The moduli space of matroids
In , Nathan Bowler and the first author introduced a category of algebraic objects called tracts and defined the notion of (weak and strong) matroids over a tract. In the first part of the paper, we summarize and clarify the connections to other algebraic objects which have previously been used in connection with matroid theory. For example, we show that both partial fields and hyperfields are fuzzy rings, that fuzzy rings are tracts, and that these relations are compatible with previously introduced matroid theories. We also show that fuzzy rings are ordered blueprints in the sense of the second author. Thus fuzzy rings lie in the intersection of tracts with ordered blueprints; we call the objects of this intersection pastures.
We then turn our attention to constructing moduli spaces for (strong) matroids over pastures. We show that, for any non-empty finite set , the functor taking a pasture to the set of isomorphism classes of rank- strong -matroids on is representable by an ordered blue scheme . We call the moduli space of rank- matroids on . The construction of requires some foundational work in the theory of ordered blue schemes; in particular, we provide an analogue for ordered blue schemes of the “Proj” construction in algebraic geometry, and we show that line bundles and their global sections control maps to projective spaces, much as in the usual theory of schemes.
Pastures themselves are field objects in a larger category which we call pasteurized ordered blueprints; roughly speaking, pastures are to pasteurized ordered blueprints as hyperfields are to hyperrings. We define matroid bundles over pasteurized ordered blue schemes and show that represents the functor taking a pasteurized ordered blue scheme to the set of isomorphism classes of rank- (strong) matroid bundles on over . This characterizes up to (unique) isomorphism.
Finally, we investigate various connections between the space and known constructions and results in matroid theory. For example, a classical rank- matroid on corresponds to a morphism , where (the “Krasner hyperfield”) is the final object in the category of pastures. The image of this morphism is a point of to which we can canonically attach a residue pasture , which we call the universal pasture of . We show that morphisms from the universal pasture of to a pasture are canonically in bijection with strong -matroid structures on . Although there is no corresponding moduli space in the weak setting, we also define an analogous weak universal pasture which classifies weak -matroid structures on . We show that the unit group of can be canonically identified with the Tutte group of , originally introduced by Dress and Wenzel. We also show that the sub-pasture of generated by “cross-ratios”, which we call the foundation of , parametrizes rescaling classes of weak -matroid structures on , and its unit group is coincides with the inner Tutte group of . As sample applications of these considerations, we show that a matroid is regular if and only if its foundation is the regular partial field (the initial object in the category of pastures), and a non-regular matroid is binary if and only if its foundation is the field with two elements. From this, we deduce for example a new proof of the fact that a matroid is regular if and only if it is both binary and orientable.
- 1 Introduction
- I Pastures, ordered blueprints, and matroids
- 2 The interplay between partial fields, hyperfields, fuzzy rings, tracts, and ordered blueprints
- 3 Comparison of matroid theories
- II Constructing moduli spaces of matroids
- 4 Projective geometry for ordered blueprints
- 5 Families of matroids and their moduli spaces
- III Applications to matroid theory
- 6 Realization spaces and the Tutte group
- 7 Cross ratios and rescaling classes
One of the most ubiquitous, and useful, moduli spaces in mathematics is the Grassmannian variety of -dimensional subspaces of a fixed -dimensional vector space. In Dress’s paper  (and much later, using a different formalism, in ), one finds that there is a precise sense in which rank- matroids on an -element set are analogous to points of the Grassmannian . More precisely, in the language of , both can be considered as matroids over hyperfields, or more generally matroids over tracts.111Tracts are more general than both hyperfields and fuzzy rings in the sense of ; see section 2 below for an in-depth discussion of the relationship between these and other algebraic structures. So it seems natural to wonder if there is a “moduli space of matroids”. More precisely, one can ask if there is some “geometric” object whose “points” over any tract are precisely the -matroids of rank on in the sense of . With some small technical caveats (such as the fact that we deal with a slightly restricted class of tracts and work with strong -matroids as opposed to weak ones), we answer this question affirmatively in the present paper. We also explore in detail how various properties of the moduli space are related to more “classical” considerations in matroid theory.
What kind of object should be? In modern algebraic geometry, one thinks of the Grassmannian as representing a certain moduli functor from schemes to sets.222More precisely, represents the functor taking a scheme to the set of isomorphism classes of surjections from onto a locally free -module of rank . This is the point of view we wish to take here, but clearly schemes would not suffice for our purposes since there is no way to encode the algebra of tracts in the language of commutative rings. It turns out that the second author’s theory of ordered blueprints and ordered blue schemes  is well-suited to the task at hand. Indeed, as we show, a certain nice subcategory of tracts – which we call pastures – contains the category of hyperfields (as well as the more general category of fuzzy rings) and embeds as a full subcategory of ordered blueprints. We can then use the theory developed in , together with a few new results and constructions, to define a suitable moduli functor and prove that it is representable by an ordered blue scheme.333We note that, in broad outline, Eric Katz had already envisioned using the theory of blueprints to represent moduli spaces of matroids in section 9.7 of .
1.1. Structure of the paper
This paper is divided into three parts, each having a different flavor: the first part is algebraic, the second geometric, and the third combinatorial. Each part is largely independent from the others except for certain common definitions. In particular, the reader who is mainly interested in the applications to matroid theory should be able to start reading sections 6 and 7 immediately after looking up the necessary definitions in sections 1.2 and 1.3. We have combined the algebraic, geometric, and combinatorial aspects of our theory into a single paper because we believe that the resulting “big picture” might lead to interesting new insights and developments in algebra/algebraic geometry and/or matroid theory.
In Part 1, which comprises sections 2 and 3, we compare various algebraic structures and different notions of matroids over these structures. The main goal of section 2 is to clarify precisely how hyperrings / hyperfields, partial fields, fuzzy rings, tracts, and pastures relate to ordered blueprints. We also describe an important reflective subcategory of ordered blueprints called pasteurized ordered blueprints, which itself contains the category of pastures; the new feature of pasteurized ordered blueprints is that they possess an element which plays the role of . (The element is needed, for example, in order to be able to write down the Plücker relations.) In section 3, we define matroids over pastures, and more generally pasteurized ordered blueprints, and compare this notion to the existing notions of matroids over tracts, fuzzy rings, etc.
In Part 2, which comprises sections 4 and 5, we construct moduli spaces of strong matroids over pasteurized ordered blueprints. These moduli spaces are constructed as ordered blue subschemes of a certain projective space, and their construction requires developing some foundational material on the “Proj” construction, line bundles, and maps to projective spaces in the context of ordered blue schemes.
More precisely, we define matroid bundles over pasteurized ordered blue schemes and show that the functor taking a pasteurized ordered blue scheme to the set of isomorphism classes of rank- matroid bundles on over is representable by a (unique up to unique isomorphism) pasteurized ordered blue scheme .
In Part 3, which comprises sections 6 and 7, we relate certain properties of moduli spaces of matroids to known constructions and results in matroid theory. For example, we use moduli spaces to associate, in a natural way, a universal pasture to each (classical) matroid . We show that morphisms from the universal pasture of to a pasture are canonically in bijection with strong -matroid structures on . Although there is no corresponding moduli space in the weak setting, we also define an analogous weak universal pasture , which classifies weak -matroid structures on , and a sub-pasture of (which we call the foundation of ) which parametrizes rescaling classes of weak -matroid structures on . The unit group of (resp. ) can be canonically identified with the Tutte group (resp. the inner Tutte group) of ; these groups were originally introduced by Dress and Wenzel via explicit presentations by generators and relations.
As sample applications of such considerations, we characterize regular and binary matroids in terms of their foundations and show that a matroid is regular if and only if it is both binary and representable over some pasture with . Examples of such pastures include fields of characteristic different from and the hyperfield of signs , so in particular we obtain a new proof of the fact that a matroid is regular if and only if it is both binary and orientable.
We now provide a more detailed overview of each of the three parts of the paper.
1.2. Part 1: Pastures, ordered blueprints, and matroids
Our first goal, which is modest but necessary, is to tame the zoo of terminology which we are forced to deal with in order to clarify the relationship between ordered blueprints and various algebraic structures which have already appeared in the literature, as well as various notions of matroids over such objects.
1.2.1. Matroids over tracts
In , Nathan Bowler and the first author introduce a new category of algebraic objects called tracts and define a notion of matroids over tracts. Examples of tracts include hyperfields in the sense of Krasner and partial fields in the sense of Semple and Whittle. For example, matroids over the Krasner hyperfield are just matroids, matroids over the hyperfield of signs are oriented matroids, matroids over the tropical hyperfield are valuated matroids, and matroids over a field are linear subspaces. Matroids over tracts generalize matroids over fuzzy rings in the sense of Dress ().
Actually, there are two different notions of matroid over a tract , called weak and strong -matroids. Over many tracts of interest, including fields and the hyperfields , and , weak and strong matroids coincide. However, the two notions are different in general. For both weak and strong -matroids, the results of  provide cryptomorphic axiomatizations of -matroids in terms of circuits, Grassmann-Plücker functions, and dual pairs. The subsequent work of Laura Anderson () also provides a cryptomorphic axiomatization of strong -matroids in terms of vectors or covectors.
More formally, a tract is a pair consisting of an abelian group (written multiplicatively), together with a subset (called the nullset of the tract) of the group semiring satisfying:
The zero element of belongs to , and the identity element of is not in .
is closed under the natural action of on .
There is a unique element of with .
One thinks of as those linear combinations of elements of which “sum to zero”. We let , and we often refer to the tract simply as .
Tracts form a category in a natural way: a morphism of tracts corresponds to a homomorphism which takes to . The Krasner hyperfield (identified with its corresponding tract) is a final object in the category of tracts.
1.2.2. Pastures and ordered blueprints: a first glance
Although the axiom (T3) suffices for establishing all of the cryptomorphisms in , from a “geometric” point of view it is more natural to replace axiom (T3) with the stronger axiom:
The nullset of is an ideal in , i.e., if and then .
We define a pasture to be a tract satisfying , i.e., a tract whose nullset is an ideal.
One advantage of working with pastures is that they can be naturally thought of as ordered blueprints. The theory of ordered blueprints, developed by the second author, has a rich geometric theory associated to it. There is a speculative remark in  to the effect that ordered blue schemes might be a suitable geometric category for defining moduli spaces of matroids over tracts.444In the special case of matroids over hyperfields, one could attempt to construct such moduli spaces as hyperring schemes in the sense of J. Jun (), but this is potentially problematic for a few reasons: (i) The category of hyperring schemes does not appear to admit fiber products; (ii) the structure sheaf of a hyperring scheme as defined by Jun has some undesirable properties, e.g., the hyperring of global sections of the structure sheaf on is not always equal to ; and (iii) the theory of hyperring schemes is not as well developed as the theory of ordered blue schemes. In any case, it is highly desirable to fit not only matroids over hyperfields but also matroids over partial fields into our theory, and for this ordered blue schemes fit the bill quite well. One of the main goals of the present paper is to turn this speculation into a rigorous theorem, at least in the case of strong matroids over pastures. The other main goal is to give applications of this algebro-geometric point of view to more traditional questions and ideas in matroid theory.
1.2.3. The relationship between various algebraic structures
Loosely speaking, the relationship between hyperfields, tracts, pastures, ordered blueprints, and other algebraic structures mentioned in this Introduction can be depicted as follows (for a more precise statement, see Theorem 2.21 and the remarks in Section 2.9):
We now turn to giving a more precise definition of (pasteurized) ordered blueprints and matroids over them.
1.2.4. Ordered blueprints
An ordered semiring is a semiring together with a partial order that is compatible with multiplication and addition. (See section 2.6 for a more precise definition.)
An ordered blueprint is a triple where is an ordered semiring and is a multiplicative subset of which generates as a semiring and contains and .
A morphism of ordered blueprints and is an order-preserving morphism of semirings with .
We denote the category of ordered blueprints by .
A hyperfield is an algebraic structure similar to a field, but where addition is allowed to be multivalued (see Section 2.3 for a precise definition). We can identify a hyperfield with an ordered blueprint as follows:
The associated semiring is the free semiring over the multiplicative group .
The underlying monoid is .
The partial order of is generated by the relations whenever .
A partial field is a certain equivalence class of pairs consisting of a commutative ring with and a subgroup containing . (See section 2.2 for a more precise definition.) We can identify a partial field with an ordered blueprint as follows:
The associated semiring is .
The underlying monoid is .
The partial order is generated by the 3-term relations whenever satisfy in .
The ordered blueprints associated to hyperfields and partial fields are in fact ordered blue fields, meaning that , where denotes the set of invertible elements of . They are also pasteurized, a notion which will be defined shortly.
The category of ordered blueprints has an initial object called with associated semiring , underlying monoid (with the usual multiplication), and partial order given by equality.
1.2.5. Some properties of ordered blueprints
The category of ordered blueprints admits pushouts: given morphisms and of ordered blueprints, one can form their tensor product , which satisfies the universal property of a fiber coproduct.
One can also form the localization of an ordered blueprint with respect to any multiplicative subset , and it has the usual universal property.
1.2.6. Pasteurized ordered blueprints
An ordered blueprint is called pasteurized if for every , there is a unique element such that . The element such that plays the role of in this theory, and is crucial for defining structures such as matroids.
We denote the category of pasteurized ordered blueprints by . The inclusion functor has a left adjoint , called pasteurization, which makes into a reflective subcategory of .
The initial object of is , which corresponds to the submonoid of together with the partial order generated by . The ordered blueprint is equal to , where is the regular partial field.
1.2.7. Pastures as pasteurized ordered blueprints
The ordered blueprints associated to hyperfields and partial fields are pasteurized ordered blue fields. More generally, if is any pasture in the sense of section 1.2.2, we can consider as a pasteurized ordered blue field as follows:
The associated semiring is .
The underlying monoid is .
The partial order is generated by the relations whenever satisfy in .
One can characterize ordered blueprints of the form for some pasture among all ordered blueprints in a simple way: they are precisely the pasteurized ordered blue fields which are purely positive, meaning that the partial order is generated by elements of the form .
1.2.8. Matroids over pasteurized ordered blueprints
Let be a pasteurized ordered blueprint, let be a finite totally ordered set, and let . We denote by the family of all -element subsets of .
A Grassmann-Plücker function of rank on with coefficients in is a function such that:
for some .
satisfies the Plücker relations
whenever and with . (We set if .)
We say that two Grassmann-Plücker functions are equivalent if for some .
A -matroid of rank on is an equivalence class of Grassmann-Plücker functions. We denote by the set of all -matroids of rank on .
If is a pasture, an -matroid of rank on in the above sense is the same thing as a strong -matroid of rank on in the sense of . In this case, we can characterize (strong) -matroids of rank on in several different cryptomorphic ways, e.g. in terms of circuits, dual pairs, or vectors (see [1, 3] or sections 3.1.4 and 3.1.6 below).
The definition of is functorial: if is a morphism of pasteurized ordered blueprints, there is an induced map .
If is a pasture and is the canonical morphism to the final object (which is shorthand for ) of the category of pastures, the push-forward is a -matroid, i.e. a matroid in the usual sense. We call the underlying matroid of .
If is a matroid, we say that is weakly (resp. strongly) representable over a pasture if for some weak (resp. strong) -matroid . This generalizes the usual notion of representability over fields, or more generally partial fields (for which the notions of weak and strong -matroids coincide).
1.3. Part 2: Constructing moduli spaces of matroids
As discussed above, we wish to construct a moduli space of rank- matroids on as a (pasteurized) ordered blue scheme which represents a certain functor. In order to formulate precisely what this means, and in particular to specify which moduli functor we wish to represent, we first provide the reader with a gentle introduction to the theory of ordered blue schemes.
1.3.1. Ordered blue schemes
One constructs the category of ordered blue schemes, starting from ordered blueprints, much in the same way that one constructs the category of schemes starting from commutative rings. We give just a brief synopsis here; see section 4.1 for further details.
Let be an ordered blueprint.
A monoid ideal of is a subset of such that and .
A prime ideal of is a monoid ideal whose complement is a multiplicative subset.
The spectrum of is constructed as follows:
The topological space of consists of the prime ideals of , and comes with the topology generated by the principal opens
The structure sheaf is the unique sheaf with the property that for all . The stalk of at a point corresponding to is .
An ordered blueprinted space is a topological space together with a sheaf in . Such spaces form a category . A morphism of ordered blueprints defines a morphism of -spaces. This defines the contravariant functor whose essential image is the category of affine ordered blue schemes.
An ordered blue scheme is an -space that has an open covering by affine ordered blue schemes . A morphism of ordered blue schemes is a morphism of -spaces. We denote the category of ordered blue schemes by .
If an ordered blue scheme has an open covering by affine ordered blue schemes with each pasteurized, we call a pasteurized ordered blue scheme.
1.3.2. Some properties of ordered blue schemes
Ordered blue schemes possess many familiar properties from the world of schemes. For example:
The global section functor defined by is a left inverse to . In particular, .
The category contains fibre products, and in the affine case .
Various familiar objects from algebraic geometry have analogues in the context of ordered blue schemes; for example, one can define an invertible sheaf on an ordered blue scheme to be a sheaf which is locally isomorphic to the structure sheaf of . There is a tensor product operation which turns the set of isomorphism classes of invertible sheaves on into an abelian group.
Similarly, one can define, for each and each ordered blueprint , the projective -space as an ordered blue scheme over .
1.3.3. Families of matroids
Let be a pasteurized ordered blue scheme. A Grassmann–Plücker function of rank on over is an invertible sheaf on together with a map such that generate and the satisfy the Plücker relations in (see Definition 5.1 for a more precise definition).
Two such functions and are said to be isomorphic if there is an isomorphism from to taking to .
A matroid bundle of rank on the set over is an isomorphism class of Grassmann–Plücker functions.
If is an affine pasteurized ordered blue scheme, it turns out that a matroid bundle over is the same thing as a -matroid.
1.3.4. The moduli functor of matroids
One can extend the (covariant) functor taking a pasteurized ordered blueprint to to a (contravariant) functor taking to the set of matroid bundles of rank on over .
We prove the following theorem, cf. Theorem 5.4:
This moduli functor is representable by a pasteurized ordered blue scheme . Furthermore, for every pasteurized ordered blue scheme there is a natural bijection
The moduli space is constructed as an ordered blue subscheme of , where . (This is analogous to the Plücker embedding of the Grassmannian .) However, making this precise requires developing some foundational material on line bundles, the “Proj” construction, etc. in the context of ordered blue schemes.
1.4. Part 3: Applications to matroid theory
We conclude this introduction by providing a more detailed overview of Part 3 of the paper, in which we connect various algebraic structures related to the moduli spaces to concepts such as realization spaces, cross ratios, rescaling classes, and universal partial fields which have been previously studied in the matroid theory literature.
1.4.1. Universal pastures
Given a (classical) matroid , we can associate to a universal pasture , which is derived from a certain “residue ordered blue field” of the matroid space .
More precisely, a classical matroid corresponds to a morphism which we call the characteristic morphism of . Topologically, is a point, and the image point in the pasteurized ordered blue scheme of the characteristic morphism has an associated residue pasture, much as every point of a (classical) scheme has an associated residue field. We call the residue pasture of the universal pasture of .
1.4.2. Realization spaces
Let be a field. The realization space over of a rank- matroid on is the subset of the Grassmannian consisting of sub-vector spaces of whose associated matroid is . Such realization spaces have been used for proving that several moduli spaces, such as Hilbert schemes and moduli spaces of curves, can have arbitrarily complicated singularities, cf. .
Given a matroid and a pasture , the realization space is the set of isomorphism classes of -matroids whose underlying matroid is . More precisely, let be a Grassmann-Plücker function, the corresponding matroid, and its characteristic morphism. The canonical map from to (which takes to and every nonzero element of to ) induces a natural map taking an -matroid to its underlying matroid. With this notation, the realization space of over is the fibre of over .
Realization spaces are functorial with respect to morphisms of pastures.
The functor from pastures to sets taking a pasture to the realization space is represented by the universal pasture . In other words, there is a canonical bijection
which is functorial in .
1.4.3. The weak matroid space
So far we have been talking more or less exclusively about strong matroids in the sense of . However, there is also a notion of weak matroids over a pasture which is quite important in many contexts.
A weak Grassmann-Plücker function of rank on with coefficients in a pasture is a function
whose support is the set of bases of a matroid and which satisfies the -term Plücker relations
for every -subset of and all with , where .
Two weak Grassmann-Plücker functions and are equivalent if for some element .
A weak -matroid of rank on is an equivalence class of weak Grassmann-Plücker functions of rank on with coefficients in . We denote the set of all weak -matroids of rank on by .
Although the functor is (presumably) not representable by an ordered blue scheme, if we fix an underlying matroid , we can still define a weak universal pasture which “morally speaking” is the residue pasture of the space of weak matroids at the point corresponding to . (See section 6.4 for a precise definition.) We can also define the weak realization space of over to be the set of all weak -matroids whose underlying matroid is .
As in the strong case, there is a canonical bijection
which is functorial in .
1.4.4. Cross ratios
Four points on a projective line over a field correspond to a point of the Grassmannian over , and their cross ratio can be expressed in terms of the Plücker coordinates of this point. This reinterpretation allows for a generalization of cross ratios to higher Grassmannians and also to non-realizable matroids.
Let be a pasture and be a matroid of rank on . The cross ratios of in are indexed by the set of -tuples for which , , and are bases of , where .
Let be a pasture and let be a weak -matroid defined by the weak Grassmann-Plücker function . The cross ratio function of is the function that sends an element of to
One checks easily that this depends only on the equivalence class of , and is thus a well-defined function of .
Let be a pasteurized ordered blueprint. A fundamental element of is an element such that for some and .
The foundation of is the subblueprint of generated by the fundamental elements of . Taking foundations is a functorial construction.
The relevance of this notion for matroid theory is that cross ratios of weak -matroids are fundamental elements of (which is a simple consequence of the 3-term Plücker relations).
If is a matroid, we define the foundation of , denoted , to be the foundation of the weak universal pasture of . Since the functor taking a pasture to the weak realization space is represented by , and cross ratios of weak -matroids are fundamental elements of , there is a natural universal cross ratio function .
We prove that the foundation of is generated by the universal cross ratios over .
We also show (cf. Theorem 7.15) that for every matroid and pasture , the following are equivalent: (a) is weakly representable over ; (b) is weakly representable over ; and (c) there exists a morphism .
1.4.6. The Tutte group and inner Tutte group
The Tutte group of a matroid was introduced by Dress and Wenzel in  as a tool for studying the representability of matroids by algebraically encoding results such as Tutte’s homotopy theorem (cf.  and ).
The Tutte group is usually defined in terms of generators and relations, and several “cryptomorphic” presentations of this group are known. Our approach allows for an intrinsic definition of the Tutte group as the unit group of the weak universal pasture (see section 6.5 for details).
Dress and Wenzel also define a certain subgroup of the Tutte group which they call the inner Tutte group. Using their results, we show that the natural isomorphism restricts to an isomorphism . In other words, the inner Tutte group of is the unit group of the foundation of .
1.4.7. Rescaling classes
Let be a matroid of rank on . The importance of the foundation of is that it represents the functor which takes a pasture to the set of rescaling classes of -matroids with underlying matroid , in the same way the universal pasture (resp. weak universal pasture) of represents the realization space (resp. weak realization space ).
Let be a pasture, and let be the group of functions . The rescaling class of an -matroid is the -orbit of in , where acts on a weak Grassmann-Plücker function by the formula
Rescaling classes are the natural generalization to matroids over arbitrary pastures of reorientation classes for oriented matroids, where two (realizable) oriented matroids are considered reorientation equivalent if they correspond to the isomorphic real hyperplane arrangements. (For non-realizable matroids, there is a similar assertion involving pseudosphere arrangements; this is part of the famous "Topological Representation Theorem" of Folkman and Lawrence, cf.  or [5, section 5.2].)
If we fix a matroid and a pasture , we define the rescaling class space to be the set of rescaling classes of weak -matroids with underlying matroid . There is a canonical bijection
which is functorial in .
As a sample motivation for considering rescaling classes over more general pastures than just the complex numbers or the hyperfield of signs, we mention that while it is true that a matroid is regular (i.e., representable over the rational numbers by a totally unimodular matrix) if and only if is representable over , there are in general many non-isomorphic -matroids whose underlying matroid is a given regular matroid . However, there is always precisely one rescaling class over . In other words, regular matroids are the same thing as rescaling classes of -matroids.
1.4.8. Foundations of binary and regular matroids
A binary matroid is a matroid that is representable over the finite field with two elements.
We show that a matroid is regular if and only if its foundation is , and binary if and only if its foundation is either or . We recover from these observations a new proof of the well-known facts that (a) a matroid is regular if and only if it is representable over every field; and (b) a binary matroid is either representable over every field or not representable over any field of characteristic different from .
We also use these observations to give new and conceptual proofs of the following facts: (a) a binary matroid has at most one rescaling class over every pasture (compare with [53, Thm. 6.9]); and (b) every matroid has at most one rescaling class over (cf. ).
In addition, we show (cf. Theorem 7.33) that a matroid is regular if and only if is binary and weakly representable over some pasture with . This implies, for example (taking to be the hyperfield of signs) the well-known fact that a matroid is regular if and only if it is both binary and orientable.
1.4.9. Relation to the universal partial field of Pendavingh and van Zwam
The universal partial field of a matroid was introduced by Pendavingh and van Zwam in . It has the property that a matroid is representable over a partial field if and only if there is a partial field homomorphism .
We show that there is a partial field naturally derived from the weak universal pasture of with the property that for every partial field there is a natural bijection
which is functorial in .
The universal partial field of Pendavingh and van Zwam is isomorphic to the partial subfield of generated by the cross ratios of . We prove that for every partial field there is a natural and functorial bijection
One disadvantage of the universal partial field is that it doesn’t always exist: there are matroids (e.g. the Vámos matroid) which are not representable over any partial field. However, every matroid is representable over some pasture, so the foundation of gives us information about representations of even when the universal partial field is undefined.
Our classification of binary and regular matroids in terms of their foundations also yields a classification of such matroids in terms of their universal partial fields: a matroid is regular if and only if its universal partial field is , and binary if and only if its universal partial field is or .
Part I Pastures, ordered blueprints, and matroids
2. The interplay between partial fields, hyperfields, fuzzy rings, tracts, and ordered blueprints
Our approach to matroid bundles utilizes an interplay between tracts and ordered blueprints, as introduced by the first author and Bowler in  and the second author in , respectively. Tracts and ordered blueprints are common generalizations of other algebraic structures that appear in matroid theory, such as partial fields, hyperfields, and fuzzy rings.
In this section, we review the definitions of all of the aforementioned notions and explain their interdependencies. Our exposition culminates in Theorem 2.21, which exhibits a diagram of comparison functors between the corresponding categories.
Since many of the following concepts are based on semirings and derived notions, we begin with an exposition of semirings. All of our structures will be commutative and, following the practice of the literature in commutative algebra and algebraic geometry, we omit the adjective “commutative” when speaking of semirings, monoids, and other structures.
A monoid is a commutative semigroup with a neutral element. A monoid morphism is a multiplicative map that preserves the neutral element. In this text, a semiring is a set together with two binary operations and and with two constants and such that the following axioms are satisfied:
is a monoid;
is a monoid;
for all ;
for all .
A morphism of semirings is a map between semirings and such that
for all . We denote the category of semirings by .
Let be a semiring. An ideal of is a subset such that , and for all and . An ideal is proper if it is not equal to .
Given any subset of elements of , we define the ideal generated by as the smallest ideal of containing , which is equal to
The group of units of is the group of all multiplicatively invertible elements of . A semifield is a semiring such that .
Note that an ideal is proper if and only if . A semiring is a semifield if and only if and are the only ideals of .
Every (commutative and unital) ring is a semiring. Examples of semirings that are not rings are the natural numbers and the nonnegative real numbers . Examples of a more exotic nature are the tropical numbers together with the usual multiplication and the tropical addition , and the Boolean numbers with , which appear as a subsemiring of the tropical numbers.
2.1.1. Monoid semirings
Let be a multiplicatively written monoid and a semiring. The monoid semiring consists of all finite formal -linear combinations of elements of , i.e. almost all are zero. The addition and multiplication of are defined by the formulas
respectively. The zero of is and its multiplicative identity is with and for .
This construction comes with an inclusion , and we often identify with its image in , i.e. we write or for the element of with and for . In the case , every element of is a sum of elements in .
Polynomial semirings are particular examples of monoid semirings. Let be a semiring and let be the monoid of all monomials in . Then is the polynomial semiring .
2.2. Partial fields
In , Semple and Whittle introduced partial fields as a tool for studying representability questions about matroids. The theory of matroid representations over partial fields was developed further by Pendavingh and van Zwam in  and ; also cf. van Zwam’s thesis . Loosely speaking, a partial field can be thought of as a set together with distinguished elements and , a map , and a partially defined map satisfying:
is a commutative monoid in which every nonzero element is invertible
is associative and commutative with neutral element
every element has a unique additive inverse
multiplication distributes over addition.
Morphisms of partial fields are defined to be structure preserving maps.
It is somewhat involved to make the requirements on rigorous; in particular, the formulation of the associativity of involves binary rooted trees with labelled leaves.
Van Zwam gives in [51, section 2.1] a simpler but equivalent description of partial fields in terms of a ring together with a subgroup of the unit group of that contains . The downside of this approach is that morphisms are not structure preserving maps; in particular, the isomorphism type of the ambient ring is not determined by a partial field. There is, however, a distinguished ambient ring for every partial field, which has better properties than other choices for , cf. [51, Thm. 2.6.11]. This latter observation leads us to the following hybrid of Semple-Whittle’s and van Zwam’s definitions.
Let be the group ring of a group , which comes together with the inclusion , sending to and to .
A partial field is a commutative group together with a surjective ring homomorphism such that
the composition is injective;
for every , there is a unique element such that ;
the kernel of is generated by all elements with such that .
We can recover the motivating properties of a partial field from these axioms:
we define as the underlying set of , i.e. is a subset of the ambient ring ;
we write for elements with , which defines the partial addition of ;
if , then , i.e. every element has a unique additive inverse with respect to the partial addition of , which we denote by .
A morphism of partial fields is a group homomorphism that extends to a ring homomorphism . Using the notation introduced above, this is the same as a map such that , , for all and if with . We denote the category of partial fields by .
A field can be identified with the partial field , where is the surjective ring homomorphism induced by the identity map . Note that with this identification a field homomorphism is the same as a morphism between the associated partial fields.
The regular partial field consists of the group and the surjective ring homomorphism mapping to the corresponding elements in . Note that it is an initial object in the category of partial fields. Later on, this partial field will be reincarnated as the ordered blue field , cf. Example 2.20.
For an extensive list of other examples, we refer to [van Zwam].
2.3. Hyperfields and hyperrings
The notion of an algebraic structure in which addition is allowed to be multi-valued goes back to Frédéric Marty, who introduced hypergroups in 1934. Later on, in the mid-1950’s, Marc Krasner developed the theory of hyperrings and hyperfields in the context of approximating non-Archimedean fields, and in the 1990’s Murray Marshall explored connections to the theory of real spectra and spaces of orderings. Subsequent advocates of hyperstructures included Oleg Viro (in connection with tropical geometry) and Connes and Consani (in connection with geometry over ).
A commutative hypergroup is a set together with a distinctive element and a hyperaddition, which is a map
into the power set of , such that:
is not empty, (nonempty sums)
, (neutral element)
there is a unique element in such that , (inverses)
if and only if (reversibility)
for all . Note that thanks to commutativity and associativity, it makes sense to define hypersums of several elements unambiguously by the recursive formula
A (commutative) hyperring is a set together with distinctive elements and and with maps and such that
is a commutative hypergroup,
is a commutative monoid,
for all where . Note that the reversibility axiom 6 for the hyperaddition follows from the other axioms of a hyperring. We will discuss an analogue 1 of this axiom for ordered blueprints in section 2.9.2.
A morphism of hyperrings is a map between hyperrings such that
for all where . We denote the category of hyperrings by .
The unit group of a hyperring is the group of all multiplicatively invertible elements in . A hyperfield is a hyperring such that and . We denote the full subcategory of hyperfields in by .
Every ring can be considered as a hyperring by defining . If is a field, the corresponding hyperring is a hyperfield.
The Krasner hyperfield is the hyperfield whose addition is characterized by . Note that all other sums and products are determined by the hyperring axioms. It is a terminal object in .
The tropical hyperfield was introduced by Viro in . Its multiplicative monoid consists of the non-negative real numbers , together with the usual multiplication, and its hyperaddition is defined by the rule if and . The tropical hyperfield has a particular importance for valuations and tropical geometry, since a nonarchimedean absolute value on a field is the same thing as a morphism of hyperfields.
The sign hyperfield is the multiplicative monoid together with the hyperaddition characterized by ,