Séminaire Lotharingien de Combinatoire 47 (2002), Article B47f

COMBINATORIAL FANS, LATTICE-ORDERED GROUPS, AND THEIR NEIGHBOURS: A SHORT EXCURSION

VINCENZO MARRA AND DANIELE MUNDICI

Abstract. Just like the celebrated dictionary between integral convex geometry and toric varieties bridged until then remote mathematical worlds, the less wellknown duality between (certain) lattice-ordered groups and rational polyhedral sets brought combinatorial techniques to bear on the investigation of these abstract algebraic structures. We offer a brief excursion into these relatively new territories, with special emphasis on the novel and admittedly unclear rôle played by ordertheoretical considerations.

1. Introduction

In an inaugural address ([25]) delivered in Florence, Italy, in 1993, G.-C. Rota maintained that

[while] one of the leading trends that is visible in present-day mathematics [is] ‘the return to concreteness’, [...] the relationship between combinatorics and algebra today [...] is not a forgetting of the past. [...] some outstanding work in combinatorics that is going on today is greatly benefiting the algebra of yesterday.

As a prominent example, Rota mentioned the discovery of toric varieties, [...] a dictionary whereby results of algebraic geometry can be translated into results of convex integral geometry in the style of Minkowski and Hadwiger. [...] The theorems of algebraic geometry now find new and unexpected life in the newly added class of toric varieties, coming from an altogether different source, a combinatorial source to be sure.

The main theme of this paper is a similar interaction between another realm of abstract algebra, the theory of lattice-ordered groups, and a version of “convex integral geometry in the style of Minkowski and Hadwiger” ([25]), the eminently combinatorial theory of rational polyhedral sets. Prima facie, ordered groups are no less far removed from polyhedra than algebraic geometry — and yet, much as in the case of toric varieties, the connection is so intimate as to be a genuine duality.

Key words and phrases. Fans, polyhedral cones, convex geometry, lattice-ordered groups, toric varieties.

We begin with a survey of basic results on complexes of polyhedral cones and their subdivisions. We then introduce lattice-ordered abelian groups, and sketch enough of their theory to establish the relationship with polyhedral complexes. Along the way, as a sort of counterpoint to our leitmotiv, we indicate the algebro-geometric significance of some of the material introduced. It is only in the last section that we bring this second theme into the foreground for a necessarily short excursion into the world of toric varieties.

2. Combinatorial Convexity: Fans

We introduce a linear version of polyhedral complexes known as fans. The terminology is due to M. Demazure, who called these objects eventails. It is quite a descriptive name, provided it is interpreted as ‘possibly torn fans’ ([23]). A comprehensive treatment may be found in [15].

Throughout this paper, we fix a free abelian group N of rank n. Upon tensoring N with ℝ we obtain a real n-dimensional vector space N_{ℝ} = ℝ ⊗ N. We let M = Hom_{ℤ}(N, ℤ) denote the ℤ-module dual to N, and M_{ℝ} = ℝ ⊗ M the vector space dual to N_{ℝ}.

In down-to-earth terms, N ≅ ℤ^{n}, N_{ℝ} ≅ ℝ^{n} and N is the group of integral n-tuples in ℝ^{n}. Similarly, M can be identified with ℤ^{n} and M_{ℝ} with ℝ^{n}. These identifications, however, only make sense if we choose a basis of N (and therefore of M, N_{ℝ}, and M_{ℝ}).^{1} Whenever convenient, we shall tacitly assume we have fixed one such basis.

Let {v⃗_{1}, …, v⃗_{m}} ⊆ N be a finite set of integral vectors. The rational polyhedral cone σ, or simply the cone σ generated by {v⃗_{1}, …, v⃗_{m}} is their positive linear hull over ℝ, i.e.,

σ = ⟨v⃗_{1}, …, v⃗_{m}⟩ = {p_{1}v⃗_{1} + ⋯ + p_{m}v⃗_{m} | 0 ≤ p_{1}, …, p_{m} ∈ ℝ}.

A cone σ is k-dimensional iff the ℝ-linear space it spans is k-dimensional. A face of σ is any nonempty convex subset τ ⊆ σ such that every line segment in σ which has an interior point in τ lies entirely in τ. Lest we burden this paper with an inordinate amount of detail, we shall assume a number of intuitive and easily proved statements about cones — e.g., the faces of a cone are finite in number, the set of faces is a (finite) lattice under inclusion (whence every face is contained in some maximal proper face, a facet), and so on.

Upon intersecting a finite number of integral half-spaces (i.e., sets of the form {(x_{1}, …, x_{n}) ∈ ℝ^{n} | z_{1}x_{1} + ⋯ + z_{n}x_{n} ≥ 0}, z_{i} ∈ ℤ) one clearly obtains a cone in N_{ℝ}. That this is no accident is a basic result in convex geometry.

The Fundamental Theorem of Polyhedra. A subset σ ⊆ N_{ℝ} is a cone iff it is the intersection of finitely many integral half spaces of N_{ℝ}.

1It turns out that this is equivalent to a choice of coordinates on an algebraic variety — see Section 4.

A cone σ = ⟨v⃗_{1}, …, v⃗_{m}⟩ is simplicial iff {v⃗_{1}, …, v⃗_{m}} is linearly independent over ℝ. A nonzero vector v⃗ ∈ N is primitive iff its coordinates (with respect to the canonical basis) are relatively prime. The vertices of a simplicial cone σ are the (uniquely determined) primitive vectors v⃗_{i} ∈ N such that σ = ⟨v⃗_{1}, …, v⃗_{m}⟩.

A rational polyhedral fan (in N_{ℝ}), in short, a fan, is a finite set Σ of polyhedral cones such that

(1) Every face of every cone of Σ belongs to Σ.

(2) Any two cones of Σ intersect in a common face. The support of Σ, denoted |Σ|, is the union of all its cones. A fan Σ is simplicial iff all its cones are simplicial.

A polyhedral cone σ carries a natural fan structure Σ_{σ} which is given by the collection of all its faces (including ∅ and σ itself). With this notation, |Σ_{σ}| = σ. We shall not be pedantic about the distinction between Σ_{σ} and σ.

Subdivisions. Subdivisions of rational polyhedral cones, besides being of interest in themselves, encode the birational geometry of toric varieties. Thus, we discuss various notions of refinement for fans (see Figure 1). Given two fans Σ and Δ in N_{ℝ}, Δ is a subdivision of Σ, or it refines Σ, iff |Σ| = |Δ| and every cone of Δ is contained in some cone of Σ. We write Δ ≤ Σ to denote the fact that Δ refines Σ. In this case, every cone of Σ is a union of cones of Δ. More generally, following Danilov ([11]) we say that Σ is inscribed in Δ iff every cone of Σ is a union of cones of Δ, whence we have the (possibly strict) inclusion |Σ| ⊆ |Δ|.

Just like a polyhedron can be triangulated without adding new vertices (say, by generalising the elementary plane construction of ‘adding diagonals’), a fan can always be subdivided into a simplicial fan without adding 1-dimensional cones. The m-skeleton Σ^{(m)} of a fan Σ is the set of m-dimensional cones of Σ. (It is not itself a fan, for it does not contain, together with a cone, all of its faces). Then there exists a simplicial fan Δ_{1} with the same 1-skeleton of Σ such that Δ_{1} ≤ Σ.

In many contexts, simplicial subdivisions are not fine enough — it turns out that a certain proper subclass of simplicial cones plays a key rôle in the theory of toric varieties. A cone σ = ⟨v⃗_{1}, …, v⃗_{m}⟩ ⊆ N_{ℝ} is unimodular (alternative terminology, regular) iff {v⃗_{1}, …, v⃗_{m}} can be completed to a basis of N. Hence, in particular, σ is simplicial and the set of its vertices is {v⃗_{1}, …, v⃗_{m}}. A fan Σ is unimodular iff all its cones are unimodular.

Clearly, not every simplicial cone is unimodular. However, it is always possible to subdivide a given cone into unimodular subcones. Further, this can always be done through stellar subdivisions. Let Σ be a fan in N_{ℝ}, and let σ ∈ Σ be an m-dimensional cone. The star in Σ of a face F ⊆ σ, denoted star F, is the set of cones in Σ having F as a face; the closed star of F is the fan cstar F = {τ | τ is a face of a cone θ ∈ star F }. The relative interior of σ, denoted relint σ, is the topological interior of σ∩V, where V is the linear subspace of N_{ℝ} spanned by σ. Let w⃗ ∈ relint σ ∩N. Then there is a unique minimal face F of σ such that w⃗ ∈ relint F, namely, the intersection of all faces containing w⃗. The stellar subdivision^{2} of Σ along σ through w⃗ is the fan obtained from Σ removing star F and joining w⃗ to the boundary (cstar F − star F) of the star of F. If Δ is obtained from Σ via a finite number (may be zero) of stellar subdivisions, we write Σ ≤^{∗} Δ. Clearly, Σ ≤^{∗} Δ implies Σ ≤ Δ, but not conversely.

Unimodular Refinements. Let Σ be a fan. There exists a simplicial fan Δ_{1} with the same 1-skeleton of Σ such that Δ_{1} ≤ Σ. Further, there exists a unimodular fan Δ_{2} such that Δ_{2} ≤^{∗} Δ_{1} ≤ Σ, that is, Δ_{2} is obtained from Δ_{1} by a finite number of stellar subdivision.

Remark. Unimodularity is the combinatorial counterpart of the central algebrogeometric notion of smoothness ([23], [16], [15]).

For self-contained proofs of the existence of simplicial and unimodular refinements, see e.g. [15].

A special sort of stellar subdivision is particularly important for the contents of this paper. Let σ = ⟨v⃗_{1}, …, v⃗_{m}⟩ be an m-dimensional cone of a fan Σ in N_{ℝ}. Suppose τ = ⟨v⃗_{i_{1}}, … v⃗_{i_{d}}⟩ is a d-dimensional face of σ. We call w⃗ = ∑ v⃗_{i_{j}} the barycenter of τ. The fan obtained from Σ by stellar subdivision along τ through w⃗ is a barycentric stellar subdivision of Σ. We write Δ ⪯ Σ to denote the fact that Δ is obtained from Σ by a finite number (maybe zero) of stellar barycentric subdivisions; if all barycenters belong to 2-dimensional cones, we write Δ ⪯_{2} Σ and speak of binary starring. It is easy to show but important to realize that if Δ ⪯ Σ and Σ is unimodular, then Δ is unimodular — barycentric subdivisions preserve unimodularity. This fails for general stellar subdivisions.^{3}

If Σ is simplicial but not unimodular, a fan Δ such Δ ⪯ Σ cannot be unimodular. Thus barycentric subdivisions cannot be used to refine a given fan into a unimodular one. On the other hand, if Σ happens to be unimodular, binary starrings produce most general subdivisions, in the following sense.

The De Concini-Procesi Lemma. Let Σ, Δ be two unimodular fans in N_{ℝ}, and assume |Σ| = |Δ|. There exists a unimodular fan Σ^{∗} in ℝ^{n} such that

(1) Σ^{∗} ≺_{2} Σ ;

(2) Σ^{∗} ≤ Δ .

The De Concini-Procesi Lemma was first established in [14], and may be interpreted as a statement on the birational geometry of toric varieties. The interested reader can find a self-contained elementary proof in Panti’s paper [24].

2Stellar subdivisions were introduced by James W. Alexander in [3] for abstract simplicial complexes, and in [2] for geometric complexes. Their importance in algebraic topology is well-known — see e.g. [4].

3The barycenter of a rational simplex is known to geometers of numbers as the (multidimensional) Farey mediant of its vertices, a classical notion intimately connected with (multidimensional) continued fraction expansions. See [10], [22] and references therein for further information.

[figure: (a); (b); (c); (d)]

Figure 1. Subdivisions of a cone. (a) A cone C. (b) A subdivision (triangulation) of C. (c) A stellar subdivision of C. (d) A barycentric stellar subdivision of C.

Morphisms of fans. Let Σ be a fan in N_{ℝ}. Let N′ be a free abelian group of rank n′, N′_{ℝ} = ℝ ⊗ N′, Σ′ a fan in N′_{ℝ}. A map of fans is an element of Hom_{ℤ}(N, N′)

— a group homomorphism ϕ : N → N′ — such that its (unique) scalar extension ϕ_{ℝ} : N_{ℝ} → N′_{ℝ} maps cones into cones; explicitly, for each σ ∈ Σ there is σ′ ∈ Σ′ such that ϕ_{ℝ}(σ) ⊆ σ′. In the fan-toric dictionary included in Section 4, maps of fans correspond to morphisms of toric varieties. The latter provided the original motivation for the definition of a map of fans.

Let us restrict attention to an endomorphism ϕ of fans with the same support — that is, assume N′ = N and |Σ′| = |Σ|. A moment’s reflection shows that ϕ induces a subdivision of Σ into Σ′. Conversely, any fan Σ′ in N_{ℝ} such that Σ′ ≤ Σ canonically induces an endomorphism ϕ of fans which fixes the support |Σ|. Thus, all forms of subdivisions we have introduced above may be regarded as morphisms (of fans or of toric varieties). The combinatorially fundamental notion of stellar subdivision gives rise to morphisms of great interest in algebraic geometry, namely blow-ups — the basic tool used to resolve singularities of algebraic varieties. As we have seen above, every fan admits a unimodular refinement. This translates to the statement that every toric variety admits a resolution of singularities. While we refer to [23], [16] and [15] for further information on the algebro-geometric aspects of desingularizations, we include a fan-theoretic treatment of resolutions of twodimensional toric singularities, both by way of illustration and with the intent of showing interesting connections with the geometry of numbers.

A cone σ ⊆ N_{ℝ} is strongly convex (alternative terminology, has apex at the origin) iff it does not contain a linear subspace of N_{ℝ}, except {0}.^{4} For the remaining part of this section, set n = 2, and let Σ be a fan in N_{ℝ} ≅ ℝ^{2}. To avoid tedious degenerate cases, we assume all cones of Σ are strongly convex. Let 𝒮_{Σ} be the set of all unimodular fans with the same support as Σ. Clearly, refinement is a partial order on 𝒮_{Σ}. We now claim that the poset ⟨𝒮_{Σ}, ≤⟩ is in fact a lattice.^{5} Furthermore, we claim we can compute finite lattice joins and meets in ⟨𝒮_{Σ}, ≤⟩. It suffices to show this much: Given unimodular fans Σ_{1},Σ_{2} ∈ 𝒮_{Σ}, one can compute Δ ∈ 𝒮_{Σ} with the property that if Σ′ ≤ Σ_{1}, Σ_{2} for some Σ′ ∈ 𝒮_{Σ}, then Δ ≤ Σ′. The argument for the existence and computability of joins is similar.

4This is an interesting notion for many reasons, amomgst which is the following characterization:

for m ≥ 2, an m-dimensional cone is strongly convex iff it is the positive linear hull of a polytope of dimension m − 1. A polytope (= bounded polyhedron) is the convex hull of a finite set of points in ℝ^{n}.

Let Σ_{1},Σ_{2} ∈ 𝒮_{Σ}, and consider the set Θ = {σ_{1} ∩ σ_{2} | σ_{i} ∈ Σ_{i}, i = 1, 2}. It is promptly seen that Θ is a fan such that |Θ| = |Σ|. Further, Θ jointly refines Σ_{1} and Σ_{2}, and it is clear that it is the coarsest such refinement. Unfortunately, Θ need not be unimodular, whence in general Θ ∉ 𝒮_{Σ}. Our claims are settled if we can show that it is possible to compute a coarsest (hence, canonical) unimodular refinement of Θ.

Figure 2. Canonical unimodular refinement of a 2-dimensional cone.

Clearly, it suffices to proceed locally and restrict attention to a single strongly convex 2-dimensional cone σ in N_{ℝ}, whence, in particular, σ is simplicial. Consider the convex hull P in N_{ℝ} of all integral points (σ ∩ N) − {0}. (See Figure 2.) This is an unbounded polygon with a finite number of compact edges. Let us display the (necessarily integral) endpoints of such edges as the finite set of vectors

{v⃗_{0}, …, v⃗_{m}}

such that v⃗_{i} and v⃗_{i+1} are adjacent^{6}, for 0 ≤ i < m. To fix ideas, suppose that the order in which we have listed the v⃗_{i}’s is counterclockwise. For each i, σ_{i} = ⟨v⃗_{i}, v⃗_{i+1}⟩ is a 2-dimensional simplicial cone whose vertices are precisely the vectors v⃗_{i}, v⃗_{i+1} — if some v⃗_{i} were not primitive, the primitive generator of ⟨v⃗_{i}⟩ would not belong to P, a contradiction. Furthermore, each σ_{i} is unimodular, as is easy to check. For 0 ≤ i < m, the collection of all cones σ_{i}, together with their 1-dimensional faces v⃗_{i} and the origin of N_{ℝ}, is a unimodular fan Σ^{σ} refining σ. A moment’s reflection shows that any unimodular subdivision of σ must refine Σ^{σ}. We have thus shown the existence of the coarsest desingularization of σ. Finally, we exhibit an explicit algorithm to compute Σ^{σ}, given a presentation σ = ⟨u⃗_{1}, …, u⃗_{t}⟩ of the cone σ. Notice that from any such presentation one can compute the vertices of σ — first compute rational vectors r⃗_{1}, r⃗_{2} such that σ = ⟨r⃗_{1}, r⃗_{2}⟩, and then use the Euclidean algorithm to compute the vertices. Without loss of generality, then, we may assume that σ is presented as σ = ⟨w⃗_{1}, w⃗_{2}⟩, where {w⃗_{1}, w⃗_{2}} is the set of its vertices, again listed in counterclockwise order. Notice that w⃗_{1} = v⃗_{0} and w⃗_{2} = v⃗_{m}. Since these vertices are primitive linearly independent vectors, another application of the Euclidean algorithm yields a primitive vector n⃗_{1} and integers p, q such that

5With maximum, but with no minimum, if Σ has at least one 2-dimensional cone.

6That is, their convex hull in N_{ℝ} does not contain any other v⃗_{k}, k ≠ i, k ≠ i + 1.

• {w⃗_{1}, n⃗_{1}} is a basis of the ℤ-module N — in other words, ⟨w⃗_{1}, n⃗_{1}⟩ is unimodular;

• 0 ≤ p < q with p and q relatively prime;

• w⃗_{2} = pw⃗_{1} + qn⃗_{1}. In case q = 1, we are done, for p = 0 and σ is unimodular. Otherwise, we consider the Hirzebruch-Jung continued fraction expansion^{7}

\frac{q}{q − p} = b_{1} − \frac{1}{b_{2} − \frac{1}{b_{3}−···}}

with b_{j} ≥ 2, which must terminate since \frac{q}{q−p} is rational. Then it can be proved that, upon inductively defining

n⃗_{i+1} = (b_{i} − 2)v⃗_{i−i} + (b_{i} − 1)n⃗_{i} ,

the following equalities hold:

v⃗_{i} = v⃗_{i−1} + n⃗_{i} ,

whence the vectors v⃗_{i} can be computed, as claimed. Thus, the Hirzebruch-Jung continued fraction expansion is precisely what is needed for the effective desingularization of 2-dimensional toric varieties.

Remark. In higher dimensions, things are far more complicated and not yet wellunderstood. Already in dimension 3, in general there is no coarsest unimodular refinement of a given strongly convex cone σ ⊆ ℝ^{3}. However, the set of maximally coarse refinements is finite, and its classification was carried out by M. Reid and V. I. Danilov (see [23] for details and references). However, the Danilov-Reid treatment is non-effective. Algorithmic desingularization of 3-dimensional toric variety was attained in the paper [1], to which we refer the interested reader for further information.

7This is different from the usual continued fraction expansion, sometimes called regular, where minus signs are replaced by plus signs. See [16].

3. Abelian lattice-ordered groups as functions on fans

This section is devoted to show that bringing together ordered algebraic structures and fans is in the nature of things — indeed, there is an intimate connection between (finitely presented) lattice-ordered groups and integral piecewise linear geometry. We begin with an admittedly cursory introduction to ordered groups. Some standard references are [17] and [13].

A partially ordered abelian group is an abelian group G together with a submonoid G^{+} of G such that the induced binary relation

a ≥ b iff a − b ∈ G^{+}

is a partial order which is translation invariant with respect to addition, i.e.,

a ≥ b implies a + t ≥ b + t for all a, b, t ∈ G

The submonoid G^{+} is the positive cone of G. Notice that 0 ∈ G^{+}. An element p ∈ G is positive iff p ∈ G^{+}, and it is strictly positive iff p ∈ G^{+} − {0} = G_{+}. Accordingly, ℤ^{+} = ℕ ∪ {0} and ℤ_{+} = ℕ.

A partially ordered abelian group G is directed iff any two elements of G have a lower and an upper bound, and it is unperforated iff ng ∈ G^{+}, n ∈ ℕ, implies g ∈ G^{+}. Further, G has the Riesz Interpolation Property iff for any x_{1}, x_{2}, y_{1}, y_{2} ∈ G, x_{i} ≤ y_{j} implies that there exists z ∈ G such that x_{i} ≤ z ≤ y_{j} (i, j = 1, 2). Homomorphisms of partially ordered abelian groups are order-preserving group homomorphisms. A subgroup J ⊆ G is an o-ideal iff it is convex (a ≤ b ≤ c and a, c ∈ J imply b ∈ J) and directed. Kernels of homomorphisms, then, are precisely o-ideals.

A lattice-ordered abelian group (ℓ-group, for short) is a partially ordered abelian group which is a lattice with respect to the order relation. Equivalently, an ℓ-group is a structure (G, +, 0, ∧, ∨) such that (G, +, 0) is an abelian group, (G, ∧, ∨) is a lattice, and + distributes over ∨ and ∧. By an ℓ-subgroup of an ℓ-group we mean a subgroup which is also a sublattice. Given a subset S ⊆ G of an ℓ-group, we denote by ⟨S⟩, [S] and [[S]] the ℓ-subgroup, the subgroup and the submonoid generated by S.

Homomorphisms of ℓ-groups (ℓ-homomorphisms) are maps preserving both the group and the lattice structure. We call their kernels ℓ-ideals. It is easy to check that ℓ-ideals are the same thing as convex ℓ-subgroups. One easily deduces appropriate versions of the usual isomorphisms theorems. Let G be an ℓ-group, J an ℓ-ideal of G, ϕ_{J} the canonical ℓ-homomorphism associated to J. We denote by G/J the ℓ-group quotient of G by J, or, equivalently, the image of G under ϕ_{J}. We use the symbol ‘≅’ to denote isomorphism of ℓ-groups.

Free ℓ-groups. Since ℓ-groups are equationally definable, they are a variety and thus admit free (more generally, finitely presented) objects. We denote by ℱ_{κ} the free ℓ-group over κ generators, for any cardinal κ. A very strong representation theorem for ℱ_{κ} holds. Its interesting but slightly involved history is more thoroughly traced in [19], where precise references to the original results can be found — here we summarize the mathematical facts. Let n ∈ ℕ. By an ℓ-function (of n variables) we mean a continuous piecewise linear homogeneous map f : ℝ^{n} → ℝ with integral coefficients.^{8} The set 𝒞_{n} of all continuous functions from ℝ^{n} to ℝ is an ℓ-group under pointwise addition and order. Let ℒ_{n} ⊆ 𝒞_{n} denote the ℓ-subgroup of all ℓfunctions of n variables. Further, let 𝒟_{n} ⊆ 𝒞_{n} denote the ℓ-subgroup generated by the projection functions π_{i} : ℝ^{n} → ℝ, i = 1, …, n. Clearly, 𝒟_{n} ⊆ ℒ_{n} ⊂ 𝒞_{n}. Besides, a trivial universal-algebraic argument shows that 𝒟_{n} is a homomorphic image of ℱ_{n}. Non-trivial geometric and algebraic arguments yield the following strengthening of the previous statements.^{9}

Weinberg-Beynon/Chang-McNaughton Representation. ℱ_{n} ≅ 𝒟_{n} = ℒ_{n}.

Remark. One half of the theorem (ℱ_{n} ≅ 𝒟_{n}, or ℤ generates the variety of abelian ℓ-groups), is due to Weinberg ([26]). The other half (𝒟_{n} = ℒ_{n}, or the ℓ-group of ℓfunctions is generated by the projection functions) is due to Beynon ([7]). As proved by the second author ([21]; also see [10] for self-contained proofs), abelian ℓ-groups with strong order unit^{10} are categorically equivalent to M(any)V(alued)-algebras, a (natural) generalization of boolean algebras introduced by C. C. Chang in his algebraic analysis of Łukasiewcz infinite-valued propositional logic. MV-algebraic versions of Weinberg’s and Beynon’s aforementioned results were proved by Chang ([9]) and McNaughton ([20]), respectively. For the interplay between ordered groups and MV-algebras see [19] and [10].

On the basis of the above representation theorem, we shall tacitly identify ℱ_{n} and ℒ_{n} (equivalently, 𝒟_{n}) whenever convenient.

Finitely presented ℓ-groups. Let G be an ℓ-group. An ℓ-ideal J of G is generated by a set S ⊆ G, written J = ⟨⟨S⟩⟩, iff the intersection of all ℓ-ideals of G containing S is precisely J. We say that J = ⟨⟨S⟩⟩ is finitely generated iff S can be chosen finite. Further, J is principal iff it is generated by a singleton {g} ⊆ G. In this case, we write J = ⟨⟨g⟩⟩. It is an exercise to check that every finitely generated ℓ-ideal is principal. We call an ℓ-group G finitely presented iff G ≅ ℱ_{n}/⟨⟨g⟩⟩ for some n ∈ ℕ. A useful description of the principal ideal ⟨⟨g⟩⟩ is the following: f ∈ ⟨⟨g⟩⟩ iff there is some n ∈ ℕ such that ng ≥ f (in ℱ_{n}). Given f ∈ ℱ_{n}, the zeroset of f is the set 𝕍(f) = {x⃗ ∈ ℝ^{n} | f(x⃗) = 0}. More generally, for any subset S ⊆ ℱ_{n}, let us define 𝕍(S) = ⋂_{f∈S} 𝕍(f). From the description of ⟨⟨g⟩⟩ given a moment ago, it follows that 𝕍(⟨⟨g⟩⟩) = 𝕍(g). We now claim that the zeroset of an ℓ-function g is the support of a fan. Let g ∈ ℱ_{n} be an ℓ-function, and let {l_{i}} (i ∈ {1, …, m}) be the set of its linear constituents (cfr. footnote 8). Let 𝒮_{m} denote the group of permutations of m letters {1, …, m} (the symmetric group). For each σ ∈ 𝒮_{m}, we define the basic cone

8In detail, a function f : ℝ^{n} → ℝ is an ℓ-function iff it is continuous and there exist finitely many homogeneous linear functions l_{i} : ℝ^{n} → ℝ (i ∈ {1, …, m}) with integer coefficients, the linear constituents of f, such that for every x⃗ ∈ ℝ^{n} there is some i ∈ {1, …, m} such that f(x⃗) = l_{i}(x⃗).

9Generalization to an arbitrary number of generators is straightforward.

10An element u ∈ G is a strong order unit of the ℓ-group G iff for every p ∈ G_{+} there exists n ∈ ℕ such that nu ≥ p.

C_{σ} = {x⃗ ∈ ℝ^{n} | l_{σ(1)}(x⃗) ≥ … ≥ l_{σ(m)}(x⃗)} .

It is clear that C_{σ} is a closed set; also notice that it may well be the singleton {0}. In fact, C_{σ} is the intersection of finitely many integral half-spaces, as is read off directly from its definition. Hence, by the Fundamental Theorem of Polyhedra, it is a (rational polyhedral) cone. Furthermore, the set of all such C_{σ} and their finite intersections C_{σ_{1}} ∩ ⋯ ∩ C_{σ_{s}}, as σ_{1}, …, σ_{s} vary in 𝒮_{m}, is a fan Σ_{g} in N_{ℝ} ≅ ℝ^{n}. Since g is linear over each C_{σ}, its zeroset 𝕍(g) can be described as the support of a fan Θ_{g} inscribed in Σ_{g}, as we claimed. In analogy with the time-honored tradition of classical algebraic geometry, it makes sense to investigate the relationship between the ℓ-group of all ℓ-functions in ℱ_{n} restricted to |Θ_{g}| — a geometric entity — and the ℓ-group ℱ_{n}/⟨⟨g⟩⟩ — an algebraic object. In order to do this at the natural level of generality, we introduce some further notions from the theory of ordered groups.

Let G be an ℓ-group and J ⊆ G an ℓ-ideal. We say that J is maximal iff for every ℓ-ideal I ⊆ G, J ⊆ I implies J = I. Further, J is proper iff J ≠ G. The intersection of any collection of ℓ-ideals is an ℓ-ideal. The radical of G is the intersection of all its maximal ℓ-ideals. Finally, G is archimedean iff for every p, q ∈ G_{+} such that p ≤ q, there is n ∈ ℕ such that np ≰ q. Assume that G is finitely generated. Then G is archimedean iff its radical is {0}, as is easy to see.

Quite generally, if J ⊆ ℱ_{n} is an ℓ-ideal of the free ℓ-group over n variables, then 𝕍(J) is a closed conical^{11} subset of N_{ℝ}. Conversely, given a closed conical subset C ⊆ N_{ℝ}, let 𝕀(C) denote the set of all ℓ-functions of n variables vanishing over C. Then 𝕀(C) is an ℓ-ideal. Clearly, 𝕍(𝕀(C)) = C for any closed conical set C. However, J ⊆ 𝕀(𝕍(J)) may well be a strict inclusion. The following result, characterizing those ℓ-ideals J such that J = 𝕀(𝕍(J)), is the analogue of Hilbert’s Nullstellensatz.

Archimedean representation.^{12} Let G be a finitely generated ℓ-group, and assume G ≅ ℱ_{n}/J. Let ℱ_{n} ↾ 𝕍(J) denote the ℓ-group of ℓ-functions of n variables restricted to 𝕍(J). Then G ≅ ℱ_{n} ↾ 𝕍(J) iff G is archimedean iff J = 𝕀(𝕍(J)).

Thus, returning to the finitely presented ℓ-group G ≅ F_{n}/⟨⟨g⟩⟩, we need to ascertain whether G can violate the archimedean condition. It cannot.

Wójcicki-Baker’s Theorem.^{13} A finitely presented ℓ-group is archimedean.

11A subset C ⊆ ℝ^{n} is conical iff it contains, together with every point x⃗ ∈ C, the ray {px⃗ | p ∈ ℝ^{+}} generated by that point.

12Fair attribution of this result would take up an amount of space we are not ready to devote to the matter here. See [10] and [19].

13This is due to Baker (ℓ-group version, [6]) and Wójcicki (MV-algebraic version, [27]).

As an immediate consequence of the above results, a finitely presented ℓ-group can be identified with the ℓ-group of ℓ-functions of n-variables restricted to the support of a fan. Conversely, given a fan Σ in N_{ℝ}, it is possible to describe its support as the zeroset of an ℓ-function g, thus obtaining a finitely presented ℓ-group ℱ_{n}/⟨⟨g⟩⟩ associated with Σ. It turns out that this duality between fans and ℓ-groups is the germ of a full-scale categorical equivalence.

Morphisms of finitely presented ℓ-groups. The maps of fans we have hitherto considered are motivated by algebro-geometric considerations — they are the direct combinatorial translation of the notion of toric morphism. We now introduce another class of morphisms of fans which is of equal importance for the contents of this paper. Let N′ be a free abelian group of rank n′, N′_{ℝ} = ℝ ⊗ N′, Σ′ a fan in N′_{ℝ}. Let Σ be a fan in N_{ℝ} and Σ′ a fan in N′_{ℝ}. A function ϕ : |Σ| → |Σ′| is an ℓ-map iff ϕ(x⃗) = (f_{1}(x⃗), …, f_{n′}(x⃗)) for every x⃗ ∈ |Σ|, where f_{i} is an ℓ-function of n variables. The fan Σ is ℓ-embedded in Σ′ iff ϕ is injective; Σ′ is an ℓ-homomorphic image of Σ iff ϕ is surjective; Σ and Σ′ are ℓ-homeomorphic iff there is an ℓ-map ψ : |Σ′| → |Σ| such that ψ ∘ ϕ is the identity function on Σ.

Baker-Beynon Duality.^{14}

(1) An abelian ℓ-group P is finitely presented iff it is isomorphic to the ℓ-group of ℓ-functions of n ∈ ℕ variables restricted to the support of some fan Σ in N_{ℝ}. We call Σ a fan associated with P.

(2) Two finitely presented ℓ-groups P_{1}, P_{2} are isomorphic iff for every two fans Σ_{1}, Σ_{2} associated with P_{1}, P_{2}, respectively, one has that Σ_{1} and Σ_{2} are ℓ-homeomorphic.

We add a few remarks on onto and into morphisms to round off the above statement.^{15} Let P ↠ Q be a surjective homomorphism of finitely presented ℓ-groups P and Q,^{16} and let Σ be a fan associated with P. Then there is a fan Σ′ associated with Q such that |Σ′| ⊆ |Σ|, the latter inclusion being an ℓ-embedding of fans. (However, it can also be shown that there exists a surjective ℓ-map of fans ρ : |Σ| ↠ |Σ′| which retracts Σ onto Σ′, i.e., ρ restricted to Σ′ is the identity.) Conversely, consider an ℓ-embedding of fans ϕ : |Σ| ↣ |Σ′|, and let P and Q be the finitely presented ℓgroups associated with Σ and Σ′. We identify |Σ| with a subset of |Σ′|. Restricting ℓ-functions on |Σ′| (= elements of Q) to the subset |Σ|, we obtain that P is a homomorphic image of Q. (However, it can indeed be shown that P is also a subalgebra of Q, because there is an ℓ-map ρ : |Σ′| ↠ |Σ| which retracts |Σ′| onto |Σ| — that is, ρ ∘ ϕ is the identity on |Σ|.) Hence we see that ℓ-embeddings in the category of fans correspond to surjections of finitely presented ℓ-groups. The analogous results for onto ℓ-maps of fans can also be established. Finally, in the light of (2) above, these statements hold for any two fans associated with P and Q. The foregoing could be easily recast into a categorical equivalence between fans and their ℓ-maps, on the one hand, and finitely presented ℓ-groups and their homomorphisms, on the other.

14The original references are [6],[7],[8].

15Some of the assertions we are about to make are not trivial to prove. Some of the material may be novel, whence the absence of detailed references. What is not new can be found in [6],[7],[8].

16Notice that it is not possible to do without the assumption that Q be finitely presented — P may have non-archimedean (let alone non-finitely presentable) homomorphic images.

Δ-linear support functions. Perusal of any textbook on toric varieties will reveal no reference to lattice-ordered groups — in its long and distinguished history, geometry has seldom taken advantage of order-theoretical considerations.^{17} We shall now argue that finitely presented ℓ-groups do arise in a natural manner from the theory of toric varieties.

Let Δ be a fan in N_{ℝ}. Let |Δ|_{ℤ} = N ∩ |Δ| be the set of integral points in |Δ|. A function f : |Δ| → ℝ is a Δ-linear support function iff it is linear on each σ ∈ Δ, and f ↾ |Δ|_{ℤ} is integer-valued. We shall denote by SF(Δ) the set of Δlinear support functions. It is an abelian group under addition. (We remark in passing that each element of the dual lattice M yields an integral linear — a fortiori Δ-linear — support function, but this correspondence is injective iff |Δ| spans the whole space N_{ℝ}.) ¿From what we expounded above, up to isomorphism there is a unique finitely presented ℓ-group P_{Δ} associated with Δ, namely P_{Δ} ≅ ℱ_{n}/𝕀(|Δ|). It is easy to check that a Δ-linear support function is the same thing as an ℓfunction on |Δ| which is linear over each σ ∈ Σ. In other words, SF(Δ) ⊆ P_{Δ}. We thus see that SF(Δ) inherits a partial order from P_{Δ} turning SF(Δ) into a partially ordered abelian group. Notice that SF(Δ) is not an ℓ-subgroup of P_{Δ} — it is not closed under lattice operations. Given s ∈ SF(Δ), to each ray (= 1-dimensional cone) ρ_{i} = ⟨r⃗_{i}⟩ ∈ Δ^{(1)} generated by its (unique) vertex r⃗_{i}, we can associate the integral value s(r⃗_{i}); if the 1-skeleton Δ^{(1)} has cardinality t, we thus obtain for each s an element (s(r⃗_{1}), …, s(r⃗_{t})) of the free abelian group ℤ^{t}. Further, the sum of two Δ-linear support functions corresponds to the coordinatewise sum of the associated elements of ℤ^{t}. In other words, we have an embedding of abelian groups ϕ : SF(Δ) ↣ ℤ^{t}. An ℓ-group is simplicial iff it is isomorphic (as an ℓgroup) to ⟨ℤ^{k}, ≤⟩, where (v_{1}, …, v_{k}) ≤ (w_{1}, …, w_{k}) iff v_{i} ≤ w_{i} for all i ∈ {1, …, k} (coordinatewise or simplicial order). Through the group embedding ϕ, the simplicial order on ℤ^{t} induces a partial order on SF(Δ). It is an exercise to check that the latter coincides with the order induced on SF(Δ) by the inclusion SF(Δ) ⊆ P_{Δ}. Thus ϕ is naturally an embedding of partially ordered abelian groups. It is not hard to show that ϕ is onto iff Δ is unimodular. To put it differently, when Δ is unimodular, for every assignment α : {r⃗_{1}, …, r⃗_{k}} → ℤ of integral values to the vertices r⃗_{i}, there is a unique Δ-linear support function s_{α} ∈ SF(Δ) such that s_{α}(r⃗_{i}) = α(r⃗_{i}) for all i ∈ {1, …, k}; whereas when Δ is not unimodular, there exists an assignment α which cannot be thus realized by any s ∈ SF(Δ).

17One notable exception being real semi-algebraic geometry.

Now assume Δ is unimodular and complete, that is, |Δ| = N_{ℝ}, whence P_{Δ} ≅ ℱ_{n} is the free abelian ℓ-group over n generators. We remark in passing that any (unimodular) fan Σ can be extended to a complete (unimodular) fan — intuitive as this may sound, the proof is not trivial, see [15].^{18} Then ϕ : SF(Δ) ↣ ℤ^{t} is an isomorphism of partially ordered groups. In any simplicial group S, there exists a unique set of strictly positive linearly independent generators. We call such a set the simplicial basis of S. Let {h⃗_{1}, …, h⃗_{t}} ⊆ ℤ^{t} be the simplicial basis of ℤ^{t}. Via the injection SF(Δ) ⊆ P_{Δ} and the isomorphism ϕ, we identify {h⃗_{i}} with a subset of P_{Δ}, of which we are now going to provide an explicit combinatorial description. Let ρ = ⟨r⃗⟩ ∈ Δ^{(1)}, where r⃗ is the vertex of ρ. The Schauder hat at ρ ([22], [10]) is the unique continuous piecewise-linear homogeneous function h_{ρ}: N_{ℝ} → ℝ such that

(1) h_{ρ}(r⃗) = 1,

(2) h_{ρ}(v⃗) = 0 for every vertex v⃗ ≠ r⃗ of any cone in Δ^{(1)},

(3) h_{ρ} is homogeneous linear on each cone of Σ. As a consequence of the unimodularity of Δ, a Schauder hat is guaranteed to have linear pieces with integral coefficients. Thus, h_{ρ} is an element of ℱ_{n}. We denote by H_{Δ} the set of all Schauder hats at the rays of the complete unimodular fan Δ, and call it a Schauder basis of ℱ_{n}.^{19} The subgroup [H_{Δ}] generated by H_{Δ} in ℱ_{n} is easily seen to be free (as a group). In fact, SF(Δ) = [H_{Δ}] ≅ ℤ^{t} as groups. But [H_{Δ}] inherits a partial order from ℱ_{n}, and SF(Δ) = [H_{Δ}] ≅ ℤ^{t} as simplicial groups. The positive cone in [H_{Δ}] is precisely the monoid [[H_{Δ}]]. Notice that from H_{Δ} it is possible to recover Δ completely, and conversely. Further, this correspondence is algorithmic — see [10] and references therein. Thus Schauder bases are the exact ℓ-group-theoretical analogue of complete unimodular fans.

Simplicial approximations. We remark that Δ-linear support functions are the combinatorial counterpart of linear systems and divisors — formal integral linear combinations of subvarieties of codimension 1 — on toric varieties (see [23] or [15]). Thus, we are considering a partial order on (what essentially is) the group of divisors SF(Δ) of a toric variety. This is not entirely foreign to algebraic geometers, who call an element of SF(Δ) effective iff, in our terminology, it is positive in the simplicial order of SF(Δ). However, we are also order-embedding^{20} SF(Δ) into a larger lattice-ordered group P_{Δ} associated with Δ, something which, to the best of our knowledge, has no established counterpart in the algebro-geometric language. We shall now show fan-theoretically that P_{Δ} — a finitely presented structure amenable to computation — completely encodes the birational geometry of the associated toric variety. We continue to use the notation introduced in the previous subsection, and to assume that Δ is complete and unimodular. Let p ∈ ℱ_{n}^{+} be a positive ℓ-function such that p ∉ SF(Δ), and suppose Σ is a complete fan in N_{ℝ} such that p ∈ SF(Σ). As we have seen, there exists a complete unimodular refinement Σ′ ≤ Σ. Clearly, p ∈ SF(Σ′). By the De Concini-Procesi Lemma, there exists a complete unimodular fan Δ_{1} ⪯_{2} Δ such that Δ_{1} ≤ Σ′ ≤ Σ, whence p ∈ SF(Δ_{1}).^{21} Equivalently, p ∈ [[H_{Δ_{1}}]]. In any ℓ-group G, every g ∈ G decomposes uniquely as g = g^{+} − g^{−} for elements g^{+}, g^{−} ∈ G^{+} such that g^{+} ∧ g^{−} = 0; thus, the fact that we have argued for p ∈ ℱ_{n}^{+} is not restrictive. Iterating this construction we obtain:

18Completeness is the exact counterpart of algebro-geometric compactness, and completion of a fan corresponds to compactification of the associated toric variety.

19Generalizing this construction to finitely presented ℓ-groups is straightforward.

20But not lattice-embedding.

Let Δ be a complete unimodular fan in N_{ℝ}. There exists a sequence of complete unimodular fans {Δ_{i}}_{i∈ℤ^{+}} such that

(1) Δ_{0} = Δ ;

(2) Δ_{i+1} ⪯_{2} Δ_{i} for all i ∈ ℤ^{+} ;

(3) For all p ∈ ℱ_{n}^{+}, there exists i ∈ ℤ^{+} such that p ∈ [[H_{Δ_{i}}]], hence for all f ∈ ℱ_{n} there is i ∈ ℤ^{+} such that f ∈ [H_{Δ_{i}}] .

We call a sequence {Δ_{i}}_{i∈ℤ^{+}} satisfying the properties of the above statement a De Concini-Procesi sequence for Δ.

A moment’s reflection shows that [[H_{Δ_{0}}]] ⊆ [[H_{Δ_{1}}]], whence [H_{Δ_{0}}] ⊆ [H_{Δ_{1}}]. Let t_{0} and t_{1} be the cardinalities of Δ_{0}^{(1)} and Δ_{1}^{(1)}, respectively. Since [H_{Δ_{0}}] ≅ ℤ^{t_{0}} and [H_{Δ_{1}}] ≅ ℤ^{t_{1}}, we obtain an embedding of free abelian groups ϕ_{1} : ℤ^{t_{0}} ↣ ℤ^{t_{1}}; since [[H_{Δ_{0}}]] ⊆ [[H_{Δ_{1}}]], ϕ_{1} is order-preserving (with respect to the simplicial order), that is, it sends positive elements to positive elements. In other words, with respect to simplicial bases in ℤ^{t_{0}} and ℤ^{t_{1}}, ϕ_{1} is represented by a unique fullrank integral t_{1} × t_{0} matrix with nonnegative entries. Given a De Concini-Procesi sequence {Δ_{i}} as in the above, we thus obtain a sequence of simplicial groups {ℤ^{t_{i}}} and injective order-preserving group homomorphisms {ϕ_{i+1} : ℤ^{t_{i}} ↣ ℤ^{t_{i+1}}}. Quite generally, given a sequence of partially ordered abelian groups {G_{i}} and orderpreserving (not necessarily injective) group homomorphisms {ϕ_{i+1} : G_{i} → G_{i+1}}, the couple

⟨G_{i}, ϕ_{i+1}⟩

is called a direct system of partially ordered abelian groups. Every such direct system admits a unique limit group G = lim G_{i}, which is again a partially ordered abelian group. When all homomorphisms are injective, the limit is simply the group G = ⋃_{i∈ℤ^{+}} G_{i}, with positive cone G^{+} = ⋃_{i∈ℤ^{+}} G_{i}^{+}. In this case, G is called ultrasimplicial. Thus:

Simplicial approximations of free ℓ-groups. Let Δ be a complete unimodular fan in N_{ℝ}. The ℓ-group P_{Δ} ≅ ℱ_{n} is ultrasimplicial. More precisely, given any De Concini-Procesi sequence {Δ_{i}} for Δ, P_{Δ} is the limit group of the direct system

21As shown in [24], such a Δ_{1} can be effectively computed.

⟨SF(Δ_{i}), ζ_{i+1}⟩

of simplicial groups of Δ_{i}-linear support functions and injective order preserving homomorphisms ζ_{i} : SF(Δ_{i}) ↣ SF(Δ_{i+1}) induced by binary starrings Δ_{i+1} ⪯_{2} Δ_{i}.

Remarks.

1. Amongst other things, simplicial approximations of free ℓ-groups via groups of support functions give rise to a very interesting combinatorics of positive integral matrices. The key point is that the connecting morphisms ζ_{i+1} : SF(Δ_{i}) ↣ SF(Δ_{i+1}) are of a very special kind indeed — geometrically, they are binary starrings. This allows for the definition of a purely matrix-theoretical analogue of binary starring which is the subject of current research, and still far from being well-understood.

2. The above result admits far-reaching generalizations. Techniques already developed in [22] allow to prove that every finitely presented ℓ-group P_{Δ} is the limit of a direct system of simplicial groups of support functions, in close analogy with what we have just shown. The first author proved that, quite generally, every abelian ℓ-group is ultrasimplicial ([18]). While the fan-theoretic interpretation of the latter result is the subject of ongoing research, applications of the ultrasimplicial property to other mathematical realms are described in [19].

3. Direct systems of simplicial groups were originally investigated by functional analysts because of their relevance for the classification of certain C^{∗}-algebras. In general, the limit need not be lattice-ordered — not even when all maps are binary starrings. However, the limit of a direct systems of simplicial groups always is a dimension group, namely a directed, unperforated partially ordered abelian group with the Riesz Interpolation Property. See [19] for more details and references.

4. Toric varieties

We briefly introduce the basic correspondence between algebraic varieties and fans. See [23], [15] and [16] for a comprehensive account. Our terminology and notation for commutative algebra follows [5].

We shall work over a fixed algebraically closed ground field k = \overline{k} of characteristic zero, even though it is known that many results remain true in positive characteristic. We continue to use our previous notation — recall that N is a free abelian group of rank n, M is its dual ℤ-module, N_{ℝ} = ℝ ⊗ N and M_{ℝ} is the dual of N_{ℝ}.

A lattice in ℝ^{n} is a free abelian group L ⊆ ℝ^{n} of rank n. A lattice point (with respect to L) in ℝ^{n} is simply an element of L. In this terminology — a legacy of Minkowski’s geometry of numbers — N is a lattice in N_{ℝ}. Let σ be a cone in N_{ℝ}, and consider the set L_{σ} = σ ∩ N ⊆ N_{ℝ} of lattice points in σ. Since σ is convex, S is a semigroup under addition. More is true.

Gordan’s Lemma. The lattice points L_{σ} within a cone σ are a finitely generated semigroup.

[figure: σ; σ^{∨}]

Figure 3. A cone σ = ⟨(0, 1), (1, 1)⟩ and its dual σ^{∨} = ⟨(1, 0), (−1, 1)⟩.

While Gordan’s Lemma is not difficult to prove (see e.g. [15]), it is an essential result for the definition of affine toric varieties, as we now show.

Again, let σ = ⟨v⃗_{1}, …, v⃗_{m}⟩ be a cone in N_{ℝ}, and set L_{σ} = σ ∩ N. By Gordan’s Lemma, L_{σ} is in an additive finitely generated semigroup; let g_{1}, …, g_{s} be a set of generators of S. (Each g_{i} is a vector in N_{ℝ}, but we drop the vector notation for simplicity.) We switch to multiplicative notation, so that an element of L_{σ} is a Laurent monomial g_{1}^{m_{1}} ⋯ g_{s}^{m_{s}} for (not necessarily unique) natural numbers m_{i} ∈ ℕ. (Since m_{i} ∈ ℕ, it might seem sufficient to speak of monomials. However, the g_{i}’s are not free variables, and because of possible relations amongst them it is actually necessary to deal with Laurent monomials — cfr. Example 4 below.) Consider the semigroup ring k[L_{σ}]. Elements of k[L_{σ}] are Laurent polynomials, that is, finite sums of Laurent monomials in L_{σ} with coefficients in k. Clearly, then, k[L_{σ}] is a k-algebra.^{22} In fact, it is a finitely generated k-algebra, because L_{σ} is finitely generated. It is almost immediate that k[L_{σ}] is reduced. (Indeed, if the n-th power of a nonzero Laurent monomial is zero, then the nm_{i}-th power of some variable g_{i} appearing in that monomial must be zero; but this implies that g_{i} is the zero vector, a contradiction. Extension to Laurent polynomials is trivial.) Thus we can define the affine toric variety

X_{σ} = MaxSpec k[L_{σ}] .

Examples.

1. If σ = ⟨e⃗_{1}, …, e⃗_{n}⟩ is the cone generated by the canonical basis of N_{ℝ}, X_{σ} = MaxSpec k[L_{σ}] = MaxSpec k[x_{1}, …, x_{n}] is affine n-space k^{n}.

2. If σ = ⟨(a, b)⟩ ⊆ ℝ^{2}, (a, b) ≠ 0, X_{σ} = MaxSpec k[L_{σ}] = MaxSpec k[x] is the affine line k.

3. If σ = ⟨(1, 0), (1, 2)⟩ ⊆ ℝ^{2}, then {(1, 0), (1, 2)} is not a set of generators for L_{σ}, because e.g. (1, 1) is not in the submonoid spanned by {(1, 0), (1, 2)}. However, {(1, 0), (1, 1), (1, 2)} is a set of generators for L_{σ}. Hence, X_{σ} = MaxSpec k[L_{σ}] = MaxSpec k[x_{1}, x_{1}x_{2}, x_{1}x_{2}^{2}]. But k[x_{1}, x_{1}x_{2}, x_{1}x_{2}^{2}] ≅ k[X, Y, Z]/J, where J is the ideal

22Recall that a ring R is a k-algebra iff k is a subring of R; that R is reduced iff it has no nilpotent elements; and that MaxSpec is a categorical equivalence between finitely generated reduced kalgebras and affine algebraic varieties over k ([5], [12]).

Combinatorial Convexity Algebraic Geometry
cone  affine toric variety
face  subvariety
lattice N ⊆ N_{ℝ}  torus = MaxSpec k[x_{1}, x_{1}^{−1}, …, x_{n}, x_{n}^{−1}]
fan  toric variety
— complete  — compact
— unimodular  — smooth
— simplicial  — quasi-smooth
— induced by a polytope  — projective
barycentric stellar subdivision equivariant blow up
unimodular map  algebraic isomorphism
map of fans  equivariant toric morphism
cardinality of Σ^{(n)}, Σ a  Euler characteristic χ(X_{Σ})
complete unimodular fan  of a smooth compact variety

Table 1. Fan-Toric Vocabulary.

of k[X, Y, Z] generated by the polynomial Y^{2} − XZ. Thus, X_{σ} is the standard cone with vertex at the origin. In particular, X_{σ} is not “smooth” (at the origin) in any reasonable sense of the word. We shall not define smoothness in this paper; as we already mentioned, an affine toric variety X_{σ} is smooth iff σ is unimodular.

4. If σ = N_{ℝ}, X_{σ} = MaxSpec k[x_{1}, x_{1}^{−1}, …, x_{n}, x_{n}^{−1}]. The latter is known as the n-dimensional algebraic torus, and it plays an important rôle in the theory of toric varieties — see [23]. If σ is not strongly convex, it contains a nontrivial linear subspace V ⊆ N_{ℝ} of dimension d ≥ 1. In this case, it is not hard to see that the d-dimensional torus X_{V} splits off as a direct factor of X_{σ}, whence the widespread technical device of restricting attention to strongly convex cones.

The next natural step is to construct general toric varieties gluing together affine toric varieties. The data for the gluing morphisms come from a fan Σ. In carrying out this programme, it turns out that it is convenient to consider dual cones of the elements of Σ.

Let σ ⊆ N_{ℝ} be an m-dimensional cone. We set

σ^{∨} = {f ∈ M_{ℝ} | f(u⃗) ≥ 0 for every u⃗ ∈ σ}

It is not hard to check that if σ = ⟨v⃗_{1}, …, v⃗_{r}⟩, then σ^{∨} is the intersection of the half-spaces {f ∈ M_{ℝ} | f(v⃗_{i}) ≥ 0} ⊆ M_{ℝ}. Thus, by the Fundamental Theorem of Polyhedra, it is a cone in M_{ℝ}, the dual cone of σ. The terminology is appropriate in that σ^{∨∨} = σ for any cone σ. Notice that faces τ of σ coincide with sets of the form τ = σ ∩ ker f, for some f ∈ σ^{∨}.

If σ ⊆ N_{ℝ} is an n-dimensional strongly convex cone, its dual σ^{∨} can be geometrically constructed as follows. For each facet (= maximal proper face) F_{i} of σ, let p⃗_{i} be the inner normal to F_{i}. Then σ^{∨} is the positive linear hull of the finite set {p⃗_{i}}, that is, σ^{∨} = ⟨p⃗_{1}, …, p⃗_{r}⟩, where r is the number of facets of σ. Further, σ^{∨} is itself n-dimensional and strongly convex (see Figure 3).

Let Σ ⊆ N_{ℝ} be a fan of strongly convex cones. For any σ ∈ Σ, we defined L_{σ} = σ ∩ N; now, dually, set L^{σ} = σ^{∨} ∩ M. For σ, τ ∈ Σ, consider the affine toric varieties X_{σ^{∨}} and X_{τ^{∨}} — again, notice that we are passing to duals. If τ is a face of σ, then L^{σ} ⊆ L^{τ}, so that k[L^{σ}] is a subalgebra of k[L^{τ}], and X_{σ^{∨}} is an open subset of X_{τ^{∨}}. Thus, in general, the common face σ ∩ τ corresponds to a common open subset X_{(σ∩τ)^{∨}} of X_{σ^{∨}} and X_{τ^{∨}}, along which we can glue these two affine varieties together. The fan Σ, then, encodes a recipe to patch together the affine pieces X_{σ^{∨}} corresponding to the dual cones of cones σ ∈ Σ, thus yielding (the most general instance of) a toric variety, denoted X_{Σ}. This construction can be carried out without passing to duals, and produces an algebraic variety, say X^{Σ}. However, the combinatorics of the faces of the cones in Σ would be contravariant with respect to the structure of subvarieties of X^{Σ} — that is, cones of smaller dimension in Σ would correspond to larger subvarieties of X^{Σ}. Dualization gives a covariant correspondence: cones of smaller dimension in Σ correspond to smaller subvarieties of X_{Σ}.

Building on this basic construction, it is possible to carry out an extensive translation of algebro-geometric concepts into fan-theoretical language, and vice-versa. This has been done in the last decades. While we refer to [15], [23] and [16] for further information, we collect the very first few entries of the fan-toric dictionary in Table 1.

References

[1] S. Aguzzoli and D. Mundici, An algorithmic desingularization of 3-dimensional toric varieties. Tohoku Math. J., 46:557-572, 1994.

[2] J. W. Alexander. Combinatorial analysis situs. Trans. Amer. Math. Soc., 28:301–329, 1926.

[3] J. W. Alexander. The combinatorial theory of complexes. Ann. of Math. (2), 31:292–320, 1930.

[4] P. S. Alexandrov. Combinatorial topology. Vol. 1, 2 and 3. Translated from the Russian, Reprint of the 1956, 1957 and 1960 translations. Dover Publications, 1998.

[5] M. F. Atiyah and I. G. Macdonald. Introduction to commutative algebra. Addison-Wesley Publishing Co., Reading, Mass., 1969.

[6] K. A. Baker. Free vector lattices. Canad. J. Math., 20:58–66, 1968.

[7] W. M. Beynon. Duality theorems for finitely generated vector lattices. Proc. London Math. Soc. (3), 31:114–128, 1975.

[8] W. M. Beynon. Applications of duality in the theory of finitely generated lattice-ordered abelian groups. Canad. J. Math., 29(2):243–254, 1977.

[9] C. C. Chang. A new proof of the completeness of the Łukasiewicz axioms. Trans. Amer. Math. Soc., 93:74–80, 1959.

[10] R. L. O. Cignoli, I. M. L. D’Ottaviano, and D. Mundici. Algebraic foundations of many-valued reasoning. Kluwer Academic Publishers, Dordrecht, 2000.

[11] V. I. Danilov. The geometry of toric varieties. Uspekhi Mat. Nauk, 33:85–134, 1978.

[12] V. I. Danilov. Algebraic varieties and schemes. In I. R. Shafarevich (ed.), Algebraic Geometry I, Encyclopaedia of Mathematical Sciences, 23. Springer-Verlag, 1994.

[13] M. R. Darnel. Theory of lattice-ordered groups. Marcel Dekker Inc., New York, 1995.

[14] C. De Concini and C. Procesi. Complete symmetric varieties. II. Intersection theory. In Algebraic groups and related topics (Kyoto/Nagoya, 1983), North-Holland, Amsterdam, 1985.

[15] G. Ewald. Combinatorial convexity and algebraic geometry. Springer-Verlag, New York, 1996.

[16] W. Fulton. Introduction to toric varieties. Ann. of Math. Studies, 131, Princeton University Press, 1993.

[17] A. M. W. Glass, Partially Ordered Groups. (1999) World Scientific Publ. Co. Inc., River Edge, NJ.

[18] V. Marra, Every abelian ℓ-group is ultrasimplicial, Journal of Algebra, 225:872-884, 2000.

[19] V. Marra and D. Mundici. MV-algebras and abelian ℓ-groups: a fruitful interaction. In J. Martinez (ed.), Ordered Algebraic Structures, Kluwer Academic Publishers, to appear.

[20] R. McNaughton. A theorem about infinite-valued sentential logic. J. Symbolic Logic, 16:1–13, 1951.

[21] D. Mundici. Interpretation of AF C^{∗}-algebras in Łukasiewicz sentential calculus. J. Funct. Anal., 65(1):15–63, 1986.

[22] D. Mundici. Farey stellar subdivisions, ultrasimplicial groups, and K_{0} of AF C^{∗}-algebras. Adv. in Math., 68(1):23–39, 1988.

[23] T. Oda. Convex Bodies and Algebraic Geometry. Springer-Verlag, 1988.

[24] G. Panti. A geometric proof of the completeness of the Łukasiewicz calculus. J. Symbolic Logic, 60(2):563–578, 1995.

[25] G.-C. Rota. Report on the present state of combinatorics. Discrete Math., 153:289–303, 1996.

[26] E. C. Weinberg. Free lattice-ordered abelian groups. Math. Ann., 151:187–199, 1963.

[27] R. Wójcicki. On matrix representations of consequence operations of Łukasiewicz’s sentential calculi. Z. Math. Logik Grundlagen Math., 19:239–247, 1973.

(V. Marra) Department of Computer Science, University of Milan, Italy

  E-mail address: marra@dsi.unimi.it

(D. Mundici) Department of Computer Science, University of Milan, Italy

  E-mail address: mundici@mailserver.unimi.it