In [1], Baker and Bowler define a category of algebraic objects called tracts which generalize both partial fields and hyperfields (in particular, they generalize fields). Given a tract F , Baker and Bowler define the notions of weak and strong matroids over F , and they prove that if F is perfect (meaning that F -vectors and F -covectors are orthogonal for every F -matroid) then the notions of weak and strong F -matroids coincide. The authors also show that partial fields and doubly distributive hyperfields are always perfect.
The fact that doubly distributive hyperfields are perfect was generalized by Bowler-Pendavingh [2] and Bowler-Su [3], who showed that every stringent hyperfield 1 is perfect and every doubly distributive hyperfield is stringent. Definition 1.2. A morphism of tracts (or pastures) is a map ϕ : F → F ′ such that ϕ(0) = 0, ϕ induces a group homomorphism from F × to (F ′ ) × , and ϕ(N F ) ⊆ N F ′ .
Given a pair (G, R) consisting of a commutative ring R with 1 and a subgroup G ≤ R × containing -1, we can associate a pasture P = P G,R with (G, R) by setting P × = G and declaring that x + y + z ∈ N P if and only if x + y + z = 0 in P. Pastures of this form are called partial fields.
Roughly speaking, a hyperfield is an algebraic structure which behaves like a field except that addition is allowed to be multivalued. More precisely, a hyperfield H consists of a multiplicative monoid with an absorbing element 0 such that H × = H \ {0} is an abelian group, an involution x ↦ → -x fixing 0, and a commutative hyperoperation which associates to each pair of elements a, b ∈ H a non-empty subset a ⊞ b of H. The multiplication and hyperaddition are required to satisfy a number of axioms including commutativity and distributivity, and we require for each a, b ∈ H that 0 ∈ a ⊞ b if and only if a = -b. There is also a reversibility axiom which says that c ∈ a ⊞ b if and only if b ∈ c ⊞ (-a).
Definition 1.3. The tract F H (resp. pasture P H ) associated to a hyperfield H has multiplicative group H × and null set defined by ∑ k i=1 x i ∈ N H iff 0 ∈ ⊞ k i=1 x i (resp. x+y+z ∈ N H if and only if 0 ∈ x⊞y⊞z). If P is a pasture and we set x ⊞ y = {z ∈ P : x + yz ∈ N P }, the pasture P corresponds to a field if and only if ⊞ is an associative binary operation. Moreover, P = P H for some hyperfield H if and only if x ⊞ y contains at least one element for all x, y ∈ P and ⊞ is associative (in the sense of set-wise addition), and P = P (G,R) for some partial field (G, R) if and only if x ⊞ y contains at most one element for all x, y ∈ P and satisfies a suitable associative law (which is a bit complicated to state, cf. [5,Section 2.2]). Pastures thus generalize (and simplify) both hyperfields and partial fields by imposing no conditions on the size of the sets x ⊞ y and no associativity conditions. Definition 1.4. A hyperfield H is stringent if |a ⊞ b| = 1 for all a, b ∈ H with a ̸ = -b.
Here are a few examples of hyperfields and their associated tracts: Example 1.6 (Sign Hyperfield). The sign hyperfield S consists of the multiplicative monoid {0, ±1}, together with the hyperaddition rule given by 1 ⊞ 1 = 1, (-1) ⊞ (-1) = -1, and 1 ⊞ (-1) = {-1, 0, 1}. As a tract, N S consists of 0 and all formal sums ∑ x i with at least one 1 and one -1.
Example 1.7 (S × S). Products exist in both the category of hyperfields and the category of tracts. As a multiplicative monoid, S ×S is given by the Cartesian product of {0, ±1} with itself, while N S×S consists of 0 and all formal sums
Example 1.8 (Phase Hyperfield). The phase hyperfield P consists of the multiplicative monoid {0}∪S 1 , where S 1 is the complex unit circle, together with the following hyperaddition rule. Given x i ∈ P, the hypersum ⊞ n i=1 x i is the set of phases of all complex numbers in the cone
As a tract,
Although one can trivially extend a pasture to a tract via the inclusion
this way of viewing pastures as tracts is not very useful in practice. Instead, it is more useful to define the tract associated to a pasture by inductively ''fusing'' together additive relations of smaller degree to generate higher-degree relations. More precisely, consider the following fusion axiom:
Given a pasture P, let P be the tract whose multiplicative group is P × and whose null set is the smallest subset of N[P × ] containing N P and satisfying the fusion axiom.
The proof of the following result is left as an exercise: Proposition 1.9. The map P ↦ → P defines a fully faithful functor from pastures to tracts. A tract F is equal to P for some pasture P if and only if F satisfies the fusion axiom (F) and every γ
In particular, there is no harm in identifying a pasture P with the corresponding tract P.
Note that the tract P associated to a pasture P is in fact an idyll (cf. [6, Section 1.2.2]), meaning that N F is an ideal in the semiring
For hyperfields, we have the following pleasant correspondence (which was in fact our motivation for the fusion axiom): Proposition 1.10. If H is a hyperfield and F H (resp. P H ) is the associated tract (resp. pasture) then
For later reference, we also define a functor from tracts to pastures: given a tract F , define the 3-term truncation of F to be the pasture whose multiplicative group is F × and whose null set is
The present paper is motivated by the observation that many tracts of interest, such as partial fields and stringent hyperfields, satisfy a property that is stronger than (F) and which turns out to be sufficient to guarantee perfection.
More precisely, consider the following strong fusion axiom:
(SF) If α +γ and β -γ are in N F with α, β, γ ∈ N[F × ] and either γ = 0 or γ ̸ ∈ N F , then α +β ∈ N F . Note that the fusion axiom is precisely the case where γ ∈ F , and in particular a tract satisfying (SF) (which we call a strongly fused tract) automatically satisfies (F).
The main result of this paper is:
Theorem 1.11. Every strongly fused tract is perfect.
In fact, we will prove a stronger version of Theorem 1.11 in which we replace (SF) with the modified axiom:
] and either γ = 0 or γ ̸ ∈ N F , and if ∥α + β∥ ≥ 4, then α + β ∈ N F .
Our proof will show more generally that a tract satisfying (MSF) is strongly perfect, a notion which will be defined in Section 3 (but it turns out to be equivalent to perfection in the usual sense; see Theorem 3.7).
As a corollary of the strengthened version of Theorem 1.11, we obtain:
Corollary 1.12. There is a rule F ↦ → σ (F ) which associates to each tract F a strongly perfect tract σ (F )
and which is the identity map on tracts satisfying (MSF).
Proof. We can take σ (F ) to be the tract whose multiplicative group is F × and whose null set is defined as follows. Let N (1) = N F , and for k ≥ 2 define N (k) to be the set of all elements of the form α + β with α, β ∈ N (k-1) or α + γ and βγ in N (k-1) for some γ ̸ ∈ N (k-1) , and such that ∥α + β∥ ≥ 4. Set N σ (F ) := ⋃ k≥1 N (k) . It is easy to see that σ (F ) = F if F satisfies (MSF). We claim that σ (F ) satisfies (MSF) for every tract F . Indeed, suppose α + γ and βγ are in N σ (P) with either γ = 0 or γ ̸ ∈ N σ (P) , and assume furthermore that ∥α + β∥ ≥ 4. Then by definition there exists k ≥ 1 such that α + γ and βγ are in N (k-1) , and if γ ̸ ∈ N σ (P) then γ ̸ ∈ N (k-1) since N (k-1) ⊂ N σ (P) . By the definition of N (k) we have α + β ∈ N (k) , hence α + β ∈ N σ (P) . □
It is easy to see that the tract embedding of a partial field satisfies the strong fusion axiom. For hyperfields, we show:
In particular, this gives a new proof of the fact, originally proved by Bowler and Pendavingh in [2], that stringent hyperfields are perfect.
Remark 1.14. If we removed the assumption that γ ̸ ∈ N F in (SF), then stringent hyperfields would no longer satisfy this property. For example, in the sign hyperfield S with α = 1, β = 1, and
It would be useful to have a natural and easily verified sufficient condition which implies perfection, is satisfied by stringent hyperfields and partial fields, and which is stable under taking finite products (since one easily shows that the product of perfect pastures is perfect.) Unfortunately, neither (SF) nor (MSF) is stable under products, as the following shows: Example 1.15. A counterexample which applies to both (SF) and (MSF) is F = S × S, where S is the sign hyperfield. Indeed, note that if γ = (1, 1)+(-1, 1), α = (1, -1)+(1, -1), and β =
The proofs of Propositions 1.10 and 1.13 are given in Section 2. In Section 3 we recall the definition of an F -matroid and define what it means for an F -matroid (resp. a tract) to be strongly perfect. We then prove that strong perfection and perfection coincide. The proof of (a strengthening of) Theorem 1.11 is given in Section 4. Finally, in Section 5 we compare our results to those of Dress-Wenzel.
In this section we prove Propositions 1.10 and 1.13.
Our goal in this section is to prove Proposition 1.10. In order to do this, we first recall the precise definition of a hyperfield.
A commutative hypergroup is a set G together with a distinctive element 0 and a hyperaddition, which is a map
for all a, b, c ∈ G.
Thanks to commutativity and associativity, it makes sense to define hypersums of several elements a 1 , . . . , a n unambiguously by the recursive formula
Definition 2.2. A (commutative) hyperring is a set R together with distinctive elements 0 and 1 and with maps ⊞ : R × R → P(R) and
A hyperfield is a hyperring H such that 0 ̸ =
We can now prove Proposition 1.10.
, that αz and β + z both belong to N F . By the inductive hypothesis, these two elements of N F are in N P . Applying the fusion axiom gives γ = α + β ∈ N P . □
Our goal in this section is to prove Proposition 1.13. The following is a more precise version of this result. Proposition 2.4. Let H be a hyperfield. Then the following are equivalent:
(1) H is stringent.
(
(3) ⇒ (1): Suppose there exist x, y ∈ H such that x ̸ = -y and |x ⊞ y| ≥ 2. Choosing z ̸ = z ′ ∈ x ⊞ y, we have 0 ∈ -z ⊞ x ⊞ y and 0 ∈ -z ′ ⊞ x ⊞ y. Since x ̸ = -y we have 0 / ∈ x ⊞ y, and thus (SF) implies that 0 ∈ z ⊞ -z ′ , which contradicts the fact that z ̸ = z ′ . □
Our goal in this section is to define what it means for a tract F to be strongly perfect, and to show that F is strongly perfect if and only if it is perfect. To do this, we need to introduce some terminology related to matroids over tracts. Definition 3.1 (Involution). Let F be a tract. An involution of F is a homomorphism τ : F → F such that τ 2 is the identity map. For an element x ∈ F , its involution is usually denoted by x instead of τ (x). Definition 3.2 (Orthogonality). Let F be a tract endowed with an involution x ↦ → x, and let E = {1, . . . , m}. The inner product of X = (x 1 , . . . ,
Note that our definition of orthogonality generalizes [1, Definition 3.4], since for us X and Y are in
] m , we denote by S ⊥ the set of all X ∈ N[F × ] m such that X ⊥ Y for all Y ∈ S. Let F be a tract endowed with an involution x ↦ → x, and let M be a (classical) matroid with ground set E. The following two definitions are taken directly from [1].
(C2) Taking supports gives a bijection from the projectivization of C to the set of circuits of M.
For the purposes of this paper, we define a (strong) F -matroid Mfoot_0 to be a matroid M (called the underlying matroid of M), together with a dual pair of F -signatures of M. The equivalence of this definition with the one given in [1] is proved in [1,Theorem 3.26].
We call C (resp. D) the set of F -circuits (resp. F -cocircuits) of M, and denote these by C (M) and C * (M), respectively.
We denote the set of all generalized vectors (resp. covectors) by V(M) (resp. V * (M)).
Note that a vector of M, in the sense of [1], is just a generalized vector belonging to F m rather than N[F × ] m , and similarly for covectors. We denote by V (M) (resp. V * (M)) the set of vectors (resp. covectors) of M. Definition 3.6. An F -matroid M is strongly perfect if V(M) ⊥ V * (M). A tract F is strongly perfect if every F -matroid is strongly perfect.
A strongly perfect tract is obviously perfect. We now show that the converse holds as well: Theorem 3.7. A tract F is perfect if and only if it is strongly perfect.
For the proof of Theorem 3.7, we will need the following straightforward lemma, whose proof we omit. (2) There is an F -matroid π e (M) on E ′ , whose underlying matroid is obtained by replacing e with two elements e 1 , e 2 in parallel, and whose F -cocircuits D ′ are given by D ′ (f ) = D(f ) for f ∈ E and D ′ (f ) = D(e) for f ∈ {e 1 , e 2 }, where D is an F -cocircuit of M. The F -circuits C ′ of π e (M) are given by either
= C (e), and C ′ (e 2 ) = 0, for C an F -circuit of M, or (iii) C ′ (f ) = 0 for f ∈ E, C ′ (e 1 ) = a ∈ F × , and C ′ (e 2 ) = -a.
Proof of Theorem 3.7. Let F be a perfect tract, let M be an F -matroid, and let X , Y ∈ N[F × ] E be elements of V(M) and V * (M), respectively. We need to show that X ⊥ Y .
To see this, for each e ∈ E let k(e) = min(1, ∥X(e)∥) and let ℓ(e) = min(1, ∥Y (e)∥). Let M ′ be the F -matroid on E ′ obtained from M by replacing each e ∈ E with k(e) series copies of a bundle of ℓ(e) parallel elements. Formally, M ′ is obtained from M as follows: for each e ∈ E, apply the operator σ e k(e) -1 times, thereby replacing e with k = k(e) elements e 1 , . . . , e k ; now, for each i = 1, . . . , k apply the operator π e i ℓ(e) -1 times.
e f M M'
, with a i (e) ∈ F , and similarly write
is the ith parallel element in any one of the k(e) bundles in series for i = 1, . . . , ℓ(e). Similarly, define
is any one of the ℓ(e) parallel elements in the jth series copy of the bundle of parallel elements replacing e for j = 1, . . . , k(e).
Using Lemma 3.8 (which by induction provides us with an explicit description of C * (M ′ ) and
The following propositions concern the behavior of generalized vectors and covectors with respect to deletion and contraction. For (non-generalized) vectors and covectors, the corresponding results are proved as Propositions 4.3 and 4.4, respectively, in Laura Anderson's paper [7]. The proofs given in [7] work mutatis mutandis for generalized vectors; alternatively, one can reduce the generalized case to the one treated in [7] using a trick similar to the one in the proof of Theorem 3.7.
In particular, the contraction of a generalized vector is again a generalized vector, and the deletion of a generalized covector is again a generalized covector.
Recall the modified strong fusion axiom:
] and either γ = 0 or γ ̸ ∈ N F , and if ∥α + β∥ ≥ 4, then α + β ∈ N F . Our goal in this section is to prove the following theorem, which generalizes Theorem 1.11: Theorem 4.1. If a tract F satisfies (MSF) then F is strongly perfect. The following is an example of a tract that satisfies (MSF) but not (SF).
Let P be the phase hyperfield and take the tract embedding (P × , N P ). Letting
] ≥4 , it is straightforward to show that P ′ = (P × , N P ′ ) satisfies the tract axiom and axiom (MSF). However, it does not satisfy the strong fusion axiom (SF). Let α = 1, β = 1 + 1, and γ = (-1) + (-1). Then, we have the following, The proof of Theorem 4.1 is fairly long and technical, so it will be broken up into a number of smaller and hopefully more digestible pieces.
] and β ∈ N F we have αβ ∈ N F .
Proof. If α = 0, this is obvious. Otherwise, write α = ∑ k i=1 x i with x i ∈ F × and inductively apply (I). □
Proof. It suffices to show that (F) is satisfied when ∥α + β∥ ≤ 3. If z = 0, then α and β are either zero or they belong to N[F × ] ≥2 . Hence, either at least one of α, β is 0 or ∥α + β∥ ≥ 4.
Assume z ∈ F × . If ∥α∥ = 1 or ∥β∥ = 1, the result is clear. Otherwise, both ∥α∥ and ∥β∥ are at least 2. □ Proposition 4.6. Let F be a tract which satisfies (MSF). Suppose α, β, γ , δ
Proof. We proceed by induction on ∥β∥. For ∥β∥ = 1, the result follows immediately from (MSF).
Assuming the result holds for ∥β∥ < k with k ≥ 2, we will prove it for β = y
As in the previous case, this implies by induction that α + βδ
] E and e ∈ E which will allow us to perform an analogue of (co)circuit elimination for generalized (co)vectors of matroids over tracts satisfying (MSF).
The following result and its proof were inspired by [4, Lemma 2.4] (which is proved in [8, Lemma 3.2]). Proposition 4.8. Let F be a tract which satisfies (MSF), and let M be an F -matroid on E = {1, 2, . . . , m}. For any X , Y ∈ V * (M) and e ∈ E we have X ∧ e Y ∈ V * (M).
We will consider the following two cases:
We have the following subcases:
It is easy to check that if any one of {C(e), X (e), Y (e)} is 0, then (X ∧ e Y ) ⊥ C . Assuming none of them is 0, we have the following subcases:
By symmetry, we may assume without loss of generality that |X ∩ C | = {e, f } and ∥X(f
And since both X (e)C (e) and X (f )C (f ) belong to F × , we must have
The following result and its proof were inspired by [4, Lemma
Proposition 4.9. Suppose F is a tract satisfying (MSF). Let X 1 , . . . ,
Proof. We may assume, without loss of generality, that X i / ∈ N F for all i and X i + X j / ∈ N F for all i ̸ = j, since otherwise
Let J be a maximum non-empty proper subset of {1, . . . , n} such that
By symmetry, we may assume without loss of generality that J ⊆ {4, 5, . . . , n} and that 3 ∈ I := J c \{1, 2}.
From the maximality of J, we have
and since X J / ∈ N F by assumption, (MSF) implies 3
3 Recall that X 1 -X 1 is not the same thing as zero in N[F × ]!
Since X 1 + X J ∈ N F and X 1 ̸ ∈ N F , (MSF) applied to (8) yields
Similarly, since X 1 + X 2 + X J ∈ N F and X J / ∈ N F , (MSF) applied to (9) yields
Finally, since
Our next goal is to prove that for any generalized vector X and any generalized covector Y such that X •Y has at most three terms, we have X ⊥ Y . We first recall the following key lemma from [1]:
Lemma 4.10. Let X be a generalized vector of M and choose e ∈ E with X (e) / ∈ N F . Then there is some circuit C with e ∈ C ⊂ X .
Although [1,Lemma 3.43] is stated in the language of fuzzy rings, the same (straightforward) proof works for generalized vectors in our sense. Note that a tract that satisfies (F) is an idyll. We therefore have X • Y = X (e)Y (e) ∈ N F .
Writing X ∩ Y = {e, f }, we observe that, since X • Y has at most 3 terms, at least three of X (e), X (f ), Y (e), Y (f ) must lie in F × (and not just N[F × ]). Without loss of generality, we may suppose that X (e), X (f ), Y (e) ∈ F × . In particular, these values are non-null. By Lemma 4.10, there exist a circuit C such that e ∈ C ⊂ X and a cocircuit D such that e ∈ D ⊂ Y . Thus e ∈ C ∩ D ⊂ X ∩ Y = {e, f } and |C ∩ D| ̸ = 1, from which it follows that C ∩D = {e, f }. We therefore have the following relations:
By the minimality of E, M ′ is perfect, and therefore
Explicitly, this means that
We also have
Since ∥X • Y ∥ ≥ 4, we may apply Proposition 4.6 with β = X (e 0 ) and γ = -∑ e∈E ′ Y (e)C (e) / ∈ N F to (11) and (12) to obtain
Y (e)X (e) + Y (e 0 )C (e 0 )X (e 0 ) ∈ N F , which means that X ⊥ Y , a contradiction.
From the previous claim, Applying Proposition 4.9 to (13) and ( 14) shows that X ⊥ Y as claimed. □
In this section we briefly compare our results with those in [4]. For ease of exposition, we work with Lorscheid's ''simplified fuzzy rings''. It is proved in [1, Appendix B] that every fuzzy ring in the sense of Dress-Wenzel is weakly isomorphic to a simplified fuzzy ring, and it is proved in [6,Theorem 2.21] that the category of simplified fuzzy rings can be identified with a full subcategory of the category of tracts. In particular, every simplified fuzzy ring can be identified in a natural way with a tract. 4A simplified fuzzy ring in the sense of Lorscheid is a tuple (K , +, •, ϵ, K 0 ) where (K , +, •) is a commutative semiring equal to N[K × ] and such that ϵ, K 0 satisfy the following axioms: (FR4) K 0 is a proper semiring ideal, i.e., K 0 + K 0 ⊆ K 0 , K • K 0 ⊆ K 0 , 0 ∈ K 0 and 1 / ∈ K 0 . (FR5) For α ∈ K * we have 1 + α ∈ K 0 if and only if α = ϵ. (FR6) If x 1 , x 2 , y 1 , y 2 ∈ K and x 1 + y 1 , x 2 + y 2 ∈ K 0 then
To give a sufficient condition for perfection, Dress and Wenzel introduce the following variant of (FR6): (FR6 ′′ ) If κ, λ 1 , λ 2 ∈ K , µ ∈ K \ K 0 , and κ + µ • λ 1 , µ + λ 2 ∈ K 0 then κ + ϵ • λ 1 • λ 2 ∈ K 0 . Proposition 5.1. A simplified fuzzy ring satisfying (FR6 ′′ ), when viewed as a tract, satisfies the strong fusion axiom (SF).
Proof. Given a simplified fuzzy ring K , let F K denote the tract associated to it. If α + γ and βγ are in N F K , then in terms of the fuzzy ring we have α + γ , βγ ∈ K 0 . If γ = 0, then α + β ∈ K 0 by (FR4). If γ / ∈ K 0 , let κ = α, µ = ϵ • γ , λ 1 = ϵ and λ 2 = β. Then α + β ∈ K 0 by (FR6 ′′ ). In the language of tracts, this means precisely that α + β ∈ N F K . □ Combining Proposition 5.1 with Theorem 1.11, we recover the following special casefoot_2 of [4, Theorem 2.7]:
Theorem. A simplified fuzzy ring which satisfies (FR6 ′′ ) is perfect.
The tract P ′ appearing in Example 4.2 comes from a simplified fuzzy ring K . Since P ′ does not satisfy (SF), Proposition 5.1 implies that K does not satisfy (FR6 ′′ ). On the other hand, Theorem 4.1 applies to P ′ since P ′ does satisfy (MSF). This shows that Theorem 4.1 is strictly stronger than [4, Theorem 2.7], at least when we restrict the latter to simplified fuzzy rings.
All F -matroids in this paper will be strong, so we sometimes omit the modifier.
For a discussion of which tracts come from simplified fuzzy rings, see[6, Example 2.11].
Dress and Wenzel prove in[4, Theorem 2.7] that this remains true with ''simplified fuzzy ring'' replaced by ''weakly distributive fuzzy ring''.