Exterior algebras in matroid theory

Ordered blueprints are algebraic objects that generalize monoids and ordered semirings, and F1±\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {F}_1^{\pm }$$\end{document}-algebras are ordered blueprints that have an element ϵ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon $$\end{document} that acts as -1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-1$$\end{document}. In this work we introduce an analogue of the exterior algebra for F1±\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {F}_1^{\pm }$$\end{document}-algebras that provides a new cryptomorphism for matroids. We also show how to recover the usual exterior algebra if the F1±\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {F}_1^{\pm }$$\end{document}-algebra comes from a ring, and the Giansiracusa Grassmann algebra if the F1±\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {F}_1^{\pm }$$\end{document}-algebra comes from an idempotent semifield.


Introduction
It is a classical theme that d-dimensional linear subspaces of the vector space K n over a field K correspond to certain elements of the exterior algebra ΛK ( n d ) , which are welldefined up to scalar multiples in K × .
The combinatorial counterpart of such linear subspaces are matroids.Baker and Bowler streamline in [BB19] this analogy in a broad sense by the theory of matroids with coefficients in so-called tracts.Fields, semifields and more generally hyperfields, can be seen as examples of tracts.
Jeffrey and Noah Giansiracusa introduce in [GG18] an exterior algebra for idempotent semifields S and exhibit a 'cryptomorphic' description of S-matroids in terms of the exterior algebra, in a formal analogy to the description of K-matroids, or linear subspaces of K n , in the case of a field K.
Somewhat puzzling, however, is that Giansiracusas' definition of the exterior algebra for idempotent semifields makes explicit use of the idempotency in the sense that for a free module with basis {e 1 , . . ., e n }, one has that e i ⊗ e i = 0, for i = 1, . . ., n, are the only defining relations, in contrast to the larger set of relations for fields.
In this paper, we give a unified approach to both the classical theory over fields and Giansiracusas' theory for idempotent semifields, which is based on Lorscheid's theory of ordered blueprints (cf.[Lor18]).Both fields and idempotent semifields can be realized as ordered blueprints in terms of faithful functors: We define an exterior algebra for ordered blueprints and show that it recovers both classical exterior algebra over rings as well as Giansiracusas' exterior algebra over idempotent semifields in terms of these functors.Moreover, we give a cryptomorphic description of matroids over F ± 1 -algebras, as introduced by Baker and Lorscheid in [BL21], by using elements of the exterior algebra, which recovers the classical viewpoint on linear subspaces of K n and Giansiracusas' interpretation of matroids over idempotent semifields.
Description of results.Let B be an F ± 1 -algebra and n be an integer.The exterior algebra ΛB n = i 0 Λ i B n of B n is a B-module whose underlying semigroup is a (typically non-commutative) B + -algebra.This exterior algebra bears properties analogous to the classical exterior algebra.
Theorem A. There are B-linear isomorphisms of ordered blue B-modules and This unifies and generalizes the classical and Giansircusas' exterior algebra in the following sense: (i) Let R be a ring and B = R mon .Then ΛR n is canonically isomorphic to (ΛB n ) hull,+ as an R-algebra.(ii) Let S be an idempotent semiring and B = S mon .Then the Giansiracusa exterior algebra ΛS n is canonically isomorphic to (ΛB n ) idem,+ as an S-algebra.(i) Let K be a field and B = K mon .Then the isomorphism ΛK ( [n]  d ) ≃ (ΛB ( [n] d ) ) hull,+ induces a bijection between K-matroids (as in [BB19]) and B-matroids.(ii) Let S be an idempotent semifield and B = S mon .Then the isomorphism ΛS ( [n]  d ) ≃ (ΛB ( [n]  d ) ) idem,+ induces a bijection between tropical Plücker vectors (as in [GG18]) and Grassmann-Plücker functions with coefficients in B in the sense of this text.(iii) Let B be an ordered blueprint.Then there is a canonical bijection between B-matroids in the sense of [BL21] and classes of B-Plücker vectors (as defined in 3.3).

Let
Remark.We draw the reader's attention to the fact that the functors (−) mon , (−) hull,+ and (−) idem,+ play the same role as in Lorscheid's approach to tropicalization as a base change from a field to the tropical hyperfield in [Lor22].This indicates that our theory is part of a larger picture that puts classical theory and idempotent analysis on an equal footing.
Acknowledgements.The author thanks Oliver Lorscheid for useful conversations and for his help with preparing this text.The present work was carried out with the support of CNPq, National Council for Scientific and Technological Development -Brazil.

Algebraic background
In this section, we review some background around F ± 1 -algebras, following [Lor18].If τ is a preorder on a set X , viewed as a subset of X ×X , we will use x τ y to denote that (x, y) is in τ and a ≡ τ b to denote that both (a, b) and (b, a) are in τ .Note that ≡ τ is an equivalence relation.If the context is clear, we will denote τ simply by .
1.1.Monoids and M-sets.A monoid is a unital semigroup.A monoid M is called pointed if it has an absorbing element, i.e., an element 0 such that 0.m = 0 = m.0 for all m in M. The neutral and the absorbing element (if they exist) are always unique.For the rest of this text, unless otherwise stated, every monoid is supposed to be commutative.
A submonoid of a monoid M is monoid N that is a subset of M, contains 1 M and whose operation • N is the restriction of • M .If M is pointed and its absorbing element is in N, we say that N is a pointed submonoid of M.
If M and W are monoids, a map f : M → W is called a morphism of monoids if f (1 M ) = 1 W and f (x).f (y) = f (xy) for all x, y in M. If M and W are pointed and f carries the absorbing element of M to the absorbing element of W , we say that f is a morphism of pointed monoids.The category of pointed monoids will be denoted by Mon * .
A preorder r on M is called multiplicative (or additive, depending on the operation of M) if, for all elements m, n and x in M, one has mx r nx whenever m r n.A congruence is a multiplicative preorder that is symmetric (thus an equivalence relation).If r is a multiplicative preorder on M, the set c r : and has an induced multiplicative partial order r : If M is a pointed monoid with absorbing element 0, a pointed M-set is a pointed set (X , p) equipped with a map M × X → X that makes X an M-set and satisfies: where [y j ] denotes the class, in are morphims of M-sets for all x 1 in X 1 and x 2 in X 2 .We denote the set of M-bilinear maps We construct the tensor product X 1 ⊗ M X 2 of X 1 and X 2 as the quotient of X 1 × X 2 by the equivalence relation generated by The map defines a morphism of M-sets.
The universal property of X 1 ⊗ M X 2 can be expressed as the fact that ϕ → ϕ is a bijection from Bil M (X 1 × X 2 ,W ) to Hom M (X 1 ⊗ M X 2 ,W ).

Semirings and modules.
A semiring is a triple (S, +, •), where (S, +) is a commutative monoid with identity 0, (S, •) is a (non-necessarily commutative) pointed monoid with identity 1 and absorbing element 0, and satisfying (a + b).c = (a.c)+ (b.c) and c.(a +b) = (c.a)+(c.b) for all a, b and c in S. We also use ab to denote a.b.A semiring is called commutative if (S, •) is commutative.If the operations are clear, we use S to denote the semiring (S, +, •).For the rest of this text, unless otherwise stated, every semiring is supposed to be commutative.
If S 1 and S 2 are semirings, a map f : S 1 → S 2 is called a morphism of semirings if it is, at the same time, a morphism of the underlying monoids f : (S 1 , +) → (S 2 , +) and f : (S 1 , •) → (S 2 , •).The category of semirings will be denoted by SRings.
A preorder r on a semiring S is called additive and multiplicative if a + z r b + z and az r bz whenever (a, b) is in r and z is in S.
An ordered semiring is a pair (S, ), where S is a semiring and is an additive and multiplicative partial order on S. A morphism of ordered semirings is a morphism of semirings that is order-preserving.
A congruence on a semiring S is an additive and multiplicative preorder that is symmetric (thus an equivalence relation).If c is a congruence on a semiring S, the quotient set S/c is naturally a semiring, with operations defined by for all s, r in S and y, z in are morphims of S-modules for all y 1 in Y 1 and y 2 in Y 2 .We denote the set of S-bilinear We construct the tensor product Y 1 ⊗ S Y 2 as the quotient of S Y 1 ×Y 2 by the congruence generated by {e (s.y 1 ,y 2 ) ≡ s.e (y 1 ,y 2 ) ≡ e (y 1 ,s.y 2 ) | s ∈ S and y i ∈ Y i } (where e (h 1 ,h 2 ) is the element of S Y 1 ×Y 2 that has 1 in the entry correponding to (h 1 , h 2 ) and 0 on the others).We denote the class of e (y 1 ,y 2 ) by y 1 ⊗ y 2 .
If ϕ : Y 1 ×Y 2 → Z is an S-bilinear map, one has a morphism of S-modules characterized by ϕ : The association ϕ → ϕ is a bijection from Bil S (Y 1 ×Y 2 , Z) to Hom S (Y 1 ⊗ S Y 2 , Z).For a number n 1 and an S-module Y , we will denote the n-fold tensor product Y ⊗ . . .⊗Y by Y ⊗n , and define Y ⊗0 as S.
For the S-module TY := ∞ l=0 (Y ) ⊗l , the map extends, by linearity, to a product of TY that makes it a (typically non-commutative) S-algebra.
(iii) B • generates B + as a semiring, i.e., every element of B + is a finite sum of elements in B • .We call B • the underlying monoid and B + the ambient semiring of B, and use B × to denote the set of invertible elements of B • .A morphism of ordered blueprints ϕ : B → C is an order-preserving morphism ϕ : If B is an ordered blueprint and r is an additive and multiplicative preorder on B + containing , one has the quotient ordered blueprint B r := {B • , B + /c r , r}, where c r and r are the congruence and the partial order induced by r, respectively, and B • is the multiplicative submonoid of B + /c r whose elements are the classes of elements in B • .
For a subset H of B + × B + , we define the preorder generated by H as H := {additive and multiplicative preorders on B + containing H and }.
where the map B • × M + → M + is the restriction of B + × M + → M + ; (iv) M • generates M + as a semigroup, i.e., every element of M + is a finite sum of elements in M • .We call M • the underlying B • -set and M + the ambient B + -module of M. A morphism of B-modules f : M → N is an order-preserving morphism f : M + → N + of B + -modules such that f (M • ) ⊆ N • .The category of B-modules will be denoted by B-Mod.
An additive B-preorder on M is an additive preorder r on the monoid M + that contains M and satisfies b and r is the partial order induced by r.For a subset L of M + × M + , we define the preorder generated by L as If M is a B-module, let g := (x, y) ∈ M + × M + | x M y or y M x .We define the algebraic hull of M as the B-module M hull := M/ /g.Note that the partial order of M hull is trivial.
There are two natural functors (−) • and (−) + that send an B-module to its underlying B • -set and ambient B + -module, respectively.
The coproduct of a family is the image of the natural map of B • -sets 1.5.Tensor product of B-modules.A B-bilinear map ϕ : The tensor product of M 1 and M 2 is defined as the B-module Let ϕ : M 1 × M 2 → N be a B-bilinear map.As ϕ is B + -bilinear, there exists a morphism φ : By the definition of B-bilinearity, φ(im ψ) ⊆ N • and φ(x 1 ⊗ y 1 ) φ(x 2 ⊗ y 2 ) for every relation Thus there exists a morphism ϕ : Proof.Let ϕ and θ two B-bilinear maps such that ϕ = θ.Thus, in particular, With this, we obtain the injectivity of Φ.
By the construction of the tensor product and from the fact that ζ is a morphism, one has that ζ is B-bilinear.As Φ( ζ) = ζ, one has that Φ is surjective.

Exterior algebra
2.1.Notations.For a blueprint B, a B-module M, a natural number n and a set I, we will use the following notations: ) n for the canonical basis; M ⊗n for the n-fold tensor product M ⊗ . . .⊗ M; and M n for the product , where ǫ 2 = 1.For the rest of this text, fix an F ± 1 -algebra B = (B • , B + , ), i.e., an ordered blueprint B equipped with a morphism F ± 1 → B or, equivalently, with a distinguished element ǫ in B • such that ǫ 2 = 1 and 0 1 + ǫ.

Construction. For a
Note that the usual product of T M + restricts to a product for im Θ.This operation turns T M into a (typically non-commutative) B-algebra.
For n ∈ N, let Lemma 2.2.The set Proof.We begin by noticing that E clearly is a generator set.This implies that there are A 1 , A 2 in T (B n ) + , both of the form ∑ d j 1 ,... j t .ej 1 ⊗ . . .⊗ e j t , such that for each set { j 1 , . . ., j t } one has (at least) two indexes u, v in [t] satisfying j u = j v ; and there are, for each index sets {i 1 , . .., i d } present in (1), two permutations σ {i 1 ,...,i d } and δ {i 1 ,...,i d } satisfying Thus, looking at the coefficient of e i 1 ⊗ . . .⊗ e i d in (1) and looking at the coefficient of e i σ {i 1 ,...,i d } (1) ⊗ . . .⊗ e i σ {i 1 ,...,i d } (d) in ( 2 The operation (α, β) → αβ defines a product on B n ) + , making it a (typically noncommutative) B + -algebra.For x, y ∈ B n ) + , we denote by x ∧ y the product of x and y.For I = {i 1 , . .., i d } with i 1 < . . .< i d , we use e I to denote the element e i 1 ∧ . . .∧ e i d .
Remark 2.4.For n 2, the exterior algebra B n may not be a B-algebra because its underlying pointed set B n • could not be multiplicatively closed, as we always have Remark 2.5.The difference between the exterior algebra B n and the B-algebra T (B n ) τ n concerns only the underlying pointed B-set.This occurs because we need sums of elements of T (B n ) τ n • to define B-Plücker vectors (cf.3.3).
But T (B n ) τ n has the following universal property (similar to the universal property of the ring-theoretic exterior algebra): given a (not necessarily commutative) Balgebra A and a morphism of B-modules ϕ : B n → A satisfying ϕ(x).ϕ(x)= 0 and ϕ(x).ϕ(y)= ǫϕ(y).ϕ(x)for all x, y ∈ B n , there exists a unique morphism of B-algebras and define the monomial ordered blueprint associated to S as S mon := S • Ω S .This is an F ± 1 -algebra, with ǫ = 1.1 S and the construction above extends to a functor (−) mon : IdempSFields → OBlpr.
For an ordered blueprint C, let C idem := C 1 ≡ 1 + 1 be the idempotent ordered blueprint associated to C. This name comes from the fact that C idem,+ is always an idempotent semiring.
Definition 2.8.Let S be an idempotent semifield.For n ∈ N, let The tropical Grassmann algebra of S n is the graded S-algebra S n := Sym S n e 2 i ≡ 0| i ∈ [n] ≃ T (S n ) Ψ n .The d th -homogeneous direct summand of S n , denoted by d S n , is called the d th tropical wedge power of S n .
The next theorem proves the second item of Theorem B by showing how to recover the tropical Grassmann algebra from our construction of the exterior algebra and exterior powers for F ± 1 -algebras.Theorem 2.9.Let S be an idempotent semifield and S := S mon .Then d S n idem,+ ≃ d S n , for all d, and S n idem,+ ≃ S n , where d S n and S n denotes the Giansiracusa d th tropical wedge power and tropical Grassmann algebra of S n , respectively.Proof.To begin with, we prove that S idem,+ ≃ S as semirings.We will denote the class of ∑ n i .si ∈ S + in S idem,+ by [∑ n i .−→ e I .
[n] = {1, . .., n} and Γ =[n]  d be the family of d-subsets of [n].We define a Bmatroid of rank d on [n] as the B × -class [ν] of an element ν = (ν I ) I∈Γ of Λ d B Γ that satisfies a certain system of relations (see 3.1) and such that ν I ∈ B × for some I ∈ Γ.This recovers the aforementioned concepts of matroids in the following sense:Theorem C. Consider 0 d n.
[a] + [b] := [a + b] and [a].[b] := [ab].If r is an additive and multiplicative preorder on a semiring S, the set c r := {(a, b) ∈ S × S| a ≡ r b} is a congruence and the quotient semiring S/c r has an induced additive and multiplicative partial order r := {([a], [b])| a r b}.An S-module is a pointed monoid (Y, +) with identity 0 equipped with a map λ : S ×Y −→ Y (s, y) −→ s.y that makes Y an (S, •)-set and satisfies (s + r).y = (s.y)+ (r.y) and s.(y + z) = (s.y)+ (s.z) we write a i b i | i ∈ I to denote H , and a i ≡ b i | i ∈ I to denote a i b i and b i a i | i ∈ I .For an ordered blueprint B, let f := a ≡ b| a b and define the algebraic hull of B as the ordered blueprint B hull := B f.Note that the partial order of B hull is trivial.If (D, •) is a pointed monoid with absorbing element 0 D , one has the semiring-algebra D := N[D] 0 ≡ 1.0 D , whose elements can be seen as formal finite sums of non-zero elements of D, with operations ∑ n d .d+ ∑ m d .d= ∑(n d + m d ).d ∑ n d .d∑ m d .d= ∑ d∈D\{0 D } ∑ a.b=d n a m b d.If H ⊆ D × D, we use D H to denote the ordered blueprint (D, D, =) H .There are two canonical functors (−) • and (−) + that sends an ordered blueprint to its underlying monoid and ambient semiring, respectively.1.4.Ordered blue modules.Let B = (B • , B + , B ) be an ordered blueprint.An ordered blue B-module, or simply B-module, is a triple M Let b I , c I in B + , I subset of [n], such that ∑ ...<i d b {i 1 ,...,i d } e i 1 ⊗ . . .⊗ e i d ≡ ∑ i 1 <...<i d c {i 1 ,...,i d } e i 1 ⊗ . . .⊗ e i d =: c.By the definition of τ n , there exists two sequences b = x 0 , . . ., x m and c = y 0 , . .., y w = x m of elements of T (B n ) + such that x ℓ+1 = x ℓ + α (resp.y ℓ+1 = y ℓ + α), where α has the form b.e a ⊗ e a ⊗ e z 1 ⊗ . . .⊗ e z f for some b in B + and a, z 1 , . . ., z f in [n]; or x ℓ+1 = ρ + β 1 and x ℓ = ρ + β 2 (resp.y ℓ+1 = ρ + β 1 and y ℓ = ρ + β 2 ), where β 1 has the form b.e i 1 ⊗ . . .⊗ e i f and β 2 = sign(σ)b.ei σ(1) ⊗ . . .⊗ e i σ( f ) , for some b in B + , ρ in T (B n ) + , i 1 , . . ., i f in [n], a permutation σ and interpreting sign(σ) as an element of F ± 1 via the identification of −1 with ǫ.
), we conclude that b {i 1 ,...,i d } = c {i 1 ,...,i d } , for each index set {i 1 , . . ., i d } present in (1).Let S d,n be the sub-B + -module of T (B n ) τ n + generated by H d,n .The B-module d B n := H d,n , S d,n , , where is induced from T (B n ) τ n , is called the d th exterior power of B n .Let H n := n l=0 H l,n and note that it generates T (B n ) τ n + as a semigroup.The B-module B n := (H n , T (B n ) τ n + , ) is called the exterior algebra of B n .
s i ].One has the natural map ϕ : S idem,+ −→ S [∑ n i .si ] −→ ∑ n i s i , which is a surjective morphism of semirings.Note that m ∑ i=1 m i .si = m ∑ i=1 1.s i in S idem,+ , for all positive integers m i and s i in S. Let a, b and c := a + b be elements of S. Note that c + b = c = c + a and c + c = a + c.Thus, in S + , one has 1.c 1.a + 1.b and 1.a 2.c.Analogously, 1.b 2.c.Therefore, 1.a + 1.b 4.c in S + , which implies 1.c 1.a + 1.b 4.c = 1.c.Thus, 1.c = 1.a + 1.b in S idem,+ .As a consequence, we obtain that ϕ is injective.Next we observe thatd S n + ≃ S ( [n] d ) + ≃ S + ( [n] d ) as S + -modules, which implies d S n idem,+ ≃ S ( [n] d ) idem,+ ≃ S idem,+ ( [n] d ) ≃ S ( [n] d ) ≃ d S nas S-modules.Thus S n idem,+ ≃ n d=0 d S n + ≃ n d=0 d S n ≃ S n , via S n idem,+ −→ S n [e I ]