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Classes of Ultrasimplicial Lattice-Ordered Abelian Groups
Journal of Algebra 213, 596᎐603 Ž1999. Article ID jabr.1998.7679, available online at http:rrwww.idealibrary.com on
Classes of Ultrasimplicial Lattice-Ordered Abelian Groups Daniele MundiciU Department of Computer Science, Uni¨ ersity of Milan, Via Comelico 39-41, 20135 Milan, Italy E-mail: [email protected] Communicated by Leonard Lipshitz Received February 23, 1998
A lattice-ordered abelian group is called ultrasimplicial iff every finite set of positive elements belongs to the monoid generated by some finite set of positive Z-independent elements. This property originates from Elliott’s classification of AF CU-algebras. Using fans and their desingularizations, it is proved that the ultrasimplicial property holds for every n-generated archimedean l-group whose maximal l-ideals of rank n are dense. As a corollary we obtain simpler proofs of results, respectively by Elliott and by the present author, stating that totally ordered abelian groups, as well as free l-groups, are ultrasimplicial. 䊚 1999 Academic Press Key Words: lattice-ordered abelian groups; ultrasimplicial property; Elliott’s classification; fan; desingularization; piecewise homogeneous linear function; regular fan.
INTRODUCTION By an l-group G we shall mean a lattice-ordered abelian group. As usual we let Gqs g g G N g G 04 . An l-group G is said to be ultrasimplicial iff for every finite set p1 , . . . , pk g Gq there is a finite set b1 , . . . , br g Gq such that Ži. each pi belongs to the monoid generated by b1 , . . . , br in G, and Žii. b1 , . . . , br form a linearly independent set in the Z-module G. Equivalently Žsee w6, Proposition 1x., G is the limit of a direct system of free abelian groups of finite rank and with product ordering, where the maps Z m ª Z n are order preserving group embeddings. Naturally enough, U
Partially supported by CNR-GNSAGA Project on Symbolic Computation. 596
0021-8693r99 $30.00 Copyright 䊚 1999 by Academic Press All rights of reproduction in any form reserved.
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the ultrasimplicial property originates from, and is of main relevance in, Elliott’s classification theory w4; 3, 7.7.2x Known classes of ultrasimplicial l-groups include free l-groups w7x, totally ordered abelian groups w4x, and 3-generated l-groupsᎏthose lgroups which are generated by at most three elements w8x. In this paper we shall concentrate on n-generated archimedean l-groups G, for an arbitrary integer n G 1. Let g 1 , . . . , g n be generators of G. Let maxŽ G . be the set of maximal l-ideals of G equipped with the hull-kernel ŽZariski. topology w2, Chaps. 10 and 13x: thus, for each g g G, upon calling support of g the set of maximal l-ideals of G to which g does not belong, a basis of open sets for this topology is given by the supports of elements of G. Since the element < g 1 < q ⭈⭈⭈ q< g n < is a strong unit in G, then maxŽ G . is a nonempty compact Hausdorff space. The archimedean property of G is equivalent to saying that the intersection of all maximal l-ideals of G only contain the zero element. By Holder’s theorem w1, 2x, for each J g maxŽ G . the ¨ quotient l-group GrJ is l-isomorphic to a subgroup of the additive group R of real numbers with the natural ordering. As a group, GrJ is free abelian, and its rank is some integer r F n. Let us agree to say that J is a maximal l-ideal of rank r.
1. MAIN RESULT Assuming familiarity with free l-groups w1x and with regular fans w5x we shall prove THEOREM 1.1. Let G be an n-generated archimedean l-group. If the maximal l-ideals of rank n are dense in maxŽ G . then G is ultrasimplicial. Proof. Let Fn be the free l-group on n generators. Then by a result of Birkhoff w1, p. 40x, Fn is an l-group of Žcontinuous. piecewise homogeneous linear functions f : R n ª R, each linear piece of f having integer coefficients. Further, for some l-ideal I of Fn the n-generated l-group G is l-isomorphic to the quotient l-group FnrI. Letting : Fn ª FnrI be the canonical quotient map, we have I s kerŽ .. Let VI be the intersection of the zero sets of all functions g g I, in symbols, V1 s gy1 Ž 0 . N g g I 4 . Then VI is a nonempty closed subspace of R n containing, together with each point x, its ray x N 0 F g R4 . Since G is archimedean, recalling,
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e.g., w2, 13.2.6x we can safely identify G with the l-group of restrictions to VI of all functions in Fn , in symbols G s Fn ° VI s f ° VI N f g Fn 4 .
Ž 1.
Let p1 , . . . , pk g Gq be given, with the intent of finding a set b1 , . . . , br g Gq satisfying conditions Ži. and Žii. above. By Ž1., there are functions f 1 , . . . , f k g Fn such that pi s f i ° VI ,
f G 0 Ž i s 1, . . . , k . .
Ž 2.
Since each f i is piecewise homogeneous linear with integer coefficients, there is a complete fan ⌺ over R n such that each cone of ⌺ is the positive span in R n of finitely many vectors, whose coordinates are integers, and each f i is homogeneous linear over . In other words, there are integers c i1 , . . . , c i n such that f i Ž x . s c i1 x 1 q ⭈⭈⭈ qc i n x n ,
for each x s Ž x 1 , . . . , x n . g .
Ž 3.
As in w5, Theorem V, 4.2x, ⌺ can be refined to the simplicial fan ⌺X Žin the sense that each cone of ⌺X is the positive span of linearly independent vectors in R n . without adding new one-dimensional cones Žs rays.. Indeed, by a routine desingularization procedure w5, Proof of Theorem VI, 8.5x, ⌺X can be refined to a regular complete fan ⌺U . In other words, Ža. ⌺U is a simplicial fan over all of R n, Žb. each n-dimensional cone of ⌺U has the form s R G 0 v1 q ⭈⭈⭈ qR G 0 vn s Ý nis1 ␣ i vi N 0 F ␣ i g R4 , where each vi is a primiti¨ e column vector in Z n Ži.e., the greatest common divisor of the coordinates of vi equals 1.; the uniquely determined primitive vectors v1 , . . . , vn are called the primiti¨ e generating ¨ ectors of , Žc. the n = n matrix M s Žv1 , . . . , vn . is unimodular, i.e., det M s "1. A fortiori, each f i is homogeneous linear over , with integer coefficients as in formula Ž3.. Let genŽ ⌺U . be the set of primitive generating vectors of Žall cones in. ⌺U . For each v g genŽ ⌺U . let starŽv. be the closed star of v in ⌺U . Thus, starŽv. is the union of all n-dimensional cones of ⌺U having v as a primitive generating vector. Let f v : R n ª R be the uniquely determined continuous function satisfying the conditions ŽU . ŽUU .
f v Žv. s 1, f v Žw. s 0, whenever v / w g genŽ ⌺U . f v is homogeneous linear over each n-dimensional cone of ⌺U .
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From the unimodularity of the integer matrix M , for each n-dimensional cone g ⌺U it follows that the homogeneous linear piece f v ° of f v has integer coefficients. As a matter of fact, in case is not contained in starŽv., then f v constantly vanishes over ; otherwise, letting v1 , . . . , vn be the primitive generating vectors of , assuming for some j s 1, . . . , n, v to coincide with vj , and letting My1 be the inverse matrix of M , the jth row of My1 yields an n-tuple of integers a j s Ž a j1 , . . . , a jn . such that for all x s Ž x1, . . . , x n . g , f v Ž x . s a j ⭈ x s a j1 x 1 q ⭈⭈⭈ qa jn x n .
Ž 4.
Since Fn coincides with the l-group of all Žcontinuous. piecewise homogeneous linear functions over R n with integer coefficients w1, Theorem 6.3x, then f v g Fn for each v g genŽ ⌺U .. Let the finite set K : Fn be defined by K s f v N v g genŽ ⌺U .4 . By Ž3. and condition ŽU ., for each i s 1, . . . , k there exists a unique linear combination g i s Ý v c iv f v with integer coefficients c iv G 0 such that f i coincides with g i at each point v g genŽ ⌺U .. Since both f i and g i are homogeneous linear over each n-dimensional cone of ⌺U , it follows that f i coincides with g i over all of R n, and hence each f i belongs to the monoid generated by K in Fn . For each v g genŽ ⌺U . we have the identity R n _ f vy1 Ž 0 . s int star Ž v . ,
Ž 5.
the latter denoting the interior of the star of v. Thus, by Ž1., for each function f g Fn we have f g I iff VI : f y1 Ž0. iff int starŽv. l VI s ⭋. Let H s K _ I s f v g K N int star Ž v . l VI / ⭋ 4
Ž 6.
B s H ° VI s f v ° VI N f v g H 4 .
Ž 7.
and
Then by Ž1., B : G, and for a suitable subset v1 , . . . , vr of genŽ ⌺U . we can write B s b1 , . . . , br 4 ,
where b1 s f v1 ° VI , . . . , br s f v r ° VI .
Ž 8.
Since each function f v g K _ H vanishes over VI and each f i is expressible as a linear combination of the functions in K with integer coefficients G 0, then by Ž2. every pi belongs to the monoid generated by b1 , . . . , br in G s Fn ° VI , and condition Ži. is satisfied.
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To prove condition Žii. we shall settle the following Claim. If c1 b1 q ⭈⭈⭈ qc r br s 0 and c1 , . . . , c r g Z then c1 s ⭈⭈⭈ s c r s 0. It is enough to prove c1 s 0. Under the above identification Ž1. the set maxŽ G . of maximal l-ideals of G is in one-one correspondence with VI via the map J g max Ž G . ¬ x J s fy1 Ž 0 . N f g J 4 . The inverse correspondence is given by x g VI ¬ Jx s f g G N f Ž x . s 0 4 . As is well known, this correspondence is in fact a homeomorphism between the space maxŽ G . with the hull-kernel topology w2, Chap. 10x and the space VI with the natural topology inherited from R n by restriction. Since by Ž6. ᎐ Ž8., int starŽv1 . l VI / ⭋, and by Ž5. the set int starŽv1 . is open in VI , then by our denseness assumption there is a point z s Ž z1 , . . . , z n . g int starŽv1 . l VI such that rank Ž GrJz . s n.
Ž 9.
Again by Holder’s theorem the evaluation map ¨ f g GrJz s G ° z4 ¬ f Ž z. g R
Ž 10 .
is an l-isomorphism of GrJz onto a subgroup of R with the natural ordering. Let 1 , . . . , n be the canonical set of free generators of Fn . Thus j : R n ª R is the jth identity function j Ž x 1 , . . . , x n . s x j . The coordinates z1 s 1Žz., . . . , z n s nŽz. are generators of GrJz , once the latter is identified with its image in R under the evaluation map Ž10.. From Ž9. we get that z1 , . . . , z n are linearly independent in the Z-module R. Thus d1 z1 q ⭈⭈⭈ qd n z n s 0,
d1 , . . . , d n g Z implies
d1 s ⭈⭈⭈ s d n s 0.
Ž 11 . Since the primitive generating vectors of every n-dimensional cone of ⌺U have integer coordinates, then z must lie in the interior of a unique
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n-dimensional cone of ⌺U containing v1 among its primitive generating vectors. For suitable primitive generating vectors wj g Z n we can write
s R G 0 w 1 q ⭈⭈⭈ qR G 0 wn , where we can safely assume v1 s w 1. By construction of ⌺U , for all i s 1, . . . , r the function bi g G is homogeneous linear over with integer coefficients, say bi Ž x . s a i1 x 1 q ⭈⭈⭈ qa i n x n
for all x s Ž x 1 , . . . , x n . g .
Ž 12 .
Assuming without loss of generality v2 s w 2 , . . . , vn s wn it follows that bj Ž z . s 0
for all j s n q 1, . . . , r .
Ž 13 .
As noted in Ž4., for each i s 1, . . . , n, the n-tuple a i s Ž a i1 , . . . , a i n . is given by the ith row of the matrix My1 s Žw 1 , . . . , wn .y1 ; further, since M is unimodular and has integer entries, then so does My1 . From the current hypothesis in our claim, together with Ž13., since z g VI we have c1 b 1Žz. q ⭈⭈⭈ qc n bnŽz. s 0. By Ž12., 0 s c1 a 1 ⭈ z q ⭈⭈⭈ qc n a n ⭈ z s Ž c1 a 1 q ⭈⭈⭈ qc n a n . ⭈ z. By Ž11. the vector c1 a 1 q ⭈⭈⭈ qc n a n g Z n must have all its coordinates equal to zero. Since the row vectors a 1 , . . . , a n of My1 are linearly independent in R n, we obtain the required conclusion c1 s ⭈⭈⭈ s c n s 0. Having thus proved our claim, we have shown that the elements b1 , . . . , br g Gq also satisfy condition Žii., and the theorem is proved. COROLLARY 1.2 w7, Lemma 2.2x.
E¨ ery free l-group F is ultrasimplicial.
Proof. It is sufficient to assume that F is finitely generated. Then F coincides with Fn for a uniquely determined integer n G 1. Given elements f 1 , . . . , f k g Fnq , the first part of the proof of the above theorem yields a complete regular fan ⌺U over R n, and a finite set K s f v N v g genŽ ⌺U .4 : Fnq such that every f i belongs to the monoid generated by K in Fn . Trivially, if a linear combination Ý v c v f v constantly vanishes over R n then for each w g genŽ ⌺U ., c w s 0 because f w is the only element of K which is nonzero at w. COROLLARY 1.3 w4, Theorem 2.2x. is ultrasimplicial.
E¨ ery totally ordered abelian group G
Proof. It is sufficient to assume G finitely generated. By Hahn’s embedding theorem w1, 2x, we can write G as a lexicographic product ª ª ª G s G1 = ⭈⭈⭈ = Gm , where = denotes lexicographic product from the left, and each Gi is a finitely generated subgroup of R with the natural total ordering. The proof is by induction on m.
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Base. Let n be the rank of the finitely generated torsion free abelian group G s G1. Then G is n-generated also as an l-group, 04 is its only maximal l-ideal, and the rank of this l-ideal Žs rank of Gr 04. is n. The desired conclusion now follows from the above theorem. ª ª ˜ s G2 = Induction Step. Let G ⭈⭈⭈ = Gm . By the induction hypothesis, ˜ are ultrasimplicial; further, G˜ is the only maximal l-ideal both G1 and G ˜ We shall tacitly identify of G and G is the group direct sum of G1 and G. ˜ G1 and G with their images in G, using the canonical embeddings ˜ ¬ Ž0, x . g G and y g G1 ¬ Ž y, 0. g G. Let p1 , . . . , pk g Gq, say xgG ˜Ž . pi s u i q ¨ i , u i g Gq 1 , ¨ i g G i s 1, . . . , k . Since G 1 is ultrasimplicial, there are Z-independent positive elements b1 , . . . , br g G1 and integers c i1 , . . . , c i r G 0 such that u i s c i1 b1 q ⭈⭈⭈ qc i r br
Ž i s 1, . . . , k . .
We can safely assume that for some t F k the elements u1 , . . . , u t are ˜q. Let ) 0, while u tq1 s ⭈⭈⭈ s u k s 0. It follows that ¨ tq1 , . . . , ¨ k g G X X Ž< < < <. e s max ¨ 1 , . . . , ¨ t , b1 s b1 y e, . . . , br s br y e. Then by definition of lexicographic product, bX1 , . . . , brX are Z-independent positive elements of G. Letting d1 , . . . , d t be defined by d i s pi y Ž c i1 bX1 q ⭈⭈⭈ qc i r brX ., by ˜q. Since G˜ is ultrasimplicial there exist construction we have d i g G ˜ such that each of Z-independent positive elements brq1 , . . . , bs g G d1 , . . . , d t , ¨ tq1 , . . . , ¨ k belongs to the monoid generated by brq1 , . . . , bs in ª ˜ In conclusion, since G s G1 = ˜ the elements bX1 , . . . , brX , brq1 , . . . , G. G, bs g Gq are Z-independent in G, and each of p1 , . . . , pk belongs to the monoid generated by bX1 , . . . , brX , brq1 , . . . , bs in G, as required.
FURTHER RESEARCH In the light of Corollary 1.2, in order to solve the problem whether e¨ ery l-group is ultrasimplicial it is sufficient to prove that the ultrasimplicial property is preserved under formation of l-homomorphic images. Unfortunately, the argument in the proof of w6, Theorem 3Žii.x isᎏ to say the least ᎏuninformative, and the problem is still open.
REFERENCES 1. M. Anderson and T. Feil, ‘‘Lattice-Ordered Groups,’’ Reidel, Dordrecht, 1988. 2. A. Bigard, K. Keimel, and S. Wolfenstein, ‘‘Groupes et Anneaux Reticules,’’ ´ ´ Lecture Notes in Mathematics, Vol. 608, Springer-Verlag, Berlin, 1971.
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3. B. Blackadar, ‘‘K-Theory for Operator Algebras,’’ MSRI Publications, Springer-Verlag, Berlin, 1986. 4. G. A. Elliott, On totally ordered groups, and K 0 , in Lecture Notes in Mathematics, Vol. 734, pp. 1᎐49, Springer-Verlag, Berlin, 1979. 5. G. Ewald, ‘‘Combinatorial Convexity and Algebraic Geometry,’’ Graduate Texts in Mathematics, Vol. 168, Springer-Verlag, New YorkrBerlin, 1996. 6. D. Handelman, Ultrasimplicial dimension groups, Arch. Math. Ž Basel . 40 Ž1983., 109᎐115. 7. D. Mundici, Farey stellar subdivisions, ultrasimplicial groups and K 0 of AF CU-algebras, Ad¨ . Math. 68 Ž1988., 23᎐39. 8. D. Mundici and G. Panti, A constructive proof that every 3-generated l-group is ultrasimplicial, in ‘‘Proceedings, Mini-semester, Warsaw, 1996 Žmemoriam Helena Rasiowa.,’’ Banach Centre Publ. Polish Acad. Sci., Warsaw, 1999.
Classes of Ultrasimplicial Lattice-Ordered Abelian Groups Daniele MundiciU Department of Computer Science, Uni¨ ersity of Milan, Via Comelico 39-41, 20135 Milan, Italy E-mail: [email protected] Communicated by Leonard Lipshitz Received February 23, 1998
A lattice-ordered abelian group is called ultrasimplicial iff every finite set of positive elements belongs to the monoid generated by some finite set of positive Z-independent elements. This property originates from Elliott’s classification of AF CU-algebras. Using fans and their desingularizations, it is proved that the ultrasimplicial property holds for every n-generated archimedean l-group whose maximal l-ideals of rank n are dense. As a corollary we obtain simpler proofs of results, respectively by Elliott and by the present author, stating that totally ordered abelian groups, as well as free l-groups, are ultrasimplicial. 䊚 1999 Academic Press Key Words: lattice-ordered abelian groups; ultrasimplicial property; Elliott’s classification; fan; desingularization; piecewise homogeneous linear function; regular fan.
INTRODUCTION By an l-group G we shall mean a lattice-ordered abelian group. As usual we let Gqs g g G N g G 04 . An l-group G is said to be ultrasimplicial iff for every finite set p1 , . . . , pk g Gq there is a finite set b1 , . . . , br g Gq such that Ži. each pi belongs to the monoid generated by b1 , . . . , br in G, and Žii. b1 , . . . , br form a linearly independent set in the Z-module G. Equivalently Žsee w6, Proposition 1x., G is the limit of a direct system of free abelian groups of finite rank and with product ordering, where the maps Z m ª Z n are order preserving group embeddings. Naturally enough, U
Partially supported by CNR-GNSAGA Project on Symbolic Computation. 596
0021-8693r99 $30.00 Copyright 䊚 1999 by Academic Press All rights of reproduction in any form reserved.
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597
the ultrasimplicial property originates from, and is of main relevance in, Elliott’s classification theory w4; 3, 7.7.2x Known classes of ultrasimplicial l-groups include free l-groups w7x, totally ordered abelian groups w4x, and 3-generated l-groupsᎏthose lgroups which are generated by at most three elements w8x. In this paper we shall concentrate on n-generated archimedean l-groups G, for an arbitrary integer n G 1. Let g 1 , . . . , g n be generators of G. Let maxŽ G . be the set of maximal l-ideals of G equipped with the hull-kernel ŽZariski. topology w2, Chaps. 10 and 13x: thus, for each g g G, upon calling support of g the set of maximal l-ideals of G to which g does not belong, a basis of open sets for this topology is given by the supports of elements of G. Since the element < g 1 < q ⭈⭈⭈ q< g n < is a strong unit in G, then maxŽ G . is a nonempty compact Hausdorff space. The archimedean property of G is equivalent to saying that the intersection of all maximal l-ideals of G only contain the zero element. By Holder’s theorem w1, 2x, for each J g maxŽ G . the ¨ quotient l-group GrJ is l-isomorphic to a subgroup of the additive group R of real numbers with the natural ordering. As a group, GrJ is free abelian, and its rank is some integer r F n. Let us agree to say that J is a maximal l-ideal of rank r.
1. MAIN RESULT Assuming familiarity with free l-groups w1x and with regular fans w5x we shall prove THEOREM 1.1. Let G be an n-generated archimedean l-group. If the maximal l-ideals of rank n are dense in maxŽ G . then G is ultrasimplicial. Proof. Let Fn be the free l-group on n generators. Then by a result of Birkhoff w1, p. 40x, Fn is an l-group of Žcontinuous. piecewise homogeneous linear functions f : R n ª R, each linear piece of f having integer coefficients. Further, for some l-ideal I of Fn the n-generated l-group G is l-isomorphic to the quotient l-group FnrI. Letting : Fn ª FnrI be the canonical quotient map, we have I s kerŽ .. Let VI be the intersection of the zero sets of all functions g g I, in symbols, V1 s gy1 Ž 0 . N g g I 4 . Then VI is a nonempty closed subspace of R n containing, together with each point x, its ray x N 0 F g R4 . Since G is archimedean, recalling,
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e.g., w2, 13.2.6x we can safely identify G with the l-group of restrictions to VI of all functions in Fn , in symbols G s Fn ° VI s f ° VI N f g Fn 4 .
Ž 1.
Let p1 , . . . , pk g Gq be given, with the intent of finding a set b1 , . . . , br g Gq satisfying conditions Ži. and Žii. above. By Ž1., there are functions f 1 , . . . , f k g Fn such that pi s f i ° VI ,
f G 0 Ž i s 1, . . . , k . .
Ž 2.
Since each f i is piecewise homogeneous linear with integer coefficients, there is a complete fan ⌺ over R n such that each cone of ⌺ is the positive span in R n of finitely many vectors, whose coordinates are integers, and each f i is homogeneous linear over . In other words, there are integers c i1 , . . . , c i n such that f i Ž x . s c i1 x 1 q ⭈⭈⭈ qc i n x n ,
for each x s Ž x 1 , . . . , x n . g .
Ž 3.
As in w5, Theorem V, 4.2x, ⌺ can be refined to the simplicial fan ⌺X Žin the sense that each cone of ⌺X is the positive span of linearly independent vectors in R n . without adding new one-dimensional cones Žs rays.. Indeed, by a routine desingularization procedure w5, Proof of Theorem VI, 8.5x, ⌺X can be refined to a regular complete fan ⌺U . In other words, Ža. ⌺U is a simplicial fan over all of R n, Žb. each n-dimensional cone of ⌺U has the form s R G 0 v1 q ⭈⭈⭈ qR G 0 vn s Ý nis1 ␣ i vi N 0 F ␣ i g R4 , where each vi is a primiti¨ e column vector in Z n Ži.e., the greatest common divisor of the coordinates of vi equals 1.; the uniquely determined primitive vectors v1 , . . . , vn are called the primiti¨ e generating ¨ ectors of , Žc. the n = n matrix M s Žv1 , . . . , vn . is unimodular, i.e., det M s "1. A fortiori, each f i is homogeneous linear over , with integer coefficients as in formula Ž3.. Let genŽ ⌺U . be the set of primitive generating vectors of Žall cones in. ⌺U . For each v g genŽ ⌺U . let starŽv. be the closed star of v in ⌺U . Thus, starŽv. is the union of all n-dimensional cones of ⌺U having v as a primitive generating vector. Let f v : R n ª R be the uniquely determined continuous function satisfying the conditions ŽU . ŽUU .
f v Žv. s 1, f v Žw. s 0, whenever v / w g genŽ ⌺U . f v is homogeneous linear over each n-dimensional cone of ⌺U .
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From the unimodularity of the integer matrix M , for each n-dimensional cone g ⌺U it follows that the homogeneous linear piece f v ° of f v has integer coefficients. As a matter of fact, in case is not contained in starŽv., then f v constantly vanishes over ; otherwise, letting v1 , . . . , vn be the primitive generating vectors of , assuming for some j s 1, . . . , n, v to coincide with vj , and letting My1 be the inverse matrix of M , the jth row of My1 yields an n-tuple of integers a j s Ž a j1 , . . . , a jn . such that for all x s Ž x1, . . . , x n . g , f v Ž x . s a j ⭈ x s a j1 x 1 q ⭈⭈⭈ qa jn x n .
Ž 4.
Since Fn coincides with the l-group of all Žcontinuous. piecewise homogeneous linear functions over R n with integer coefficients w1, Theorem 6.3x, then f v g Fn for each v g genŽ ⌺U .. Let the finite set K : Fn be defined by K s f v N v g genŽ ⌺U .4 . By Ž3. and condition ŽU ., for each i s 1, . . . , k there exists a unique linear combination g i s Ý v c iv f v with integer coefficients c iv G 0 such that f i coincides with g i at each point v g genŽ ⌺U .. Since both f i and g i are homogeneous linear over each n-dimensional cone of ⌺U , it follows that f i coincides with g i over all of R n, and hence each f i belongs to the monoid generated by K in Fn . For each v g genŽ ⌺U . we have the identity R n _ f vy1 Ž 0 . s int star Ž v . ,
Ž 5.
the latter denoting the interior of the star of v. Thus, by Ž1., for each function f g Fn we have f g I iff VI : f y1 Ž0. iff int starŽv. l VI s ⭋. Let H s K _ I s f v g K N int star Ž v . l VI / ⭋ 4
Ž 6.
B s H ° VI s f v ° VI N f v g H 4 .
Ž 7.
and
Then by Ž1., B : G, and for a suitable subset v1 , . . . , vr of genŽ ⌺U . we can write B s b1 , . . . , br 4 ,
where b1 s f v1 ° VI , . . . , br s f v r ° VI .
Ž 8.
Since each function f v g K _ H vanishes over VI and each f i is expressible as a linear combination of the functions in K with integer coefficients G 0, then by Ž2. every pi belongs to the monoid generated by b1 , . . . , br in G s Fn ° VI , and condition Ži. is satisfied.
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To prove condition Žii. we shall settle the following Claim. If c1 b1 q ⭈⭈⭈ qc r br s 0 and c1 , . . . , c r g Z then c1 s ⭈⭈⭈ s c r s 0. It is enough to prove c1 s 0. Under the above identification Ž1. the set maxŽ G . of maximal l-ideals of G is in one-one correspondence with VI via the map J g max Ž G . ¬ x J s fy1 Ž 0 . N f g J 4 . The inverse correspondence is given by x g VI ¬ Jx s f g G N f Ž x . s 0 4 . As is well known, this correspondence is in fact a homeomorphism between the space maxŽ G . with the hull-kernel topology w2, Chap. 10x and the space VI with the natural topology inherited from R n by restriction. Since by Ž6. ᎐ Ž8., int starŽv1 . l VI / ⭋, and by Ž5. the set int starŽv1 . is open in VI , then by our denseness assumption there is a point z s Ž z1 , . . . , z n . g int starŽv1 . l VI such that rank Ž GrJz . s n.
Ž 9.
Again by Holder’s theorem the evaluation map ¨ f g GrJz s G ° z4 ¬ f Ž z. g R
Ž 10 .
is an l-isomorphism of GrJz onto a subgroup of R with the natural ordering. Let 1 , . . . , n be the canonical set of free generators of Fn . Thus j : R n ª R is the jth identity function j Ž x 1 , . . . , x n . s x j . The coordinates z1 s 1Žz., . . . , z n s nŽz. are generators of GrJz , once the latter is identified with its image in R under the evaluation map Ž10.. From Ž9. we get that z1 , . . . , z n are linearly independent in the Z-module R. Thus d1 z1 q ⭈⭈⭈ qd n z n s 0,
d1 , . . . , d n g Z implies
d1 s ⭈⭈⭈ s d n s 0.
Ž 11 . Since the primitive generating vectors of every n-dimensional cone of ⌺U have integer coordinates, then z must lie in the interior of a unique
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n-dimensional cone of ⌺U containing v1 among its primitive generating vectors. For suitable primitive generating vectors wj g Z n we can write
s R G 0 w 1 q ⭈⭈⭈ qR G 0 wn , where we can safely assume v1 s w 1. By construction of ⌺U , for all i s 1, . . . , r the function bi g G is homogeneous linear over with integer coefficients, say bi Ž x . s a i1 x 1 q ⭈⭈⭈ qa i n x n
for all x s Ž x 1 , . . . , x n . g .
Ž 12 .
Assuming without loss of generality v2 s w 2 , . . . , vn s wn it follows that bj Ž z . s 0
for all j s n q 1, . . . , r .
Ž 13 .
As noted in Ž4., for each i s 1, . . . , n, the n-tuple a i s Ž a i1 , . . . , a i n . is given by the ith row of the matrix My1 s Žw 1 , . . . , wn .y1 ; further, since M is unimodular and has integer entries, then so does My1 . From the current hypothesis in our claim, together with Ž13., since z g VI we have c1 b 1Žz. q ⭈⭈⭈ qc n bnŽz. s 0. By Ž12., 0 s c1 a 1 ⭈ z q ⭈⭈⭈ qc n a n ⭈ z s Ž c1 a 1 q ⭈⭈⭈ qc n a n . ⭈ z. By Ž11. the vector c1 a 1 q ⭈⭈⭈ qc n a n g Z n must have all its coordinates equal to zero. Since the row vectors a 1 , . . . , a n of My1 are linearly independent in R n, we obtain the required conclusion c1 s ⭈⭈⭈ s c n s 0. Having thus proved our claim, we have shown that the elements b1 , . . . , br g Gq also satisfy condition Žii., and the theorem is proved. COROLLARY 1.2 w7, Lemma 2.2x.
E¨ ery free l-group F is ultrasimplicial.
Proof. It is sufficient to assume that F is finitely generated. Then F coincides with Fn for a uniquely determined integer n G 1. Given elements f 1 , . . . , f k g Fnq , the first part of the proof of the above theorem yields a complete regular fan ⌺U over R n, and a finite set K s f v N v g genŽ ⌺U .4 : Fnq such that every f i belongs to the monoid generated by K in Fn . Trivially, if a linear combination Ý v c v f v constantly vanishes over R n then for each w g genŽ ⌺U ., c w s 0 because f w is the only element of K which is nonzero at w. COROLLARY 1.3 w4, Theorem 2.2x. is ultrasimplicial.
E¨ ery totally ordered abelian group G
Proof. It is sufficient to assume G finitely generated. By Hahn’s embedding theorem w1, 2x, we can write G as a lexicographic product ª ª ª G s G1 = ⭈⭈⭈ = Gm , where = denotes lexicographic product from the left, and each Gi is a finitely generated subgroup of R with the natural total ordering. The proof is by induction on m.
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Base. Let n be the rank of the finitely generated torsion free abelian group G s G1. Then G is n-generated also as an l-group, 04 is its only maximal l-ideal, and the rank of this l-ideal Žs rank of Gr 04. is n. The desired conclusion now follows from the above theorem. ª ª ˜ s G2 = Induction Step. Let G ⭈⭈⭈ = Gm . By the induction hypothesis, ˜ are ultrasimplicial; further, G˜ is the only maximal l-ideal both G1 and G ˜ We shall tacitly identify of G and G is the group direct sum of G1 and G. ˜ G1 and G with their images in G, using the canonical embeddings ˜ ¬ Ž0, x . g G and y g G1 ¬ Ž y, 0. g G. Let p1 , . . . , pk g Gq, say xgG ˜Ž . pi s u i q ¨ i , u i g Gq 1 , ¨ i g G i s 1, . . . , k . Since G 1 is ultrasimplicial, there are Z-independent positive elements b1 , . . . , br g G1 and integers c i1 , . . . , c i r G 0 such that u i s c i1 b1 q ⭈⭈⭈ qc i r br
Ž i s 1, . . . , k . .
We can safely assume that for some t F k the elements u1 , . . . , u t are ˜q. Let ) 0, while u tq1 s ⭈⭈⭈ s u k s 0. It follows that ¨ tq1 , . . . , ¨ k g G X X Ž< < < <. e s max ¨ 1 , . . . , ¨ t , b1 s b1 y e, . . . , br s br y e. Then by definition of lexicographic product, bX1 , . . . , brX are Z-independent positive elements of G. Letting d1 , . . . , d t be defined by d i s pi y Ž c i1 bX1 q ⭈⭈⭈ qc i r brX ., by ˜q. Since G˜ is ultrasimplicial there exist construction we have d i g G ˜ such that each of Z-independent positive elements brq1 , . . . , bs g G d1 , . . . , d t , ¨ tq1 , . . . , ¨ k belongs to the monoid generated by brq1 , . . . , bs in ª ˜ In conclusion, since G s G1 = ˜ the elements bX1 , . . . , brX , brq1 , . . . , G. G, bs g Gq are Z-independent in G, and each of p1 , . . . , pk belongs to the monoid generated by bX1 , . . . , brX , brq1 , . . . , bs in G, as required.
FURTHER RESEARCH In the light of Corollary 1.2, in order to solve the problem whether e¨ ery l-group is ultrasimplicial it is sufficient to prove that the ultrasimplicial property is preserved under formation of l-homomorphic images. Unfortunately, the argument in the proof of w6, Theorem 3Žii.x isᎏ to say the least ᎏuninformative, and the problem is still open.
REFERENCES 1. M. Anderson and T. Feil, ‘‘Lattice-Ordered Groups,’’ Reidel, Dordrecht, 1988. 2. A. Bigard, K. Keimel, and S. Wolfenstein, ‘‘Groupes et Anneaux Reticules,’’ ´ ´ Lecture Notes in Mathematics, Vol. 608, Springer-Verlag, Berlin, 1971.
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3. B. Blackadar, ‘‘K-Theory for Operator Algebras,’’ MSRI Publications, Springer-Verlag, Berlin, 1986. 4. G. A. Elliott, On totally ordered groups, and K 0 , in Lecture Notes in Mathematics, Vol. 734, pp. 1᎐49, Springer-Verlag, Berlin, 1979. 5. G. Ewald, ‘‘Combinatorial Convexity and Algebraic Geometry,’’ Graduate Texts in Mathematics, Vol. 168, Springer-Verlag, New YorkrBerlin, 1996. 6. D. Handelman, Ultrasimplicial dimension groups, Arch. Math. Ž Basel . 40 Ž1983., 109᎐115. 7. D. Mundici, Farey stellar subdivisions, ultrasimplicial groups and K 0 of AF CU-algebras, Ad¨ . Math. 68 Ž1988., 23᎐39. 8. D. Mundici and G. Panti, A constructive proof that every 3-generated l-group is ultrasimplicial, in ‘‘Proceedings, Mini-semester, Warsaw, 1996 Žmemoriam Helena Rasiowa.,’’ Banach Centre Publ. Polish Acad. Sci., Warsaw, 1999.