where and is any containing .
Thus, we can refer to the rank weak (resp. strong) matroid over the hyperfield (often abbreviated ) with ground set , rank weak (resp. strong) Grassmann–Plücker function and weak (resp. strong) . In particular, in this case . We call this matroid the underlying matroid of . Another way to obtain the underlying matroid is via the push-forward with respect to the unique map .
In the setting of matroids over hyperfields, duality depends on the choice of an involution of the hyperfield. To be more precise, for a , any involution of gives rise to a matroid “dual” to as explained in the following results.
Theorem 1.19 ([4, Theorem 3.20]).
Given a finite ground set with , a hyperfield , an involution of and a rank weak (resp. strong) on with weak (resp. strong) and rank weak (resp. strong) Grassmann–Plücker function , there exists a rank weak (resp. strong) on , called the dual of with respect to , that satisfies the following properties:
The set of of are the elements of , where denotes the elements of of minimal support;
A weak (resp. strong) Grassmann–Plücker function for is defined by the formula
where is any ordering of ;
The underlying matroid of is the dual of that of ;
The weak (resp. strong) of are called the weak (resp. strong) of with respect to , and vice versa.
Definition 1.20 (Dual pairs; [4, Definition 3.21, Definition 3.23]).
Let be a matroid with ground set . We say that a collection is a circuit coloring of (with values in ) if:
For all and all , ;
For all with for some ;
The set is the set of circuits of .
We say that is a cocircuit coloring of if is a circuit coloring of , the dual matroid to . Moreover, given a circuit coloring and a cocircuit coloring of we say that , form a weakly dual pair with respect to if for all , with . Similarly, we say that is a strongly dual pair with respect to if for all , .
Theorem 1.21 ([4, Theorem 3.22, Theorem 3.23]).
Given a matroid with ground set and a hyperfield with an involution , let be a circuit coloring and be a cocircuit coloring of . Then and are the set of weak (resp. strong) and with respect to of a weak (resp. strong) on with underlying matroid if and only if they are a weak (resp. strong) dual pair with respect to .
Matroids over the hyperfields listed in Example 1.11 are all well-studied combinatorial objects. In fact, matroids over hyperfields provide a common framework for several notions of matroids that appear in the literature.
A (weak or strong) matroid over the Krasner hyperfield is the same as a matroid in the usual sense;
A (weak or strong) matroid over the hyperfield of signs is the same as an oriented matroid;
A weak matroid over the phase hyperfield is the same as the notion of complex matroid introduced by Anderson and Delucchi in [3, Definition 2.4]. Notice that in this context the standard duality theory is given by taking the involution of the hyperfield to be complex conjugation (compare [3, Definition 2.12]). As pointed out by Baker and Bowler in [4, Appendix A], both notions of weak (compare [3, Definition 2.4]) and strong (compare [3, Definition 2.3, Definition 2.15]) matroids over the phase hyperfield are introduced in , but they are mistakenly asserted to be equivalent. However, the arguments in the proof of [3, Proposition 5.6] still hold for the weak case;
A (weak or strong) matroid over the tropical hyperfield is the same as a valuated matroid in the sense of Dress and Wenzel .
In the context of matroids and oriented matroids this dependence of the duality theory on the involution is hidden, since the Krasner hyperfield and the hyperfield of signs have the identity as unique involution.
As pointed out by Baker and Bowler, the notions of weak and strong matroids over hyperfields do not agree in general. In particular, they provide in [4, Section 3.10] the following counterexamples:
2. Rescaling classes of matroids over hyperfields
Let be a rank matroid with ground set and let be a given hyperfield. We want to study the set of weak (resp. strong) with underlying matroid and the space of “rescaling classes” of such , generalizing methods of  to the context of hyperfields.
In this section we will start with the more immediate case — weak (resp. strong) defined in terms of weak (resp. strong) Grassmann–Plücker functions — and then offer a reformulation of the axiomatization in terms of weak (resp. strong) that is more convenient for our later purposes. The cornerstone will be the proof that the cryptomorphisms of Theorem 1.18 and Theorem 1.21 induce a one-to-one correspondence between the spaces of with underlying matroid defined in terms of weak (resp. strong) Grassmann–Plücker functions and, respectively, in terms of weak (resp. strong) , which in turn determines a one-to-one correspondence between the respective spaces of rescaling classes.
2.1. Grassmann–Plücker functions
Let (resp. ) be the set of weak (resp. strong) Grassmann–Plücker functions with underlying matroid . As a subset of we consider it with the induced topology.If is the equivalence relation among weak (resp. strong) Grassmann–Plücker functions introduced in Definition 1.14, we can state the following definition.
A type weak (resp. strong) with underlying matroid is an equivalence class of the relation on (resp. ). The space of type weak (resp. strong) with underlying matroid is (resp. ). On this space we consider the quotient topology.
We now proceed to define the space of rescaling classes of weak (resp. strong) defined in terms of weak (resp. strong) Grassmann–Plücker functions.
Two weak (resp. strong) Grassmann–Plücker functions and (resp. ) are called if there is a function such that, for all ,
A straightforward computation shows that is an equivalence relation between weak (resp. strong) Grassmann–Plücker functions.
Two type weak (resp. strong) and with underlying matroid are (denoted by ) if there exist weak (resp. strong) Grassmann–Plücker functions and such that .
We can now define a rescaling class as an equivalence classes of .
Given a matroid we define the space of rescaling classes of type weak (resp. strong) with underlying matroid as the set (resp. ) of classes. Again, we endow the space of Grassmann-Plücker functions with the quotient topology.
2.2. Hyperfield circuit and cocircuits signatures
A signature of a matroid is a collection of functions , one for each circuit of . In the same way, a signature of a matroid is a set of functions , one for each cocircuit of . We say that a signature and a signature are weak orthogonal (resp. strong orthogonal) with respect to — denoted by — if, for any circuit and cocircuit with (resp. for any circuit and cocircuit), we have
We denote by (resp. ) the space of pairs of and signatures of that are weak (resp. strong) orthogonal with respect to . This is a subset of and we topologize it with the induced topology.
Two pairs and of and signatures of that are weak (resp. strong) orthogonal with respect to are called (denoted by ) if there exist functions , , and , , such that:
for any circuit and any ;
for any cocircuit and any .
One readily verifies that is an equivalence relation on the set (resp. ).
The function that associates to a pair (resp. ) the set
induces a bijection between the quotient set (resp. ) and the family of all sets of circuits of weak (resp. strong) with underlying matroid .
Thus, we are led to the following definition.
Definition 2.8 (See Definition 1.20).
A type weak (resp. strong) is an equivalence class of the relation on the set (resp. ).
The space of type weak (resp. strong) with underlying matroid is the set (resp. ). On this space we consider the quotient topology.
We now define an equivalence relation on the set of type weak (resp. strong) with underlying matroid , in order to obtain the counterpart of the set of rescaling classes of weak (resp. strong) Grassmann–Plücker functions with underlying matroid .
Two pairs and of and signatures of that are weak (resp. strong) orthogonal with respect to are called (denoted ) if there exists a function such that:
for any circuit and any ;
for any cocircuit and any .
Here stands for the inverse of in the multiplicative group .
It is easy to see that is an equivalence relation on (resp. ).
Two type weak (resp. strong) and with underlying matroid are (denoted by ) if there are and such that .
Again, is obviously an equivalence relation. As previously done, we conclude this section with the definition of the space of rescaling classes of type weak (resp. strong) with underlying matroid .
The space of rescaling classes of type weak (resp. strong) with underlying matroid is the set (resp. ). Again, we endow the space of rescaling classes with the quotient topology.
If and are involutions of the hyperfield , it is not hard to see that the sets and are in one-to-one correspondence. To verify this it suffices to consider the map
that associates to a pair of the pair of defined by
for any and for any ;
for any and for any .
Moreover, a straightforward check of definitions shows that this one-to-one correspondence induces one-to-one correspondences between the quotients and as well as between the quotients and . To be more precise, we have a commutative diagram
where the quotient maps and are bijections. With the same arguments these results hold for the strong case.
Proof of Proposition 2.7.
We only prove the weak case. The strong one will follow from the same arguments.
First, let us show that given a pair , the collection is the set of weak of a weak with underlying matroid . To see this, it suffices to consider the set
The pair belongs to . This implies that and form a weakly dual pair with respect to (compare Definition 1.20). To check this, let us consider and . From the definition of the sets and there exist a circuit , a cocircuit and elements , such that:
From the distributive law we then find
Hence, Theorem 1.21 implies that is the collection of weak of a weak with underlying matroid .
Now, we have to prove that the map is surjective. To see this, let be a weak with underlying matroid . Let us denote by the set of weak of and let be the set of weak of with respect to . Fix an arbitrary order of the ground set . Given a circuit let us consider the function
where is any weak of such that and is the first element of with . Axiom 4 implies that for any circuit the map is properly defined. In the same way, given a cocircuit let us consider the function
where is any weak of with respect to such that and is the first element of with . Since , is a weakly dual pair with respect to , the pair