CONFLUENCE AND COMBINATORICS IN FINITELY

CONFLUENCE AND COMBINATORICS IN FINITELY GENERATED UNITAL LATTICE-ORDERED ABELIAN GROUPS MANUELA BUSANICHE ‡ , LEONARDO CABRER ‡ , AND DANIELE MUNDICI†

Abstract. A unital `-group (G, u) is an abelian group G equipped with a translation-invariant lattice-order and a distinguished element u, called order-unit, whose positive integer multiples eventually dominate each element of G. It is shown that, for direct systems S and T of finitely presented unital `-groups, confluence is a necessary condition for lim S ∼ = lim T . (Sufficiency is an easy byproduct of a general result). When (G, u) is finitely generated we equip it with a sequence W(G,u) = (W0 , W1 , . . .) of weighted abstract simplicial complexes, where Wt+1 is obtained from Wt either by the classical Alexander binary stellar operation, or by deleting a maximal simplex of Wt . We show that the map (G, u) 7→ W(G,u) has an inverse. A confluence criterion is given to recognize when two sequences arise from isomorphic unital `-groups.

1. Introduction This paper deals with an abelian group G equipped with a translation-invariant lattice-order and a distinguished order-unit u, i.e., an element whose positive integer multiples eventually dominate each element of G. For brevity, (G, u) will be called a unital `-group. We refer to [5, 15] for background. Morphisms in the category of unital `-groups are known as unital `-homomorphisms: they preserve the lattice, the group structure and map order-units into order-units. Whenever the context is clear, we will write for short “isomorphism” instead of “unital `-isomorphism”. Since a categorical equivalence Γ exists between unital `-groups and the equational class of MV-algebras, [20], one can naturally define free unital `-groups (Theorem 2.1), as well as finitely presented unital `-groups. The latter are defined as usual as the quotients of free finitely generated unital `-groups modulo a finitely generated congruence. Then every finitely generated unital `group is the direct limit (=filtered colimit, in categorical language) of a countable direct system of finitely presented unital `-groups with surjective connecting unital `-homomorphisms. And conversely, the direct limit of any such sequence is a finitely generated unital `-group. Two sequences of unital `-groups (1)

(G0 , u0 )  (G1 , u1 )  . . . and (H0 , v0 )  (H1 , v1 )  . . .

are said to be confluent if there are indices i(1) < j(1) < i(2) < j(2) < . . . and surjective unital `-homomorphisms fi(k) : (Gi(k) , ui(k) )  (Hj(k) , vj(k) ) and gj(k) : (Hj(k) , vj(k) )  (Gi(k+1) , ui(k+1) ) such that the composite map gj(k) ◦ fi(k) coincides with the map (Gi(k) , ui(k) )  (Gi(k+1) , ui(k+1) ) and conversely, fi(k+1) ◦ gj(k) coincides with (Hj(k) , vj(k) )  (Hj(k+1) , vj(k+1) ) in (1). Then by a standard argument [12, 2, VIII, 4.13-4.15], the confluence of the two sequences above is sufficient for their direct limits to be isomorphic. While in general categories confluence is not a necessary condition for direct limits to be isomorphic, in Theorems 3.1 and 3.3 it is proved that direct systems of unital `-groups and unital `-homomorphisms with isomorphic limits are necessarily confluent. Date: March 17, 2014. 2000 Mathematics Subject Classification. Primary: 06F20, 52B11 Secondary: 52B20, 57Q15, 46L05. Key words and phrases. Lattice-ordered abelian group, rational polyhedron, order-unit, confluence, direct system, confluent direct system, simplicial complex, abstract simplicial complex, weighted abstract simplicial complex, stellar subdivision, Alexander starring, regular fan, De Concini-Procesi theorem, piecewise linear function, Elliott classification, AF C ∗ -algebra. 1

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We next deal with finitely generated unital `-groups. In Section 4 we recall the definition of a weighted abstract simplicial complex, i.e., an (always finite) abstract simplicial complex K enriched with a weight function from the set of vertices of K into {1, 2, 3, . . .}. Using Alexander stellar operations we introduce suitable sequences of weighted abstract simplicial complexes, called stellar sequences. In Section 5 we construct a map assigning to each stellar sequence a unital `group, and in Theorem 5.1 we prove that, up to isomorphism, every finitely generated unital `-group arises from some stellar sequence. In Corollary 5.4 a necessary and sufficient condition is given to recognize when two stellar sequences yield isomorphic unital `-groups. 2. Unital `-groups, polyhedra and regular complexes Lattice-ordered abelian groups with order-unit. A lattice-ordered abelian group (`-group) is a structure (G, +, −, 0, ∨, ∧) such that (G, +, −, 0) is an abelian group, (G, ∨, ∧) is a lattice, and x + (y ∨ z) = (x + y) ∨ (x + z) for all x, y, z ∈ G. An order-unit in G (“unit´e forte” in [5]) is an element u ∈ G having the property that for every g ∈ G there is 0 ≤ n ∈ Z such that g ≤ nu. A unital `-group (G, u) is an `-group G with a distinguished order-unit u. By an `-ideal I of (G, u) we mean the kernel of a unital `-homomorphism. Any such I determines the (quotient) unital `-homomorphism (G, u) → (G, u)/I in the usual way [5, 15]. We let M([0, 1]n ) denote the unital `-group of piecewise linear continuous functions f : [0, 1]n → R, such that each piece of f has integer coefficients, with the constant function 1 as a distinguished order-unit. The number of pieces is always finite; “linear” is to be understood in the affine sense. More generally, for any nonempty subset X ⊆ [0, 1]n we denote by M(X) the unital `-group of restrictions to X of the functions in M([0, 1]n ), with the constant function 1 as the order-unit. For every f ∈ M(X) we let Z(f ) = f −1 (0). For every `-ideal I of M(X) we let Z(I) = {Y ⊆ X | ∃g ∈ I with Y = Z(g)}. The coordinate functions πi : [0, 1]n → R, (i = 1, . . . , n), together with the unit 1, generate the unital `-group M([0, 1]n ). They are said to be a free generating set of M([0, 1]n ) because they have the following universal property: Theorem 2.1. [20, 4.16] Let {g1 , . . . , gn } ⊆ [0, u] be a set of generators of a unital `-group (G, u). Then the map πi 7→ gi can be uniquely extended to a unital `-homomorphism of M([0, 1]n ) onto (G, u). Corollary 2.2. Up to isomorphism, every finitely generated unital `-group has the form M([0, 1]n ) /I for some n = 1, 2, . . . and `-ideal I of M([0, 1]n ). Rational polyhedra, complexes and regularity. Following [25, p.4], by a polyhedron P ⊆ Rn we mean a finite union of convex hulls of finite sets of points in Rn . A rational polyhedron is a finite union of convex hulls of finite sets of rational points in Rn , n = 1, 2, . . . . An example of rational polyhedron P ⊆ [0, 1]n is given by the zeroset Z(f ) of any f ∈ M([0, 1]n ). In Propositions 2.4 and 2.6 below we will see that this is the most general possible example. As an immediate consequence of the definitions we have Lemma 2.3. If P = P1 ⊇ P2 ⊇ P3 ⊇ . . . is a sequence of nonempty rational polyhedra in the n-cube, then the set hPi = {f ∈ M([0, 1]n ) | Z(f ) ⊇ Pi for some i = 1, 2, . . .} is an `-ideal of M([0, 1]n ). For any rational point y ∈ Rn we denote by den(y) the least common denominator of the coordinates of y. The integer vector y˜ = den(y)(y, 1) ∈ Zn+1 is called the homogeneous correspondent of y. For every rational m-simplex T = conv(v0 , . . . , vm ) ⊆ Rn , we will use the notation T ↑ = R≥0 v˜0 + · · · + R≥0 v˜m for the positive span of v˜0 , . . . , v˜m in Rn+1 .

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We refer to [14] for background on simplicial complexes. Unless otherwise specified, every complex K in this paper will be simplicial, and the adjective “simplicial” will be omitted. For every complex K, its support |K| is the pointset union of all simplexes of K. We say that the complex K is rational if all simplexes of K are rational: in this case, the set K↑ = {T ↑ | T ∈ K} is known as a simplicial fan [14]. A rational m-simplex T = conv(v0 , . . . , vm ) ⊆ Rn is regular if the set {˜ v0 , . . . , v˜m } is part of a basis in the free abelian group Zn+1 . A rational complex ∆ is said to be regular if every simplex T ∈ ∆ is regular. In other words, the fan ∆↑ is regular [14, V, §4]. For later use we recall here some results about regular complexes and rational polyhedra. For the proofs we refer to [18] and [22], where regular complexes are called “unimodular triangulations”. Proposition 2.4. [18, 5.1] A set X ⊆ [0, 1]n coincides with the support of some regular complex ∆ iff X = Z(f ) for some f ∈ M([0, 1]n ). Proposition 2.5. [18, 5.2] A unital `-group (G, u) is finitely presented iff there is a rational polyhedron P ∈ [0, 1]n such that (G, u) ∼ = M(P ) for some n ∈ {1, 2, . . .}. Proposition 2.6. [22, p.539] Any rational polyhedron P ⊆ [0, 1]n is the support of some regular complex ∆. Subdivision, blow-up, Farey mediant. Given complexes K and H with |K| = |H| we say that H is a subdivision of K if every simplex of H is contained in a simplex of K. For any point p ∈ |K| ⊆ Rn , the blow-up K(p) of K at p is the subdivision of K given by replacing every simplex T ∈ K that contains p by the set of all simplexes of the form conv(F ∪ {p}), where F is any face of T that does not contain p (see [14, III, 2.1] or [26, p. 376]). For any regular 1-simplex E = conv(v0 , v1 ) ⊆ Rn , the Farey mediant of E is the rational point v of E whose homogeneous correspondent v˜ coincides with v˜0 + v˜1 . If E belongs to a regular complex ∆ and v is the Farey mediant of E then the blow-up ∆(v) , called binary Farey blow-up, is a regular complex. Proposition 2.7. Suppose we are given rational polyhedra Q ⊆ P ⊆ [0, 1]n and a regular complex ∆ with support P . Then there is a subdivision ∆\ of ∆ obtained by binary Farey blow-ups, such S that Q = {T ∈ ∆\ | T ⊆ Q}. Proof. We closely follow the argument of the proof in [22, p.539]. Let us write Q = T1 ∪ . . . ∪ Tt for suitable rational simplexes. Let H = {H1 , . . . , Hh } be a set of rational half-spaces in Rn such that every Tj is the intersection of halfspaces of H. Using the De Concini-Procesi theorem [14, p.252], we obtain a sequence of regular complexes ∆ = ∆0 , ∆1 , . . . , ∆r where each ∆k+1 is obtained by blowing-up ∆k at the Farey mediant of some 1-simplex of ∆k , and for each i = 1, . . . , h, the convex polyhedron Hi ∩ [0, 1]n is a union of simplexes of ∆r . It follows that each simplex T1 , . . . , Tt is a union of simplexes of ∆r . Now ∆\ = ∆r yields the desired subdivision of ∆.  The following proposition states that every `-ideal I of M(P ) is uniquely determined by the zerosets of all functions in I: Proposition 2.8. Let P ⊆ [0, 1]n be a rational polyhedron and I an `-ideal of M(P ). Then for every f ∈ M(P ) we have: f ∈ I iff Z(f ) ⊇ Z(g) for some g ∈ I. Proof. For the nontrivial direction, suppose Z(f ) ⊇ Z(g) and without loss of generality, g ≥ 0, and f ≥ 0. We must find 0 ≤ m ∈ Z such that mg ≥ f . By Proposition 2.6, P is the support of some regular complex Λ. By suitably subdividing Λ, we obtain a rational (simplicial but not necessarily regular) complex ∆ with |∆| = P , such that over every T ∈ ∆ both f and g are linear. Let {v1 , . . . , vs } be the vertices of ∆. Since f (vi ) 6= 0 implies g(vi ) 6= 0, there exists an integer mi > 0 such that mi g(vi ) ≥ f (vi ) for each i = 1, . . . , s. Letting m = max(m1 , . . . , ms ), the desired result follows from the linearity of f and g over each simplex of ∆.  Proposition 2.9. Let I be an `-ideal of M([0, 1]m ) and P ∈ Z(I). Let further

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(i) I |`P = {f |`P | f ∈ I}, (ii) Z(I)∩P = {X ∩ P | X ∈ Z(I)}, (iii) Z(I)⊆P = {X ∈ Z(I) | X ⊆ P }. Then Z(I |`P ) = Z(I)∩P = Z(I)⊆P . Proof. [Z(I)∩P ⊆ Z(I)⊆P ]: Let X ∈ Z(I)∩P . By definition of Z(I)∩P , there exists f ∈ I such that X = Z(f ) ∩ P . Combining Propositions 2.4 and 2.6 there exists g ∈ M([0, 1]m ) such that P = Z(g). Since P ∈ Z(I), g ∈ I by Proposition 2.8. Therefore, |f | + |g| ∈ I and X = Z(f ) ∩ P = Z(|f |) ∩ Z(|g|) = Z(|f | + |g|) ∈ Z(I)⊆P . The inclusions [Z(I)⊆P ⊆ Z(I)∩P ], [Z(I |`P ) ⊆ Z(I)∩P ], and [Z(I)∩P ⊆ Z(I |`P )] immediately follow by definition.  3. Z-homeomorphism of rational polyhedra Given rational polyhedra P ⊆ Rm and Q ⊆ Rn , a piecewise linear homeomorphism η of P onto Q is said to be a Z-homeomorphism, in symbols, η : P ∼ =Z Q, if all linear pieces of η and η −1 have integer coefficients. The following first main result of this paper highlights the mutual relations between Z-homeomorphisms of polyhedra and isomorphisms of finitely generated unital `-groups, as represented by Corollary 2.2: Theorem 3.1. For any `-ideals I of M([0, 1]m ) and J of M([0, 1]n ) the following conditions are equivalent: (i) M([0, 1]m ) /I ∼ = M([0, 1]n ) /J. (ii) For some P ∈ Z(I), Q ∈ Z(J) and Z-homeomorphism η of P onto Q, the map X 7→ η(X) sends Z(I)∩P one-one onto Z(J)∩Q . Proof. (i) → (ii) Let ι : M([0, 1]m ) /I ∼ = M([0, 1]n ) /J, and  = ι−1 . Let idm denote the identity (π1 , . . . , πm ) over the m-cube, and idn the identity over the n-cube. Each element πi /I ∈ M([0, 1]m ) /I is sent by ι to some element ai /J of M([0, 1]n ) /J. Writing [0, 1] 3 ((ai /J) ∨ 0) ∧ 1 = ((ai ∨ 0) ∧ 1)/J, and replacing, if necessary, ai by (ai ∨ 0) ∧ 1, it is no loss of generality to assume that ai belongs to the unit interval of M([0, 1]n ), i.e., the range of ai is contained in the unit interval [0, 1]. Thus for a suitable m-tuple a = (a1 , . . . , am ) of functions ai ∈ M([0, 1]n ) we have a : [0, 1]n → [0, 1]m . Symmetrically, for some b = (b1 , . . . , bn ) : [0, 1]m → [0, 1]n , bj ∈ M([0, 1]m ) we can write (2)

ι : idm /I 7→ a/J and  : idn /J 7→ b/I.

For any f ∈ M([0, 1]m ) and g ∈ M([0, 1]n ), arguing by induction on the number of operations in f and g in the light of Theorem 2.1, we get the following generalization of (2): (3)

ι : f /I 7→ (f ◦ a)/J and  : g/J 7→ (g ◦ b)/I.

It follows that

f f f ◦a f ◦a◦b f = ( ◦ ι) = (ι( )) = ( )= . I I I J I By definition of the congruence induced by I, for each i = 1, . . . , m the function |πi − ai ◦ b| = |πi − πi ◦ a ◦ P b| belongs to I. Here, as usual, | · | denotes absolute value. It follows that the m function e = i=1 |πi − ai ◦ b | belongs to I, and its zeroset Z(e) belongs to Z(I). The set P = Z(e) satisfies the identity P = {x ∈ [0, 1]m | (a ◦ b)(x) = x}. One similarly notes that the set Q = {y ∈ [0, 1]n | (b ◦ a)(y) = y} belongs to Z(J). By construction, the restriction of b to P provides a Z-homeomorphism η of P onto Q, whose inverse θ is the restriction of a to Q. In symbols, (4) b |`P = η : P ∼ =Z Q, a |`Q = θ : Q ∼ =Z P. Suppose X ∈ Z(I)∩P , with the intent of proving η(X) ∈ Z(J)∩Q . By Proposition 2.9, we can write X = Z(k |`P ) for some k ∈ I. By (3), the composite function k ◦ a belongs to J. Thus

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η(X) = η(Z(k |`P )) = Z((k |`P ) ◦ θ) = Z(k ◦ a |`Q) = Q ∩ Z(k ◦ a) ∈ Z(J)∩Q . Reversing the roles of η and θ we have the required one-one correspondence X 7→ η(X) between Z(I)∩P and Z(J)∩Q . (ii) → (i) Let IP (resp., let JQ ) be the `-ideal of M([0, 1]m ) (resp., of M([0, 1]n )) given by all functions identically vanishing over P (resp., over Q). By [18, 5.2], we have isomorphisms ∼ M([0, 1]m ) /IP with α(I |`P ) = I/IP (5) α : M(P ) = and (6)

β : M(Q) ∼ = M([0, 1]n ) /JQ

with

β(J |`Q) = J/JQ .

As a particular case of a general algebraic result (sometimes called “the second isomorphism m m /IP ) /IP ) theorem”), the map fI/I 7→ fI is an isomorphism of M([0,1] onto M([0,1] . From (5)-(6) we I/IP I P have isomorphisms (7)

M([0, 1]m ) ∼ M([0, 1]m ) /IP ∼ M(P ) = = I I/IP I |`P

and M([0, 1]n ) ∼ M([0, 1]n ) /JQ ∼ M(Q) . = = J J/JQ J |`Q Letting θ = η −1 , we have θ : Q ∼ =Z P and the map λ : k 7→ k ◦ θ is an isomorphism of M(P ) onto M(Q). Further, the map Y 7→ θ(Y ) sends Z(J)∩Q = Z(J |`Q) one-one onto Z(I)∩P = Z(I |`P ). (8)

Claim. The restriction of λ to the `-ideal I |`P of M(P ) maps I |`P one-one onto J |`Q. Thus the map k λ(k) 7→ I |`P λ(I |`P ) defines an isomorphism of M(P )/(I |`P ) onto M(Q)/(J |`Q). By Proposition 2.9, for each l ∈ M(P ) if l ∈ I |`P then Z(l) ∈ Z(I |`P ) = Z(I)∩P . Thus by definition of λ, Z(λ(l)) = Z(l◦θ) = η(Z(l)) ∈ Z(J)∩Q . By Proposition 2.8, λ(l) ∈ J |`Q. Reversing the roles of λ and λ−1 , our claim is settled. Combining (7)-(8) and our claim above we have isomorphisms M([0, 1]m ) ∼ M(P ) ∼ M(Q) ∼ M([0, 1]n ) , = = = I I |`P J |`Q J as required to conclude the proof.



Using Theorem 3.1, in Theorem 3.3 below we will show that confluence is a necessary condition for two direct systems of finitely presented unital `-groups to have isomorphic direct limits. For the proof we prepare Corollary 3.2. Let P ⊆ [0, 1]m and Q ⊆ [0, 1]n be rational polyhedra. (i) M(P ) ∼ = M(Q) if and only if P ∼ =Z Q. (ii) If η is a Z-homeomorphism of Q onto some rational polyhedron R ⊆ P , the map f 7→ f ◦ η is a unital `-homomorphism of M(P ) onto M(Q). (iii) For every unital `-homomorphism h of M(P ) onto M(Q) there exists a unique Z-homeomorphism θ of Q onto some rational polyhedron R ⊆ P such that h(f ) = f ◦ θ for each f ∈ M(P ). Proof. (i) Let IP = {f ∈ M([0, 1]m ) | Z(f ) ⊇ P } and JQ = {g ∈ M([0, 1]n ) | Z(g) ⊇ Q}. By [18, 5.2], the maps α : f |`P 7→ f /IP and β : g |`Q 7→ g/JQ are isomorphisms of M(P ) onto M([0, 1]m ) /IP and of M(Q) onto M([0, 1]n ) /JQ , respectively. An application of Theorem 3.1 now settles (i). (ii) By (i), M(R) ∼ = M(Q). Let us define now the map ι : M(R) → M(Q) by ι : f 7→ f ◦ η.

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Then the proof of Theorem 3.1 shows that ι is an isomorphism of M(R) onto M(Q). The map λ : g 7→ g |`R is an `-homomorphism of M(P ) onto M(R). Thus ι ◦ λ(f ) = (f |`R) ◦ η = f ◦ η for each f ∈ M(P ), and the map f 7→ f ◦ η is a unital `-homomorphism of M(P ) onto M(Q). (iii) With reference to (i), let the unital `-homomorphism h0 of M([0, 1]m ) onto M([0, 1]n ) /JQ be defined by h0 (f ) = β(h(f |`P )). Letting I denote the kernel of h0 , it follows that IP ⊆ I and the map ι : f /I 7→ h0 (f ) is an isomorphism of M([0, 1]m ) /I onto M([0, 1]n ) /JQ . By Theorem 3.1, there exist S ∈ Z(I), T ∈ Z(JQ ) and a Z-homeomorphism η of S onto T such that the map X 7→ η(X) sends Z(I)∩S one-one onto Z(JQ )∩T . By definition of JQ and Proposition 2.6, Q is the smallest element of Z(JQ ), whence R = η −1 (Q) is the smallest element of Z(I)∩S . In the proof of Theorem 3.1, a map a : [0, 1]n → [0, 1]m is introduced having the property that ι(f /I) = (f ◦a)/JQ and η −1 = a |`T for each f ∈ M([0, 1]m ). Since Q ⊆ T , for each f ∈ M([0, 1]m ) we can write h(f |`P ) = β −1 (h0 (f )) = β −1 (ι(f /I)) = β −1 ((f ◦ a)/JQ ) = (f ◦ a) |`Q = f ◦ (η −1 |`Q). Let us define θ = η −1 |`Q. Then θ : Q ∼ =Z R, R ⊆ P ∩ S ⊆ P and h(f |`P ) = f ◦ θ. Finally, the uniqueness of θ follows from the separation property [20, 4.17], stating that for any two distinct points x, y ∈ P there is f ∈ M(P ) with f (x) = 0 and f (y) > 0.  Theorem 3.3. Given direct systems S and T of finitely presented unital `-groups with surjective connecting unital `-homomorphisms f1

f2

g1

g2

S = (G0 , u0 )  (G1 , u1 )  (G2 , u2 ) . . . T = (H0 , v0 )  (H1 , v1 )  (H2 , v2 ) . . . , let (G, u) and (H, v) denote their respective direct limits. Then the following conditions are equivalent: (i) (G, u) ∼ = (H, v). (ii) S and T are confluent. Proof. (ii)→(i) was dealt with in the Introduction. For the converse implication, Proposition 2.5 yields rational polyhedra P0 , P1 , . . . such that M(Pi ) ∼ = (Gi , ui ) for each i = 0, 1, 2, . . .. Let θ i : Pi ∼ =Z θi (Pi ) ⊆ Pi−1 be the Z-homeomorphism associated to each fi , as given by Corollary 3.2. Let the sequence P be defined by P = P00 ⊇ P10 ⊇ P20 ⊇ . . . , where P00 = P0 ⊆ [0, 1]m and Pi0 = θ1 ◦ . . . ◦ θi (Pi ) for each i = 1, 2, . . .. Once more from Corollary 3.2 we get ∼ M(Pi ) ∼ (9) (Gi , ui ) = = M(P 0 ). i

It follows that (G, u) ∼ = M([0, 1]m ) /hPi. Applying the same construction to T we obtain a sequence η1

η2

η3

[0, 1]n ⊇ Q0 ← Q1 ← Q2 ← . . . , ∼ M(Qi ) and ηi is a Z-homeomorphism of Qi onto ηi (Qi ) ⊆ Qi−1 . Let where for each i, (Hi , vi ) = Q = Q00 ⊇ Q01 ⊇ Q02 ⊇ . . . , where Q00 = Q0 and Q0i = η1 ◦ . . . ◦ ηi (Qi ) for each i = 1, 2, . . . . It follows that (10) (Hi , vi ) ∼ = M(Qi ) ∼ = M(Q0 ) and (H, v) ∼ = M([0, 1]n ) /hQi. i

By hypothesis, M([0, 1] ) /hPi ∼ = (G, u) ∼ = (H, v) ∼ = M([0, 1]n ) /hQi. By Theorem 3.1, there exist P ∈ hPi, Q ∈ hQi and a Z-homeomorphism φ : P ∼ =Z Q sending Z(hPi)∩P one-one onto Z(hQi)∩Q . By definition of hPi and hQi, there exist Pk0 and Q0l such that Pk0 ⊆ P and Q0l ⊆ Q. Thus, for each i ≥ k there exists i0 such that φ−1 (Q0i0 ) ⊆ Pi0 . Reversing the roles of φ and φ−1 it follows that for each j ≥ l there exists j 0 such that φ(Pj00 ) ⊆ Q0j . Summing up, there are indices 0 0 i(1) < j(1) < i(2) < j(2) < . . . such that φ(Pi(k) ) ⊆ Q0j(k) and φ−1 (Q0j(k) ) ⊆ Pi(k+1) for each k = 1, 2, . . . . The desired result now follows from (9) and (10), in view of Corollary 3.2.  m

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4. Weighted abstract simplicial complexes Let us recall that a (finite) abstract simplicial complex is a pair H = (V, Σ) where V is a finite nonempty set, whose elements are called the vertices of H, and Σ is a collection of subsets of V whose union is V, and with the property that every subset of an element of Σ is again an element of Σ. Following Alexander [2, p. 298], given a two-element set {v, w} ∈ Σ and a 6∈ V we define the binary subdivision ({v, w}, a) of H as the abstract simplicial complex ({v, w}, a)H obtained by adding a to the vertex set, and replacing every set {v, w, u1 , . . . , ut } ∈ Σ by the two sets {v, a, u1 , . . . , ut } and {a, w, u1 , . . . , ut } and their subsets. A weighted abstract simplicial complex is a triple W = (V, Σ, ω) where (V, Σ) is an abstract simplicial complex and ω is a map of V into the set {1, 2, 3, . . .}. For {v, w} ∈ Σ and a 6∈ V, the binary subdivision ({v, w}, a)W is the abstract simplicial complex ({v, w}, a)(V, Σ) equipped with the weight function ω ˜ : V ∪ {a} → {1, 2, 3, . . .} given by ω ˜ (a) = ω(v) + ω(w) and ω ˜ (u) = ω(u) for all u ∈ V. For every regular complex Λ, the skeleton of Λ is the weighted abstract simplicial complex WΛ = (V, Σ, ω) given by the following stipulations: (1) V = vertices of Λ. (2) For every vertex v of Λ, ω(v) = den(v). (3) For every subset W = {w1 , . . . , wk } of V, W ∈ Σ iff conv(w1 , . . . , wk ) ∈ Λ. Given two weighted abstract simplicial complexes W = (V, Σ, ω) and W 0 = (V 0 , Σ0 , ω 0 ) we write γ: W ∼ = W 0, and we say that γ is a combinatorial isomorphism between W and W 0 , if γ is a one-one map from V onto V0 such that ω 0 (γ(v)) = ω(v) for all v ∈ V, and {w1 , . . . , wk } ∈ Σ iff {γ(w1 ), . . . , γ(wk )} ∈ Σ0 for each subset {w1 , . . . , wk } of V. Definition 4.1. Let W be a weighted abstract simplicial complex and ∇ a regular complex. Then a ∇-realization of W is a combinatorial isomorphism ι between W and the skeleton W∇ of ∇. We write ι : W → ∇ to mean that ι is a ∇-realization of W . For any regular complex Λ, the identity function over the set of vertices of Λ is a Λ-realization of WΛ , called the trivial realization of the skeleton WΛ . Symmetrically, let W = (V, Σ, ω) be a weighted abstract simplicial complex with vertex set V = {v1 , . . . , vn }. For e1 , . . . , en the standard basis vectors of Rn , let ∆W be the complex whose vertices are v10 = e1 /ω(v1 ), . . . , vn0 = en /ω(vn ), and whose k-simplexes (k = 0, . . . , n) are given by 0 0 conv(vi(0) , . . . , vi(k) ) ∈ ∆W

iff

{vi(0) , . . . , vi(k) } ∈ Σ.

Note that ∆W is a regular complex and |∆W | ⊆ [0, 1]n . The function (11)

˜ι : vi ∈ V 7→ vi0 ∈ [0, 1]n

is a ∆W -realization of W , called the canonical realization of W . The dependence on the order in which the elements {v1 , . . . , vn } are listed, is tacitly understood. For later purposes, we record here the following trivial property of linear Z-homeomorphisms: Lemma 4.2. Let T = conv(v0 , . . . , vk ) ⊆ Rm and U = conv(w0 , . . . , wk ) ⊆ Rn be regular k-simplexes. If den(vi ) = den(wi ) for all i = 0, . . . , k, then there is precisely one linear Zhomeomorphism ηT of T onto U such that ηT (vi ) = wi for all i. Lemma 4.3. Let Λ and ∇ be regular complexes, with |Λ| ⊆ Rm and |∇| ⊆ Rn . We then have: (i) If θ : WΛ ∼ = W∇ is a combinatorial isomorphism between the skeletons of Λ and ∇, then there is a Z-homeomorphism ηθ of |Λ| onto |∇| such that ηθ (v) = θ(v) for each vertex v of Λ, and ηθ is linear over each simplex of Λ. (ii) Letting ∇ = ∆WΛ , it follows that the combinatorial isomorphism ˜ι of (11) between WΛ and W∇ uniquely extends to a Z-homeomorphism η˜ι of |Λ| onto |∇| such that η˜ι is linear over each simplex of Λ.

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Stellar transformations. Let W = (V, Σ, ω) and W 0 be two weighted abstract simplicial complexes. A map [ : W → W 0 is called a stellar transformation if [ is either a deletion of a maximal set of Σ, or a binary subdivision, or else [ is the identity map. A sequence W = (W0 , W1 , . . .) of weighted abstract simplicial complexes is stellar if Wj+1 is obtained from Wj by a stellar transformation. Recalling Definition 4.1 we have Lemma 4.4. Let W = (V, Σ, ω) and W 0 = (V 0 , Σ0 , ω 0 ) be two weighted abstract simplicial complexes, ∆ a regular complex, and ι a ∆-realization of W , ι : W → ∆. Suppose that [ : W → W 0 is a stellar transformation. (i) In case [ deletes a maximal set M ∈ Σ, let [(ι) : ∆ → ∆0 delete from ∆ the corresponding maximal simplex conv(ι(M )). Then the map ι0 = ι |`V 0 is a ∆0 -realization of W 0 . (ii) In case [ is the binary subdivision W 0 = ({a, b}c)W at some two-element set E = {a, b} ∈ Σ, and c 6∈ V, let e be the Farey mediant of the 1-simplex conv(ι(E)). Let [(ι) be the Farey blow-up ∆0 = ∆(e) of ∆ at e. Then the map ι0 = ι ∪ {(c, e)} is a ∆0 -realization of W 0 . Further, we have a commutative diagram [

0 W  0 yι ∆0

W  −→ ι y [(ι) ∆ −→

We say that [(ι) is the ∆-transformation of [. (It is tacitly understood that if [ is the identity map, then [(ι) : ∆ → ∆0 is the identity function.) 5. Construction of the map W 7→ G(W) In this section we will construct a map W 7→ G(W), from stellar sequences to unital `-groups and prove that the map is onto all finitely generated unital `-groups. Main Construction. Let W = W0 , W1 , . . . be a stellar sequence. For each j = 0, 1, . . . let [j be the corresponding stellar transformation sending Wj to Wj+1 . For some n ≥ 1 and regular complex ∆0 in the n-cube let ι0 be a ∆0 -realization of W0 . Then Lemma 4.4 yields a commutative diagram [

(12)

0 W 0 −→ ι y0 [0 (ι0 ) ∆0 −→

[

1 W 1 −→ ι y1 [1 (ι1 ) ∆1 −→

W 2 . . . ι y2 ∆2 . . .

The sequence of supports |∆0 | ⊇ |∆1 | ⊇ · · · is called the ∆0 -orbit of W and is denoted O(W, ∆0 ) (the role of ι0 being tacitly understood). As in Lemma 2.3, the filtering set O(W, ∆0 ) determines the `-ideal I(W, ∆0 ) = hO(W, ∆0 )i of M([0, 1]n ), as well as the unital `-group G(W, ∆0 ) = M([0, 1]n ) /I(W, ∆0 ). In the particular case when ι0 is the canonical realization of W0 we write O(W), I(W), G(W) instead of O(W, ∆W0 ), I(W, ∆W0 ), G(W, ∆W0 ). Theorem 5.1. For every finitely generated unital `-group (G, u) there is a stellar sequence W such that G(W) ∼ = (G, u). As a preliminary step for the proof we need the following immediate consequence of the definitions: Lemma 5.2. For any weighted abstract simplicial complex W and regular complexes ∇ and ∆, let ι be a ∇-realization of W , and  a ∆-realization of W . Let ηγ : |∇| → |∆| be the Z-homeomorphism of Lemma 4.3 corresponding to the combinatorial isomorphism γ =  ◦ ι−1 . Suppose the stellar transformation [ transforms W into W 0 . Let the commutative diagram

FINITELY GENERATED UNITAL `-GROUPS

[()

∆ x −→   [ W  −→ ι y [(ι) ∇ −→

(13)

9

0 ∆ x 0  0 W  0 yι ∇0

be as in Lemma 4.4. Let further γ 0 = 0 ◦ ι0−1 , and ηγ 0 be the Z-homeomorphism of |∇0 | onto |∆0 | given by Lemma 4.3. Then ηγ |`|∇0 | = ηγ 0 , whence in particular ηγ 0 is linear over each simplex of |∇0 |. We next prove Lemma 5.3. Let W = W0 , W1 , . . . be a stellar sequence. Let 0 be a ∆0 -realization of W0 and ι0 be a ∇0 -realization of W0 . Then G(W, ∆0 ) ∼ = G(W, ∇0 ). Proof. Let us write for short I = I(W, ∆0 ), J = I(W, ∇0 ). By definition of realization, there is a combinatorial isomorphism ξ of W∆0 onto W∇0 . By Lemma 4.3 (i), ξ can be extended to a Z-homeomorphism η of |∆0 | onto |∇0 |, which is linear over each simplex of ∆0 . Lemma 5.2 now yields Z-homeomorphisms ηi = η |` |∆i | : |∆i | ∼ =Z |∇i |, i = 0, 1, 2, . . . , with η |`|∆i | linear on every simplex of ∆i . In other words, we have a commutative diagram i1

i2

j1

j2

|∆ x2 | . . . x1 | ←- |∆ x0 | ←- |∆      −1 −1 η2 yη −1 η1 yη η0 yη 2 1 0 |∇0 |

←- |∇1 | ←- |∇2 | . . .

where, for each k = 1, 2, . . ., ik : |∆k | ,→ |∆k−1 | and jk : |∇k | ,→ |∇k−1 | are the inclusion maps. Corollary 3.2 ensures that the following diagram is commutative:

(14)

g1

g2

h1

h2

M(|∆ x0 |)  M(|∆ x1 |)  M(|∆ x2 |) . . . −1 α −1 α −1 α α1 y 1 α2 y 2 α0 y 0 M(|∇0 |)  M(|∇1 |)  M(|∇2 |) . . .

Here gk : M(|∆k−1 |)  M(|∆k |) (resp., hk : M(|∇k−1 |)  M(|∇k |)) are defined by gk (f ) = f |`|∆k | (resp., hk (f ) = f |`|∇k |), and αk : M(|∆k |) ∼ = M(|∇k |) are the isomorphisms defined by αk (f ) = f ◦ ηk = f ◦ η |`|∆k |. To conclude the proof we observe that G(W, |∆0 |) and G(W, |∇0 |) respectively are the direct g1

g2

h1

h2

limits of the direct systems M(|∆0 |)  M(|∆1 |)  M(|∆2 |) . . . and M(|∇0 |)  M(|∇1 |)  M(|∇2 |) . . .. From (14) it follows that G(W, |∆0 |) ∼  = G(W, |∇0 |), and the proof is complete. Proof of Theorem 5.1. By Corollary 2.2, there exists an integer n > 0 such that (G, u) is isomorphic to M([0, 1]n ) /I for some `-ideal I of M([0, 1]n ). We list the elements of I in a sequence Ti f0 , f1 , . . . Let Pi = j=0 Z(fi ), for each i = 0, 1, 2, . . . Since Z(fi ) ∈ Z(I) and Z(I) is closed under finite intersections, Pi belongs to Z(I). Moreover, for each f ∈ I there is j = 0, 1, 2, . . . such that Pj ⊆ Z(f ). Thus, (15)

h{P0 , P1 , . . .}i = I.

By Proposition 2.6, P0 is the support of a regular complex ∆0 . Proposition 2.7 yields a finite sequence of regular complexes ∆0,0 , ∆0,1 , . . . , ∆0,k0 having the following properties: (1) ∆0,0 = ∆0 ; (2) for each t = 1, 2, . . . , ∆0,t is obtained by blowing-up ∆0,t−1 at the Farey mediant of some 1-simplex E ∈ ∆0,t−1 ; (3) P1 is a union of simplexes of ∆0,k0 .

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M.BUSANICHE, L.CABRER, AND D.MUNDICI

Let the sequence of regular complexes ∆0,k0 , ∆0,k0 +1 , . . . , ∆0,r0 be obtained by the following procedure: for each i > 0, delete in ∆0,k0 +i−1 a maximal simplex T which is not contained in P1 ; denote by ∆0,k0 +i the resulting complex; stop when no such T exists. Then the sequence of skeletons W∆0,0 , . . . , W∆0,k0 , . . . , W∆0,r0 is a finite initial segment of a stellar sequence and |∆0,r0 | = P1 . Let us write ∆1,0 instead of ∆0,r0 . Proceeding inductively, we obtain a sequence S of regular complexes S = ∆0,0 , . . . , ∆1,0 , . . . , ∆2,0 , . . . , ∆j,0 , . . . such that Pj = |∆j,0 | for each j = 0, 1, 2, . . . . To conclude the proof, let W be the stellar sequence given by the skeletons of the regular complexes in S. Let ρ be the trivial ∆0 -realization of the skeleton W∆0 of ∆0 . Recalling (15) we get I(W, ∆0 ) = hO(W, ∆0,0 )i = h{|∆0,0 |, . . . , |∆1,0 |, . . . , }i = h{P0 , P1 , . . .}i = I. An application of Lemma 5.3 yields G(W) ∼ = G(W, ∆0 ) = M([0, 1]n ) /I(W, ∆0 ) = M([0, 1]n ) /I ∼ = (G, u), which concludes the proof of Theorem 5.1.



The following is an immediate consequence of Theorem 3.3: ¯ let us write O(W) = |∆0 | ⊇ |∆1 | ⊇ . . ., Corollary 5.4. For any two stellar sequences W and W ¯ = |∆¯0 | ⊇ |∆¯1 | ⊇ . . . Then the following conditions are equivalent: and O(W) ¯ (i) G(W) ∼ = G(W). (ii) For some integer i ≥ 0 there is a Z-homeomorphism η of |∆i | such that h{η(|∆i |), η(|∆i+1 |), . . .}i = ¯ hO(W)i. 6. Concluding remarks 6.1. Relations with Beynon’s work. In his Ph.D. thesis, [4, Lemma 1, pp.173-174], Beynon proves that confluence is a necessary condition for the isomorphism of the direct limit of two sequences of finitely presented `-groups. From the 20 lines of his self-contained proof we have been unable to extract any simplifying argument for our Theorems 3.1-3.3. This should come as no surprise: the proofs of several results in the theory of finitely presented `-groups need not have an analog for finitely presented unital `-groups—and vice-versa. Here are some typical examples: • By Baker-Beynon duality theory, finitely generated projective `-groups are the same as finitely presented `-groups. As shown in [8], finitely generated projective unital `-groups are a tiny fragment of finitely presented ones. • Baker-Beynon duality also yields a correspondence between abstract simplicial complexes A and finitely presented `-groups G, such that G is isomorphic to G0 iff A and A0 are connected by a path of Alexander stellar moves. This follows from the main result of Alexander’s classical paper [2]. Stellar moves are a generalization of the binary subdivisions considered in this paper, and their inverses. By contrast, the results of this paper yield, as a particular case, a correspondence between finitely presented unital `-groups (G, u) and weighted abstract simplicial complexes W , in such a way that (G, u) is isomorphic to (G0 , u0 ) iff the regular fans corresponding to W and W 0 are connected by a path of regular blow-ups and blow-downs. This follows from the proof of the weak Oda conjecture by Wlodarczyk-Morelli, [26, 19]. • As proved in [22], every finitely presented unital `-group has a faithful invariant positive unital homomorphism into R, but no finitely presented `-group G has a faithful invariant positive homomorphism into R, unless G is a finite product of integers with the product ordering. • The isomorphism problem of finitely presented `-groups is undecidable. The (un)decidability of the isomorphism problem for finitely presented unital `-groups is open. As shown in [1] for finitely presented unital `-groups with one-dimensional maximal spectral space, weighted abstract simplicial complexes and their connectability may be a key tool to settle this problem (also see [23]).

FINITELY GENERATED UNITAL `-GROUPS

11

6.2. Relations with Elliott classification. Up to isomorphism, every stellar sequence W determines a unique AF C*-algebra A = AW via the map W 7→ G(W) 7→ K0−1 (G(W)), where K0 (A) is the unital dimension group of A, [16]. Combining Elliott classification [13, 16] with Theorem 5.1, it follows that the range of the map W 7→ AW coincides (up to isomorphism) with the class of unital AF C*-algebras A whose dimension group K0 (A) is lattice-ordered and finitely generated. Various important AF C*-algebras existing in the literature belong to this class, including the Behnke-Leptin algebra with a two-point dual [3], the Effros-Shen algebras [11], and various algebras considered in [10] and [24], the universal AF C*algebra M1 of [21] (= the algebra A of [6], see [23]). Corollary 5.4 provides a simple criterion to recognize when two stellar sequences W and W 0 determine isomorphic AF C*-algebras AW and AW 0 . This criterion is a simplification of the equivalence criterion for Bratteli diagrams, [7, 2.7]. The proof of Theorem 5.1 crucially uses Proposition 2.7, which is an affine variant of the De Concini-Procesi theorem on the elimination of points of indeterminacy in toric varieties. 7. Acknowledgement We are very grateful to the referee for his careful reading of an earlier version of this paper, for his illuminating remarks, and for drawing our attention to references [1], [4] and [12]. Now he might be considered as a fourth co-author of this paper. References [1] S. Aguzzoli, V. Marra, Finitely presented MV-algebras with finite automorphism group, Journal of Logic and Computation, In press. Advance Access published on November 14, 2008; doi:10.1093/logcom/exn080 [2] J.W. Alexander, The combinatorial theory of complexes, Annals of Mathematics, 31:292-320, 1930. [3] H. Behncke, H. Leptin, C*-algebras with a two-point dual, Journal of Functional Analysis, 10:330-335, 1972. [4] W.M.Beynon, Geometric aspects of the theory of partially ordered systems, Ph. D. Thesis, Dept, of Mathematics, King’s College, The University of London, 1973. [5] A. Bigard, K. Keimel, and S. Wolfenstein, Groupes et Anneaux R´ eticul´ es, Lecture Notes in Mathematics, Springer-Verlag, Berlin, volume 608, 1971. [6] F. Boca, An AF algebra associated with the Farey tessellation, Canad. J. Math, 60:975-1000, 2008. [7] O. Bratteli, Inductive limits of finite-dimensional C*-algebras, Trans. Amer. Math. Soc., 171:195-234, 1972. [8] Cabrer, L. M., Mundici, D.: Projective MV-algebras and rational polyhedra. Algebra Universalis, Special issue in memoriam P. Conrad, (J.Mart´ınez, Ed.) DOI:10.1007/s00012-010-0039-6. [9] R. Cignoli, D.Mundici, An invitation to Chang’s MV-algebras, In Advances in Algebra and Model Theory, M. Droste, R. G¨ obel, Eds., Gordon and Breach Publishing Group, Reading, UK, 171-197, 1997. [10] R. Cignoli, G.A. Elliott, D.Mundici, Reconstructing C ∗ -algebras from their Murray von Neumann orders, Advances in Mathematics, 101:166-179, 1993. [11] E.G. Effros, C.L. Shen, Approximately finite C*-algebras and continued fractions, Indiana J. Math., 29:191204, 1980. [12] S.Eilenberg, N.Steenrod, Foundations of algebraic topology, Princeton University Press, Princeton, NJ, 1952. [13] G.A. Elliott, On the classification of inductive limits of sequences of semisimple finite-dimensional algebras, J. Algebra, 38:29-44, 1976. [14] G. Ewald, Combinatorial convexity and algebraic geometry, Springer-Verlag, New York, 1996. [15] A.M.W. Glass, W.C. Holland, (Eds.) Lattice-ordered groups, Kluwer Academic Publishers, 1989. [16] K.R. Goodearl, Notes on real and complex C∗ -algebras, Birk¨ auser, Boston, Inc. Shiva Math. Series, 5, 1982. [17] P.M. Gruber, C.G. Lekkerkerker, Geometry of numbers, 2nd edition, North-Holland, Amsterdam, 1987. [18] V. Marra, D. Mundici, The Lebesgue state of a unital abelian lattice-ordered group, Journal of Group Theory, 10:655-684, 2007. [19] R. Morelli, The birational geometry of toric varieties, Journal of Algebraic Geometry, 5:751-782, 1996. [20] D. Mundici, Interpretation of AF C ∗ -algebras in Lukasiewicz sentential calculus, Journal of Functional Analysis, 65:15-63, 1986. [21] D. Mundici, Farey stellar subdivisions, ultrasimplicial groups, and K0 of AF C ∗ -algebras, Advances in Mathematics, 68:23-39, 1988. [22] D. Mundici, The Haar theorem for lattice-ordered abelian groups with order-unit, Discrete and Continuous Dynamical Systems, 21:537-549, 2008. [23] D. Mundici, Recognizing the Farey-Stern-Brocot AF algebra, Rendiconti Lincei Mat. Appl., 20:327-338, 2009. [24] D. Mundici, C. Tsinakis, G¨ odel incompleteness in AF C*-algebras, Forum Mathematicum, 20:1071-1084, 2008. [25] J.R. Stallings, Lectures on Polyhedral Topology, Tata Institute of Fundamental Research, Mumbay, 1967.

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[26] J. Wlodarczyk, Decompositions of birational toric maps in blow-ups and blow-downs. Trans. Amer. Math. Soc., 349:373-411, 1997. ´ tica Aplicada del Litoral- FIQ, CONICET-UNL, Guemes 3450, (M.Busaniche) Instituto de Matema S3000GLN-Santa Fe, Argentina E-mail address: [email protected] ´ ticas – Facultad de Ciencias Exactas, Universidad Nacional (L.Cabrer) CONICET, Dep. de Matema del Centro de la Provincia de Buenos Aires, Pinto 399 – Tandil (7000), Argentina E-mail address: [email protected] ` degli Studi di Firenze, viale Mor(D.Mundici) Dipartimento di Matematica “Ulisse Dini”, Universita gagni 67/A, I-50134 Firenze, Italy E-mail address: [email protected]