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Embedding finitely generatedAbelian lattice-ordered groups:
Higman’s Theorem and arealisation of π
A. M. W. Glass and Vincenzo Marra
April 22, 2003
AbstractGraham Higman proved that a finitely generated group can
be embedded in a finitely presented group iff it has a recur-sively enumerable set of defining relations. We consider the ana-logue for lattice-ordered groups. Clearly, the finitely generatedlattice-ordered groups that can be `-embedded in finitely pre-sented lattice-ordered groups must have recursively enumerablesets of defining relations. We prove the converse direction for aspecial class of lattice-ordered groups:
Theorem. Every finitely generated Abelian lattice-ordered group that has finite rank and a recursively enu-merable set of defining relations can be `-embedded in afinitely presented lattice-ordered group.
If ξ is a real number, let D(ξ) be the Abelian rank 2 group Z2
with order (m,n) > 0 iff m+ nξ > 0.Corollary. D(ξ) can be `-embedded in a finitely pre-
sented lattice-ordered group iff ξ is a recursive real num-ber.
Thus we obtain an algebraic characterisation of recursive realnumbers. In particular, π is “`-algebraic” in that it can be cap-tured by finitely many relations in this language.1
1AMS Classification 06F15, 06F20, 20B27, 20F60.Keywords: lattice-ordered groups, presentations, direct systems, polyhedral geom-
etry, recursive functions.
1
1 Introduction
In 1961, Graham Higman proved that every finitely generated group that hasa recursively enumerable set of defining relations can be embedded in a finitelypresented group [17]. The analogous result for lattice-ordered groups has beensought for the past 30 years (see [7], no. 11, [20], 12.12, and [14], Chapter 11Question 12).
The purpose of this article is to provide the first results in this directionfor a whole class of lattice-ordered groups. We achieve this by amalgamatingfive disparate areas: (1) continued fractions, (2) recursion theory, (3) recentwork on direct limits of Abelian ordered groups of finite rank, (4) simplicialgeometry, and (5) coding techniques developed in establishing the existence ofa finitely presented lattice-ordered group with insoluble word problem. Specif-ically, we prove:
Theorem A Every finitely generated Abelian lattice-ordered group that hasfinite rank (as an Abelian group) and a recursively enumerable set of definingrelations can be `-embedded in a finitely presented lattice-ordered group.
Let ξ be any irrational real number and D(ξ) = Z×Z ordered by: (m,n) >(0, 0) iff m+ nξ > 0 in R. Then D(ξ) is defined by a set of relations given byeither the finite continued fraction or the finite decimal approximations to ξ.If either (whence both) expansion is recursively definable, then we obtain asan immediate consequence of Theorem A:
Corollary 1.1 Let ξ be a positive irrational real number with recursively enu-merable decimal expansion. Then there is a finitely presented lattice-orderedgroup L(ξ) and commuting elements x, y ∈ L(ξ) such that xmyn > 1 in L(ξ)iff m+ nξ > 0 in R.
We will call a real number ξ `-algebraic if D(ξ) can be `-embedded in afinitely presented lattice-ordered group.
With this terminology, we can rephrase the corollary:
Corollary 1.2 A real number is recursive iff it is `-algebraic.
In particular, since√
2, π, e, . . . are recursive (see Section 2.2), we have that√2, π, e, . . . are `-algebraic and so are each “obtainable” from a finite number
of equations (in fact, a single equation) in the language of lattice-orderedgroups. This may seem somewhat surprising since π and e are transcendentalnumbers, but the finitely presented lattice-ordered groups L(π) and L(e) are,necessarily, non-Abelian.
2
We provide the necessary definitions and background in Section 2. In Sec-tion 3 we show how to associate, with any irrational real number ξ, a sequenceof unimodular matrices Mn : n ∈ N with non-negative integer entries suchthat D(ξ) is the direct limit of the system Z
2 M0→ Z2 M1→ Z
2 . . .. Moreover, thissequence of matrices is recursively enumerable if the decimal expansion of ξ is.In Section 4 we generalise the ideas in Section 3 to get an analogous direct limitrepresentation for arbitrary finitely generated Abelian lattice-ordered groupsG of finite rank. As a consequence, we obtain a strengthening of the mainresult in [27].
Theorem B Every finitely generated Abelian lattice-ordered group of finiterank which has a recursively enumerable set of defining relations has solubleword problem.
Finally, in Section 5 we show how to obtain a finitely presented lattice-ordered group containing a given finitely generated Abelian lattice-orderedgroup of finite rank defined by a recursively enumerable set of relations.
We have constructed the article so that the reader who is only interestedin π (or his favourite recursively obtainable irrational real number) can skipSection 4 and the polyhedral geometry entirely; he may return to it if hisappetite is sufficiently whetted.
2 Background
We will write R for the additive group of reals with the standard total order,Z for the additive group of integers with the standard total order, and N forthe set of non-negative integers.
2.1 Continued Fractions
We outline the standard continued fraction algorithm for a positive irrationalreal number α. For more details, see [2], [9] and especially [31] where thegeometry is explicitly given.
Let (q−2, p−2) = (1, 0) = e1 and (q−1, p−1) = (0, 1) = e2. Let (q0, p0) =a0(q−1, p−1) + (q−2, p−2) where a0 ∈ N is maximal such that (q0, p0) lies belowthe line y = αx. So (q0, p0) and (q−2, p−2) lie below the line y = αx and(q−1, p−1) lies above it. Assume (q0, p0), . . . , (qn−1, pn−1) have been defined sothat
(i) q−1 < . . . < qn−1,
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(ii) |α− pmqm| < 1
q2m
and
(iii) (qm, pm) lies above the line y = αx if m is odd and below the liney = αx if m is even.
Since (qn−1, pn−1) is closer to the line y = αx than (qn−2, pn−2) is, we candefine an to be the largest (strictly) positive integer a such that a(qn−1, pn−1)+(qn−2, pn−2) lies on the same side of the line y = αx as (qn−2, pn−2), and let
(qn, pn) = an(qn−1, pn−1) + (qn−2, pn−2).
The sequence pn/qn : n ∈ N is called the set of convergents of α and wewrite α = [a0, a1, a2, . . .]. So p0/q0 = a0, p1/q1 = a0 + 1
a1, etc..
Note thatlimn→∞
pnqn
= α.
Given the decimal expansion of α accurate to enough decimal places, wecan calculate a0, a1, . . . , ak exactly; and given the expression for α as a con-tinued fraction to enough entries, we can explicitly calculate the first k placesof the decimal expansion for α.
Although the above is stated (as is standard) for α > 0, the reader shouldhave no difficulty in adapting the technique when α < 0.
2.2 Recursion Theory
We assume that the reader has a minimal knowledge of recursive function the-ory (see [29]) as well as the following less familiar (but very important) proposi-tion. Let f, g, h, u, v be functions from N into N. Then f is said to be obtainedfrom g, h, u, v by general recursion if fg = u, fh = vf, and
⋃range(hkg) :
k ∈ N = N.
Proposition 2.1 (Julia Robinson, [28]) The class of recursive functions ofone variable is the smallest class of numerical functions that is closed undercomposition and general recursion, and contains the zero function θ (θ(n) ≡ 0)and the successor function s (s(n) = n+ 1).
In the Introduction, we stated that√
2, e and π are recursive real numbers.To see the first of these, take the largest whole number whose square is less than2, namely 1. Take the largest of the numbers 1.1, 1.2, . . . , 1.9 whose squareis less than 2, viz: 1.4. Take the largest of the numbers 1.41, 1.42, . . . , 1.49
4
whose square is less than 2, namely 1.41. Continuing with this algorithm givesthe decimal approximation for
√2 recursively.
For e =∑∞n=0
1n! , note that e is irrational and that
∞∑n=m+1
1n!≤ (m+ 2)/(m!)(m+ 1)2.
We can therefore use the partial sums∑mn=0
1n! to calculate the decimal expan-
sion for e as follows. Calculate∑10n=0
1n! to one decimal place; this gives the
decimal expansion of e to one decimal place. Next calculate∑11n=0
1n! to two
decimal places. Continue. At stage m, we have the correct decimal expansionof e to m places unless the correct decimal expansion for e to m places endsin a 0 whereas that of the approximation obtained (to m places) ends in a 9.In this case, continue the process until the decimal approximation obtainedby the partial sums does not end in a 9. This must occur (since e is irrationaland so its decimal expansion does not terminate in all 0s) . Then we havethe correct decimal expansion for e to k decimal places where k = k(m) > m.Although we know of no bound on the number of times this occurs (as a func-tion of m) or how long the sequence of 9s will continue, we are guaranteedthat this process will correct itself and give the right decimal expansion for e.
A similar technique can be applied to π = 2 + 4∑∞n=1
11−4n2 (see [1], (13)
page 190 with z = 1/2), eπ, etc., though far better recursive techniques areknown for π (see [4], especially articles 70, 56, 62 and 64).
2.3 Lattice-ordered groups
As is standard, in any group G we write fg for g−1fg and [f, g] for f−1g−1fg.We will also employ the shorthand x(m,n) for xm1 x
n2 when m,n ∈ Z.
A lattice-ordered group is a group which is also a lattice that satisfies theidentities x(y ∧ z)t = xyt ∧ xzt and x(y ∨ z)t = xyt ∨ xzt. Throughout wewrite x ≥ y as a shorthand for x ∧ y = y, etc., `-group as a shorthand forlattice-ordered group and o-group if the `-group is totally ordered. If G andH are `-groups, then the cardinal ordering on G ⊗ H is given by (g, h) ≥ 1iff g ≥ 1 and h ≥ 1; it is a lattice ordering. A simplicial group is the `-groupobtained by the natural extension of this cardinal ordering to Zn (n ∈ N). IfG is an o-group and H is an `-group, then we get another lattice order onG ⊗H by (g, h) ≥ 1 if g > 1 or (g = 1 & h ≥ 1). In this case, G ⊗H is saidto be lexicographically ordered and G is said to be a lexicographic extensionof H; we write this `-group as G⊗→H.
Lattice-ordered groups are torsion-free and f∨g = (f−1∧g−1)−1; moreovereach element of G can be written in the form fg−1 where f, g ∈ G+ = h ∈
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G : h ≥ 1— see, e.g., [14], Corollary 2.1.3, Lemma 2.3.2 & Lemma 2.1.8. Foreach g ∈ G, let |g| = g ∨ g−1. Then |g| ∈ G+ iff g 6= 1, where G+ = G+\1.Therefore, (w1 = 1 & . . . & wn = 1) iff |w1|∨ . . .∨|wn| = 1 [ibid, Lemma 2.3.8& Corollary 2.3.9]. Consequently, in the language of lattice-ordered groups(and in sharp contrast to group theory) any finite number of equalities can bereplaced by a single equality.
An `-homomorphism from one `-group to another is a group and a latticehomomorphism. Kernels are precisely the normal sublattice subgroups thatare convex (If k1, k2 belong to the kernel and k1 ≤ g ≤ k2, then g belongs tothe kernel.)
Free `-groups on finite sets of generators exist by universal algebra. Finitelygenerated lattice-ordered groups are the `-homomorphic images of free `-groups on finitely many generators. If the kernel is finitely generated asa convex normal sublattice subgroup, then we call the `-homomorphic im-age finitely presented; if the kernel is generated by a recursively enumerableset of elements (as a convex normal sublattice subgroup), then we say thatthe finitely generated lattice-ordered `-homomorphic image has a recursivelyenumerable set of defining relations.
The previous paragraph applies mutatis mutandis for Abelian `-groups,free Abelian `-groups, etc.. We will write An for the free Abelian `-groupon n generators (n ∈ N). It can be realised as follows: Let F (Rn,R) be theadditive group of all real-valued functions on Rn with the pointwise ordering(so f ∈ F (Rn,R)+ iff xf ≥ 0 for all x ∈ Rn). Let π1, . . . , πn be the n projectionmaps (xπj = xj). Then An is the sublattice subgroup of F (Rn,R) generatedby π1, . . . , πn ([3] or see [14], Theorem 5.A), and every integral homogeneouspiecewise linear function from R
n to R belongs to An. Thus every Abelian`-group G with generators g1, . . . , gn is `-isomorphic to An/K(R) where Ris a set of generators of the kernel K(R) of the natural map given by πj 7→gj (j = 1, . . . , n). We will also write this as G ∼=` 〈π1, . . . , πn;R〉`.
Let u ∈ Am. Then Z(u) = r ∈ Rm : u(r) = 0 is a finite union of
closed convex polyhedral cones with vertex at the origin, each defined byhyperplanes with integer coefficients. If further v ∈ An, G ∼=` 〈π1, . . . , πm;u〉`and H ∼=` 〈π1, . . . , πn; v〉`, then G ∼=` H iff Z(u) and Z(v) are piecewise linearhomeomorphic over Z. This is known as the Baker-Beynon Duality (see [5] or[14], Theorem 5.B).
Throughout finite rank will only refer to the standard definition of rank asan Abelian group; that is, we write that an Abelian (possibly lattice-ordered)group G has rank m if there are g1, . . . , gm ∈ G such that each element of G canbe written in the form n1g1 + . . .+ nmgm for uniquely determined coefficientsn1, . . . , nm ∈ Z.
6
Caution: When given a finitely generated Abelian lattice-ordered group,we assume that the generators are given explicitly. So, in our results thatinclude the hypothesis “finitely generated Abelian lattice-ordered group offinite rank”, we mean that there is a given finite set of generators for theAbelian lattice-ordered group and additionally the Abelian (`)-group has finiterank (though we may NOT necessarily be able to obtain a finite set of elementsof the finitely generated Abelian `-group that generate it as an Abelian group— or even deduce that such a set exists — algorithmically from the definingrelations). The finite rank hypothesis is thus deus ex machina. This distinctionwill only be used in the algorithmic portion of section 4; namely, in Proposition4.1(2) and Lemma 4.4.
Let (Ω,≤) be a totally ordered set. Then Aut(Ω,≤) is an `-group whenthe group operation is composition and the lattice operations are just thepointwise supremum and infimum (α(f ∨ g) = maxαf, αg, etc..) Thereis an analogue of Cayley’s Theorem for groups, namely the Cayley-HollandTheorem ([14], Theorem 7.A):
Proposition 2.2 (Holland, [18]) Every lattice-ordered group can be `-embeddedin Aut(Ω,≤) for some totally ordered set (Ω,≤). Moreover, if the originallattice-ordered group is countable, then (Ω,≤) can be taken to be (R,≤).
If h ∈ Aut(R,≤), then the support of h, supp(h), is the set β ∈ R : βh 6=β. Since each real interval (α, β) is order-isomorphic to (R,≤) we obtain:
Corollary 2.3 Let α, β ∈ R with α < β. Then every finitely generated `-groupG can be `-embedded in Aut(R,≤) so that supp(g) ⊆ (α, β) for all g ∈ G.
Finally, by considering intervals of support, it is easy to establish the well-known fact that
Proposition 2.4 For all f, g ∈ Aut(R,≤), supp(fg) = supp(f)g. Hence iffg ∧ f = 1 and g ≥ 1, then fn ≤ g for all n ∈ N.
2.4 Direct limits
We recall the definition of a direct limit over N in the category of partiallyordered Abelian groups.
Let Hii∈N be an ascending chain of partially ordered Abelian groups (i.e.,Hi is a subgroup of Hi+1 and H+
i ⊆ H+i+1, for all i ∈ N). Let +i denote the
group operation in Hi. Clearly, the set-theoretical union H =⋃i∈NHi is an
Abelian group with respect to the binary operation⊕ defined by : a⊕b = a+i0b
7
where i0 = mini : a, b ∈ Hi. The Abelian group H is partially ordered by where:
a b iff there exists i0 ∈ N such that a i0 b
and i denotes the partial order of Hi, for all i ∈ N. Thus, H is a partiallyordered Abelian group with respect to ⊕ and . We call H the direct unionor the (direct) limit of the direct system Hii∈N.
A dimension group is an Abelian group with a directed partial order sat-isfying the Riesz interpolation property, whose positive cone is semi-isolated(unperforated). If H is a countable partially ordered Abelian group, then His a dimension group iff it is a direct limit of simplicial groups Hii∈N, tran-sition maps being compatible order-preserving group homomorphisms (notnecessarily embeddings as in the previous paragraph), see [11] and compare[22]. However, H need not be a lattice-ordered, even when each transition mapis an embedding and each Hi has rank 3. For example, let Hii∈N, be the rank3 simplicial group Z3 with canonical basis Bi = e1 + ie3, e2 + ie3, e3; that is,m(e1+ie3)+n(e2+ie3)+ke3 ≥ (0, 0, 0) iff m,n, (m+n)i+k ≥ 0. Define mapsφi : Hi → Hi+1 by (e1+ie3)φi = e1+(i+1)e3, (e2+ie3)φi = e2+(i+1)e3, ande3φi = e3. We regard each φi as an inclusion map, and thus obtain the directunion H of the direct system Hii∈N. Now e1, e2 is a canonical basis for thegroup Z2 with the simplicial order, and e3 generates Z. Since e1, e2 > ne3 forall n ∈ N, we have that H is `-isomorphic to Z2 ⊗→Z, and the order on Z2 issimplicial. Hence H is not lattice-ordered. For more details, see [23], Chapter8 Example 9.
For 1 ≤ i, j ≤ n with i 6= j, let Ej,i denote the n × n integer matrixobtained from the identity matrix In by replacing the i-th column by the sumof the i-th and j-th columns. Let En = Ei,j : i 6= j & 1 ≤ i, j ≤ n ∪ In.We call En the set of standard matrices of rank n. Note that |En| = n2−n+1.Let M be an n × n integer matrix. We write M ≥ 0 iff mi,j ≥ 0 for alli, j ∈ 1, . . . , n.
Let Sii∈N be a direct system of simplicial groups of fixed rank n ∈ N;thus Si ∼=` Z
n for all i ∈ N. For each i ∈ N, take the set of n atoms of S+i as
the basis for the Z-module Si. This basis is denoted by Bi and is called thecanonical basis of Si. For each inclusion map Si ⊆ Si+1, there is a unique n×ninteger matrix Mi representing this map with respect to the bases Bi and Bi+1.Since inclusion maps are injective and order preserving, det (Mi) 6= 0 andMi ≥ 0. If Mi ∈ En for all i ∈ N, we call Sii∈N a standard (rank n) system.We apply the same terminology to the limit S =
⋃i∈N Si. Finally, we say that
the standard system Sii∈N is recursively enumerable iff the sequence Miis recursively enumerable.
8
2.5 Polyhedral geometry
Consider Rn as the vector space of real n-tuples (the transpose of usual nota-tion, adopted here to conform to conjugation notation used in Section 5). Letv1, . . . ,vm ⊆ Zn be linearly independent over R. The (rational) simplicialcone σ generated by v1, . . . ,vm is their positive linear hull over R, i. e.,
σ = 〈v1, . . . ,vm〉 = p1v1 + · · ·+ pmvm | 0 ≤ p1, . . . , pm ∈ R.
A face of a simplicial cone σ = 〈v1, . . . ,vm〉 is the simplicial cone spanned by asubset of v1, . . . ,vm; its dimension is the dimension (as a real vector space)of the linear subspace of Rn spanned by its vertices. A (rational simplicial)fan is a finite set Σ of simplicial cones such that every face of every cone in Σbelongs to Σ, and any two cones in Σ intersect in a common face. The supportof Σ (denoted by |Σ|) is the union of all its cones. A fan in Rn is complete iff itssupport is Rn. An integral vector is primitive iff its coordinates are relativelyprime. Let σ ⊆ R
n be a simplicial cone. Then σ can always be writtenas σ = 〈v1, . . . ,vm〉 for uniquely determined primitive linearly independentintegral vectors vi; such vectors are called the vertices of σ. We say thata simplicial n-dimensional cone σ in Rn is unimodular iff the integer matrixwhose rows are the vertices of σ has determinant ±1. A complete fan is calledunimodular iff all its n-dimensional cones are. For more details and furtherbackground, see [12] (and, to a lesser extent, [13]).
Let v be a vertex of a complete unimodular fan Σ in Rn. The Schauder hatat v in Σ is the unique continuous piecewise-linear homogeneous functionhv:Rn → R such that
1. vhv = 1,
2. uhv = 0 for every vertex u 6= v of (any cone in) Σ,
3. hv is linear homogeneous on each cone of Σ.
These are special cases of the support functions defined in [12]. Their use hasbeen pivotal in the study of finitely generated Abelian lattice-ordered groupsand their application was the great insight of D. Mundici (see [26]) — byunimodularity, the linear pieces of a Schauder hat have integer coefficientsand so belong to An.
We denote by HΣ ⊆ An the set of all Schauder hats at the vertices ofthe complete unimodular fan Σ, and call it a Schauder basis of An. By stan-dard piecewise linear geometry, the projection functions π1, . . . , πn are lattice-ordered group expressions over any Schauder basis. By the Baker-BeynonDuality (Section 2.3), any Schauder basis is a set of generators for An.
9
Let Σ,∆ be fans with the same support. We say that Σ refines ∆ (writtenΣ ≤ ∆) iff every cone of ∆ is a union of cones of Σ. If Σ and ∆ are completeunimodular fans in Rn, then H∆ is contained in the integral positive span ofHΣ iff Σ ≤ ∆. We denote this by HΣ ≤ H∆.
We now introduce a more specialised refinement. Let σ = 〈v1, . . . ,vm〉be an m-dimensional cone of a complete unimodular fan Σ in Rn where thevi’s are the vertices of σ. Then τ = 〈vi,vj〉, for 1 ≤ i < j ≤ m, is a 2-dimensional face of σ. We call w = vi + vj the Farey mediant of τ . Let σi =〈w,v1, . . . ,vi−1,vi+1, . . . ,vm〉, σj = 〈w,v1, . . . ,vj−1,vj+1, . . . ,vm〉. DefineΣ∗ as the fan obtained from Σ by replacing each cone ρ ∈ Σ of which τ is aface by the two cones σi, σj , along with all their faces. Then we say that Σ∗ isobtained from Σ by stellar subdivision along the Farey mediant τ , or simplyby binary starring. Note that binary starring preserves the unimodularity offans. For further results and background, see [16], [30], [24], [25], [26] and [23].
Let Σ be a complete unimodular fan in Rn. If Σ∗ is obtained from Σ by asingle application of binary starring, we write
Σ∗ 2 Σ .
If ∆ is obtained from Σ by a finite number of binary starrings, we write
∆ Σ .
We use the same notation for the corresponding Schauder bases HΣ, HΣ∗ andH∆.
Remark. Note that HΣ∗ 2 HΣ iff HΣ∗ is obtained from HΣ by replacingtwo hats h1, h2 ∈ HΣ by three hats
h1 − (h1 ∧ h2) , h1 ∧ h2 , h2 − (h1 ∧ h2) .
Lemma 2.5 (The De Concini-Procesi Lemma, [10]) Let H be a Schauderbasis in Rn. There exists a recursively enumerable sequence Hii∈N of Schauderbases satisfying the following properties:
DP1. H0 = H .
DP2. For all i ∈ N, Hi+1 2 Hi .
DP3. Given any Schauder basis K in Rn, there exists i0 ∈ N such that Hi0 ≤K .
10
3 D(ξ) as a limit of a standard system
The easiest Abelian `-groups to obtain as limits of standard rank n systemsare the simplicial groups Zn (n ∈ N): simply take each Si to be Zn with thecanonical basis B0 of S0 and each standard matrix to be In. (We only includedIn in En to obtain this `-group.)
Our second set of examples are D(ξ) for ξ an irrational real number. Theconstruction uses a “slow” version of the continued fraction algorithm for ξ —instead of calculating an at stage n, break this step into an steps all obtainedby adding (qn−1, pn−1) one at a time — see Section 2.1.
D(ξ) can be regarded as the set of integer points in R2 and D(ξ)+ as thesubset of all integer points above the real line x + yξ = 0. Hence the firstquadrant of R2 is contained in D(ξ)+. Note that the first quadrant is mappedinto itself by both E1,2 and E2,1.
Let B0 = e1, e2 where e1 = (1, 0) and e2 = (0, 1). We write a > b inD(ξ) if the vector a−b (written with respect to the standard basis e1, e2) liesabove the line x+yξ = 0. We will inductively define sequences of bases Bii∈Nand standard matrices Mii∈N. Let Si be Z2 with basis Bi = bi,1,bi,2. Ifbi,1 > bi,2 in D(ξ), let Si+1 be Z2 with basis Bi+1 = bi,1 − bi,2,bi,2 andMi = E1,2; and if bi,1 < bi,2 in D(ξ), let Si+1 be Z2 with basis Bi+1 =bi,1,bi,2 − bi,1 and Mi = E2,1. Let L be the limit of this standard rank 2system.
Since L and D(ξ) are both isomorphic to Z2 as groups, they are certainlygroup-isomorphic to each other. We now show that they are `-isomorphic.
Observe that if we coalesce the matrices into a sequence
Z2Ea01,2→ Z
2Ea12,1→ Z
2Ea21,2→ Z
2 . . . ,
we have applied an analogue of the continued fraction algorithm to approxi-mate ξ. Hence, by the continued fraction algorithm (see Section 2.1), we havethat L+ = D(ξ)+. Consequently, D(ξ) is `-isomorphic to the limit L of thestandard system.
It is immediate that the sequence of matrices is recursively enumerable iffthe sequence of continued fraction approximations for ξ is; equivalently, iff thesequence of finite decimal approximations to ξ is.
11
4 Representations of Abelian `-groups of
finite rank as limits of standard systems
Our goal in this section is to establish the following connection between stan-dard systems and finitely generated Abelian `-groups (whence Theorem B asan easy consequence).
Proposition 4.1 Let G be an Abelian `-group and n ∈ N.
1. G has rank n iff G is the limit of a standard rank n system Sii∈N.
2. If G is finitely generated and has rank n, then G is definable by a re-cursively enumerable set of relations iff G is the limit of a recursivelyenumerable standard rank n system Sii∈N.
We will use the following structure theorem on Abelian `-groups of finiterank which is an easy consequence of the Birkhoff-Jaffard-Conrad theory oforthofinite `-groups (see [6], [19] and [8]).
Proposition 4.2 Let G be an Abelian `-group of rank n ∈ N. Then there ex-ists a finite set O = O1, . . . , Ok of finitely generated Archimedean o-groups,rank (Oi) = ri with
∑ki=1 ri = n, such that G can be built from O using a fi-
nite number of cardinal sums and lexicographic extensions, each Oi being usedexactly once.
With the above notation, we call the Oi’s the constituents of G. A maximallinearly independent subset of O+
i will be called a basis of Oi and will bedenoted by Bi. Taking the union over all i provides a basis B of G. We willalways assume that B is given with the partition Bi : i = 1, . . . , k.
To prove Proposition 4.1 we need two lemmata.
Lemma 4.3 Let G be an Abelian `-group of rank n. Let B be a basis of G.Let B0, . . . , Bs be a finite sequence of subsets of G+ with B0 = B and Bi+1
is obtained from Bi by replacing two elements a,b ∈ Bi such that a > bwith the two elements a − b,b. Then the following property holds for eachi ∈ 1, . . . , s:
If a,b ∈ Bi with a 6= b, then either a ∧ b = 0 or a > b or a < b (1)
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Proof. Let O1, . . . , Ok be the constituents of G. Then (1) holds for B0 = Bas is easily verified by induction on k. If g ∈ G, then g = g1 + · · · + gk withgi ∈ Oi for i = 1, . . . , k. We say that Oj is a maximal component of g if gj 6= 0and gi = 0 if O+
i > O+j (i.e., c > d for all c ∈ O+
i , d ∈ O+j ). Note that each
element of B0 has a unique maximal component. Further, if a,b ∈ B0, thena > b implies that a−b has the same maximal component as a and replacinga by a−b in B0 does not change the linear independence in the constituents.Clearly these properties are preserved in passing from Bi to Bi+1 and ensurethat (1) holds by induction on i (using the linear independence). //
Remark. Suppose that H is a Schauder basis for An and φ maps H onto abasis B of a rank n Abelian `-group G. Let φ also denote the unique extensionto an `-homomorphism from An into G. Let H1 be the Schauder basis of Anobtained by replacing h1, h2 by h1− (h1∧h2), h1∧h2, h2− (h1∧h2). Since Hφis a basis for G, one of the elements h1−(h1∧h2), h1∧h2, h2−(h1∧h2) belongsto the kernel of φ by Lemma 4.3. Thus h1φ and h2φ are either comparable ordisjoint, and |H1φ \ 0| = |Hφ|.
We can now prove most of Proposition 4.1.
Proof of Proposition 4.1. (1) By hypothesis, G satisfies Proposition 4.2.We adopt the notation employed there. Let B = b1, . . . , bn, with the appro-priate partition provided. Let π1, . . . , πn be the free generators of An. Themap πj 7→ bj extends uniquely to a surjective `-homomorphism φ : An → G .Let K = kerφ. Consider the simplicial cone σ = 〈e1, . . . , en〉 ⊆ R
n, wheree1, . . . , en is the canonical basis of Rn. Clearly, there exists a complete uni-modular fan Σ0 such that σ ∈ Σ0. By the De Concini-Procesi Lemma thereexists a recursively enumerable sequence Hii∈N of Schauder bases of Ansatisfying (DP1) – (DP3).
Set HKi = Hiφ\0. By the remark following Lemma 4.3, |HK
i | = n andHKi is linearly independent in G for all i ∈ N. Let Si be the free Abelian group
generated by HKi ; order Si simplicially by choosing HK
i as the canonical basis.Then Si ⊆ Si+1 and S+
i ⊆ S+i+1. If Mi is the n×n integer matrix representing
the inclusion map Si ⊆ Si+1, it is clear by Lemma 4.3 that Mi is a standardmatrix. Let L =
⋃i∈NSi.
We claim that L ∼=` G.Indeed, L ∼= G as a group, for G = S1
∼= Zn, and L is generated (as a
group) by S1. We now show that L+ = G+. Clearly L+ ⊆ G+. Let g ∈ G+.Then there are z1, . . . , zn ∈ Z such that
g = z1(π1φ) + · · ·+ zn(πnφ) ,
13
since G = S1 as a group. The function
z1π1 + · · ·+ znπn : Rn → R,
a preimage of g, can be viewed as a homogeneous integral hyperplane Pg inRn+1. Choose a complete unimodular fan ∆ in Rn such that the linear space
ZPg given by Pg = 0 is a union of simplicial cones of ∆. By Lemma 2.5, thereexists i0 ∈ N such that Hi0 ≤ H∆. Then HK
i0must span Pgφ with non-negative
coefficients as we now show.Consider the partition Hi0 = H(+)∪H(−)∪H(z) defined as follows. Let
h ∈ Hi0 . The coefficient of h in the unique integral linear span of Pg by Hi0
is either positive, negative, or zero. Accordingly, let h ∈ H(+), h ∈ H(−),or h ∈ H(z). Notice that if h1 ∈ H(+) and h2 ∈ H(−), then h1 ∧ h2 = 0 inAn by our choice of ∆ — the supports of h1 and h2 are separated by ZPg .Hence, H(+) and H(−) are pairwise disjoint subsets of An. Now partitionHKi0
= Hi0φ similarly, according to the signs appearing in the unique integralspan of Pgφ in terms of HK
i0; say, HK
i0= H(+)∪H(−)∪H(z) . By the definition
of HKi0
, we get H(z) = ∅, H(+) = H(+)φ\0, and H(−) = H(−)φ\0. Itfollows that H(+) and H(−) are pairwise disjoint sets in G. Since g ∈ G+,we deduce that H(+) 6= ∅ = H(−), as desired.
Consequently, g ∈ L+ as claimed.
Conversely, suppose G =⋃i∈N Si is the limit of a standard rank n system,
and let B0 denote the canonical basis of S0. Since standard matrices areunimodular, B0 generates G as a group, whence G has rank n. A fortiori, B0
also generates G as an `-group.
(2) Suppose that G is presented as G ∼=` 〈x1, . . . , xm;R〉` , where R is a recur-sively enumerable set of relations in the language of `-groups, in the variablesx = (x1, . . . , xm). We may assume that R is the set of all `-group wordsw(x1, . . . , xm) that are the identity in G; i.e., the set of all `-group wordsw(x1, . . . , xm) for which w(g1, . . . , gm) = 1 for every g1, . . . , gm ∈ G. Withthe above notation, by (1) we have G ∼=` L =
⋃i∈NSi, the latter a standard
rank n system. We identify G with L. Since x1, . . . , xm generates G, Bcan be written in terms of the elements x. Assume that such a writing, whichobviously consists of a finite amount of information, is given. (Caution: we areonly claiming the existence of such formulæ, not their effective computabilityfrom the given presentation of G). To complete the proof, it is enough toshow that the sequence HK
i constructed in (1) is recursively enumerable.We prove this by induction on i. If i = 1, then HK
1 = π1φ, . . . , πnφ = H1φand there is nothing to show. Suppose we can effectively enumerate HK
j up
14
to HKi−1. As pointed out in the remark immediately prior to Lemma 2.5,
the set Hi is obtained from Hi−1 by replacing h1, h2 ⊆ Hi−1 by the set
H[ = h1 − (h1 ∧ h2) , h1 ∧ h2 , h2 − (h1 ∧ h2) .
By the remark following Lemma 4.3, exactly one of the elements of H[ belongsto kerφ = K. We claim that we can decide which. Indeed, since there is anexpression for B in terms of x and the elements of Hi are effectively computableexpressions in terms of π1, . . . , πn, we can effectively obtain the elements ofH[φ in terms of x. Enumerating R, one of these expressions must appear inthe list. The corresponding element of H[ is the only one that belongs to K,establishing our claim. Thus we can effectively obtain HK
i .
To prove the converse, we need the second lemma:
Lemma 4.4 Let G be an Abelian `-group that is the limit of a recursivelyenumerable standard rank n system Sii∈N. Let B0 = b1, . . . , bn be thecanonical basis for S0. Then there is an effective procedure which, when givenan arbitrary `-group word w(b), produces a group word u(b) equal to it in G.
Proof: Since standard matrices are unimodular, B0 generates G as a group.Hence G satisfies Proposition 4.2. Let O1, . . . , Ok be the constituents of G. Itis easy to check that B0 is a basis B of G. Assume we are given the partitioningB =
⋃Bi.
Note that w1∧w2 = ((w1−w2)∧0)+w2 and w1∨w2 = −((−w1)∧(−w2)).It therefore suffices to show that for every group word g(b), there is a groupword u(b) such that u(b) = g(b) ∧ 0.
Assume that k = 1. Then either g(b) > 0 or g(b) < 0 or g(b) = 0(and u(b) = 0, g(b), or 0, respectively). We can effectively rewrite g(b) asz1b1 + · · · + znbn. Let z = (z1, . . . , zn). Since B0 is linearly independent, thecase g(b) = 0 is decidable; namely, g(b) = 0 iff z = 0. Let Mi be the matrixrepresenting the i-th inclusion map Si ⊆ Si+1. Then for i0 sufficiently large,either zM1 · · ·Mi0 >i0 0 or 0 >i0 zM1 · · ·Mi0 , where ≥i0 denotes the simplicialorder of Si0 . Thus the remaining two cases are decidable. Hence the lemmais proved when k, the number of constituents of G, is equal to 1.
Assume by induction that the statement holds for all natural numberssmaller than k. Write G = Ok ∗ H, where H is a finitely generated Abelian`-group of finite rank m < n, and ∗ = ⊗→ or ∗ is the cardinal sum. It is clearthat H, being a sublattice subgroup of G, has a recursively enumerable set ofdefining relations. Hence the direct implication in Proposition 4.1(1) holds.Thus H is the limit of a standard recursively enumerable system
⋃i∈NTi of
finite rank m. Let C0 be the canonical basis of T0. It is easy to see that we
15
may assume C0 ⊆ B0. Thus B0 = Bk ∪ C0, with Bk ∩ C0 = ∅. We writebk and c to denote the elements of Bk and C0, respectively. Then we mayeffectively rewrite g(b) = g1(bk) + g2(c) for unique group words g1(bk) ∈ Okand g2(c) ∈ H. If G = Ok ⊗ H with the cardinal ordering, then g(b) ∧ 0 =(g1(bk)∧ 0) + (g2(c)∧ 0), and the lemma follows by induction. If G = Ok⊗
→H,
we can effectively decide whether g1(bk) = 0, g1(bk) > 0 or g1(bk) < 0. Inthe latter two cases, g(b) > 0 or g(b) < 0, respectively, and we are done. Ifg1(bk) = 0, then we can effectively decide whether g2(c) = 0, g2(c) > 0 org2(c) < 0, and the proof is complete. //
We can now complete the proof of Proposition 4.1 by establishing theconverse direction in (2):
Proof: Let B0 = b1, . . . , bn be the canonical basis of S0 and L =〈x1, . . . , xn;R〉`, where R is the set of all `-group words w(x) such thatw(b) = 0 holds in G; thus L ∼=` G. We claim that R is a recursively enu-merable set. Indeed, let wj(x)j∈N be an effective enumeration of all `-groupwords in the variables x. By Lemma 4.4, for each j we can compute a groupword uj(b) such that uj(b) = wj(b) in G. It follows that uj(x) − wj(x) be-longs to R. We can effectively rewrite uj(x) in the form z1x1 + · · · + znxnfor z1, . . . , zn ∈ Z. Since B0 is linearly independent in G, so is x1, . . . , xn.Hence uj(x) = 0 iff z1 = · · · = zn = 0, proving our claim. This completes theproof of the proposition. //
We now prove Theorem B.Proof: SupposeG = 〈x1, . . . , xm;R〉`, whereR is a recursively enumerable
set of relations in the language of `-groups, involving only the variables x. ByProposition 4.1, G is `-isomorphic to an Abelian `-group L that is the limit ofa recursively enumerable standard (rank n) system Sii∈N. We identify G andL. Let B0 be the canonical basis of S0. For each xj , there exists a group worduj(b) such that xj = uj(b) in G. Assume that such a set uj(b) : j = 1, . . . ,m,which clearly consists of a finite amount of information (namely, an integermatrix), is given. Let w(x) be an `-group word in the variables x. Let v(b)denote the corresponding `-group word under the transformation xj = uj(b).By Lemma 4.4, we may effectively compute a group word u(b) such thatu(b) = v(b) holds in G. We may effectively reduce u(b) to the form
u(b) = z1b1 + · · ·+ znbn , zi ∈ Z .
Since B0 is linearly independent, w(x) = v(b) = u(b) = 0 in G iff zi = 0 forall i ∈ 1, . . . , n, and the theorem is proved. //
Remark. The above decision procedure is not uniform. As far as we know,uniform solubility for this class of `-groups remains open.
16
5 Coding of recursive sequences of matri-
ces
In group theory, embeddings into finitely presented groups are accomplished byHigman-Neumann-Neumann extensions or equivalents thereof (see [21]). TheHNN result is invalid for `-groups ([14], Theorem 7.C), so we resort insteadto permutation groups. The idea is to construct a finite set of elements T =t1, . . . , tn of Aut(R,≤) satisfying a finite set of relations R(t1, . . . , tn). Itwill follow that if X = x1, . . . , xn is a finite set of formal symbols, then thesublattice subgroup of Aut(R,≤) generated by T is an `-homomorphic imageof the finitely presented abstract `-group G = 〈X;R(x1, . . . , xn)〉`. Hence ifw(t1, . . . , tn) 6= 1 in Aut(R,≤), then w(x1, . . . , xn) 6= 1 in G.
By the Cayley-Holland Theorem (Proposition 2.2 above), every finitelygenerated `-group G can be `-embedded in Aut(R,≤), and hence as elementswhose support is contained in (0, 1) (see Corollary 2.3 above). The interplayof the permutation approach and the formal abstract approach is crucial inour completion of the proof of our main theorem.
This section is quite technical. We have tried to make it sufficiently self-contained by quoting the necessary results from [15].
Specifically, in [15] we constructed elements a0, b1, c1, d1 ∈ Aut(R,≤)+
each having bounded support, so thata0 maps 0 to 1,b1 maps inf(supp(a0)) above sup(supp(a0)),c1 maps inf(supp(b1)) above sup(supp(b1)), andd1 maps inf(supp(c1)) above sup(supp(c1)),
where inf and sup are in R.
Since we are assuming that supp(g) ⊆ (0, 1) for all g ∈ G, it follows thatαg < 1 = 0a0 < αa0 for all g ∈ G and α ∈ supp(g); whence g < a0 inAut(R,≤) for all g ∈ G.
As shown in the appendix of [15], there is a4 ∈ Aut(R,≤) such that
a−14 a0a4 = a0, a−1
4 b1a4 = c1, a−14 c1a4 = d1 (I),
and a4 is the identity on supp(a0).Using these relations (among others) it was shown in [15] that for each recur-sive function f : N→ N, there is a finitely presented `-group G(f) containingelements a0, b1, c1, d1, a4, af such that, for all m ∈ N, in G(f)
(∗m) a0 ∗ (cm1 af ) = a0 ∗ (bf(m)1 cm1 ),
17
where, as in [15], we write x ∗ y for xy when y is complicated (as a concessionto the reader’s eyesight — and our own.). The relations for G(f) included,inter alia, [af , b1] = [af , d1] = 1 = af (af ∧ d0)−1 ∧ d0; and the finite set ofdefining relations for G(f) all had a natural interpretation in Aut(R,≤).
In keeping with our original intention, and for simplicity of exposition, wefirst prove our main theorem for two generator Abelian `-groups of rank 2.
We need to modify the construction of G(f) by adjoining two extra gen-erators x1, x2 and extra relations xa0
j ∧ xj = 1 (j = 1, 2), and wherever therewas a defining relation involving a0 we also adjoin the same relation with x1
in place of a0 and the same relation with x2 in place of a0. Call the resultingfinitely presented `-group H(f). Note that xnj ≤ a0 for all n ∈ N by Proposi-tion 2.4 (j = 1, 2). Since a4 could be interpreted as being the identity on thesupport of a0 and all powers of x1 and x2 are less than a0, no collapsing isintroduced. In particular, we have
xj ∗ a4 = xj (j = 1, 2) (II).
By precisely the same method as in [15], we obtain that if f is recursive,then (∗m) hold in H(f) for all m ∈ N. Moreover, so do
(∗∗m) xj ∗ (cm1 af ) = xj ∗ (bf(m)1 cm1 ) (j = 1, 2)
for all m ∈ N. Further, if f is obtained by composition or general recur-sion from other functions, the generators and relations associated with these“constituent” functions are included in H(f).
Specifically, we start with the finitely presented `-group H with generatorsx1, x2, a0, . . . , a7 and relationsxa0j ∧ xj = 1 (j = 1, 2), [x1, x2] = 1, [xj , a4] = [xj , a5] = 1 (j = 1, 2)a0 ∧ a2 = 1, a1 ≥ 1, aa1
0 ≤ a2, aa12 ≤ a2, a0 ≤ b0, a1 ≤ b1, [a1, b0] =
1, [a1, b1] = 1, [a1, c1] = 1, [a1, d1] = 1, da30 ∧ d1(d0)−a3 = 1, [a0, a4] =
1, aa41 = b1, ba4
1 = c1, ca41 = d1, [a0, a5] = 1, ba5
0 = c0, [a5, c1] =1, a1b
−10 ∧ b0 = 1, a6 ∧ a7 = 1, [c1, a6 ∨ a7] = 1, a0 ∧ a6a
−10 = 1, ac16 ≤
a6, a0 ≤ ac17 , a7 ≤ ac17 , where bi ≡ aa3i , ci ≡ a
a23i , di ≡ a
a33i (for i = 0, 1).
Without x1, x2, these were the initial defining relations in [15], and all wererealised in Aut(R,≤); and in any case, all the generators and defining relationsof H are contained in those of H(f) when f is recursive. By Corollary 2.3, wecan `-embed any finitely generated Abelian `-group, A, in Aut(R,≤) so thatsupp(y) ⊂ supp(a0) for all y ∈ A with α|y|n ≤ αa0 for all n ∈ N and α ∈ R.Thus we can realise any two generator rank 2 Abelian `-group in this way so
18
that all the relations of H hold; in particular, D(ξ) for any positive irrationalreal number ξ.
Since we are considering a finitely generated recursively related Abelian `-group of rank 2, we can obtain it from a recursive sequence of linear maps fromZ
2 to Z2 (each of which is given by the unimodular 2×2 matrix I2, E1,2 or E2,1)as described in Section 3. Let M0,M1,M2 be these three matrices. Given anyfunction f : N → N, let Mf(n) = Mk where k ∈ 0, 1, 2 and f(n) ≡ k (mod3).
Let f be a function from N to N. We call f presentable if there are afinite number of generators including f as well as those of H(f), and a finitenumber of defining relations including those of H(f) as well as [f , d1] = 1 =f(f ∧ d0)−1 ∧ d0, so that, in a resulting finitely presented `-group, L(f), forall m ∈ N we have (∗m), (∗∗m), and
(†m) xj ∗ (cm1 f) = x ∗ (ejMf(m)cm1 ) (j = 1, 2)
where e1 = (1, 0) and e2 = (0, 1). Moreover, all these relations hold in thenatural interpretation in Aut(R,≤) given in [15] as modified above.
Lemma 5.1 Every recursive function f : N→ N is presentable.
Proof: Consider first the zero function θ. We adjoin to H(θ) the generatorθ and the relation θ = 1. Let L(θ) be the resulting finitely presented `-group.Then (∗m) and (∗∗m) hold in L(θ) for all m since they hold in H(θ), and,since M0 = I2, we have for j = 1, 2 and all m ∈ N,
xj ∗ (cm1 θ) = xj ∗ cm1 = x ∗ (ejM0cm1 ) = x ∗ (ejMθ(m)c
m1 ).
Thus (†m) also holds in L(θ) for all m ∈ N. Hence θ is presentable.
Next consider the successor function s. To H(s) we adjoin a new generator,s and relations [s, c3
1] = [s, d1] = 1 = s(s ∧ d0)−1 ∧ d0, and for j = 1, 2,
xj ∗ s = x ∗ (ejM1), xj ∗ (c1s) = x ∗ (ejM2c1), xj ∗ (c21s) = x ∗ (ejM0c
21).
Let L(s) be the resulting finitely presented `-group. So (∗m) and (∗∗m) holdin L(s) for all m ∈ N since they hold in H(s). Then
xj ∗ (c3n1 s) = xj ∗ (sc3n
1 ) = x ∗ (ejMs(3n)c3n1 ),
xj ∗ (c3n+11 s) = xj ∗ (c1sc
3n1 ) = x ∗ (ejMs(3n+1)c
3n+11 ),
andxj ∗ (c3n+2
1 s) = xj ∗ (c21sc
3n1 ) = x ∗ (ejMs(3n+2)c
3n+21 ).
19
Hence (†m) also holds in L(s) for all m ∈ N. Therefore s is presentable.
Before proceeding further, we need to appreciate a consequence of thedefining relations of H (and so of H(f)).
Now a0 ∗ cm1 ≤ a6 for all m ∈ N ([15], Lemma 1(x)). Since xj ≤ a0, itfollows that xj ∗ cm1 ≤ a6 (j = 1, 2). Therefore if m ∈ N and j = 1, 2, then
if zt−1 ∧ a6 = 1, then xj ∗ (cm1 z) = xj ∗ (cm1 t) (III).
[For if α ∈ supp(xj ∗ cm1 ), then α(xj ∗ cm1 )a6 > α(xj ∗ cm1 ). Thus,
α(xj ∗ cm1 ) = α(xj ∗ cm1 )(zt−1 ∧ a6) = α(xj ∗ cm1 )zt−1 = α(xj ∗ (cm1 zt−1)),
by Proposition 2.4. Hence xj ∗ (cm1 z) = xj ∗ (cm1 t) for all m ∈ N and j = 1, 2.]
Next we consider the composition of two presentable recursive functions fand g. Let L(f, g) be a finitely presented `-group containing the presentationL(f) ∪ L(g) ∪ H(fg) in any appropriate way. We add to L(f, g) a furthergenerator fg and relations fg(aga4fa
−14 a−1
g )−1 ∧ a6 = 1 and [fg, d1] = 1 =fg(fg ∧ d0)−1 ∧ d0. This is L(fg). By (III),
xj ∗ (cm1 fg) = xj ∗ (cm1 aga4fa−14 a−1
g ) for all m ∈ N, j = 1, 2.
Then (∗m) and (∗∗m) hold in L(fg) for all m ∈ N since they hold in H(fg).For j = 1, 2 and m ∈ N, we have
xj ∗ (cm1 fg) = xj ∗ (cm1 aga4fa−14 a−1
g ) = xj ∗ (bg(m)1 cm1 a4fa
−14 a−1
g ) =
xj ∗ (a4a−14 b
g(m)1 cm1 a4fa
−14 a−1
g ).
By (I) and (II), we deduce that
xj ∗ (cm1 fg) = xj ∗ (cg(m)1 dm1 fa
−14 a−1
g ) = xj ∗ (cg(m)1 fdm1 a
−14 a−1
g ) =
x ∗ (ejMf(g(m))cg(m)1 dm1 a
−14 a−1
g ) = x ∗ (ejMf(g(m))a−14 a4c
g(m)1 dm1 a
−14 a−1
g ) =
x ∗ (ejMf(g(m))a−14 b
g(m)1 cm1 a
−1g ).
Now x ∗ejMf(g(m)) is just xr1(j)1 x
r2(j)2 where (r1(j), r2(j)) = ejMf(g(m)). Since
a4 commutes with x1 and x2, we have
x ∗ (ejMf(g(m))a−14 ) = x ∗ ejMf(g(m)).
Therefore (since xj ∗ (bg(m)1 cm1 a
−1g ) = xj ∗ cm1 ), the same considerations con-
cerning (x ∗ (ejMf(g(m))) give
x∗(ejMf(g(m))a−14 b
g(m)1 cm1 a
−1g ) = x∗(ejMf(g(m))b
g(m)1 cm1 a
−1g ) = x∗(ejMf(g(m))c
m1 ).
20
Hencexj ∗ (cm1 fg) = x ∗ (ejMf(g(m))c
m1 ).
This establishes (†m), whence composition of presentable recursive functionsis presentable.
Finally we must consider the general recursion case. Let g, h, u, v be pre-sentable recursive functions with the desired relations holding in the finitelypresented `-group L(g, h, u, v). If f is obtained from g, h, u, v by general re-cursion, then we adjoin to L(g, h, u, v) ∪ H(f) the new generator f and therelations that ensure that f obeys the required commutator laws and cor-responds appropriately with respect to composition when restricted to d0.That is, [f , d1] = 1 = f(f ∧ d0)−1 ∧ d0, u(aga4fa
−14 a−1
g )−1 ∧ a6 = 1 and(aha4fa
−14 a−1
h )(afa4va−14 a−1
f )−1 ∧ a6 = 1.Let L(f) be the resulting finitely presented `-group.Then (∗m) and (∗∗m) hold in L(f) for all m ∈ N since they hold in H(f).Now if m ∈ range(h0g), say m = g(n), then for j = 1, 2, by (III) we get
xj ∗ (cn1aga4f) = xj ∗ (cn1 uaga4).
Using similar steps to those given for the proof of composition of pre-sentable functions, we obtain
xj ∗ (cm1 fdn1 ) = x ∗ (ejMu(n)c
m1 d
n1 ) (j = 1, 2).
Therefore,
xj ∗ (cm1 f) = x ∗ (ejMf(m)cm1 ) (j = 1, 2),
as desired.Now assume (†n) holds if n ∈ range(hkg) and let m = h(n) for such an n.
Then, by (III),
xj ∗ (cn1aha4f) = xj ∗ (cn1afa4va−14 a−1
f aha4) (j = 1, 2).
That is,
xj ∗ (cm1 fdn1 ) = x ∗ (ejMv(f(n))c
f(n)1 dn1a
−14 a−1
f aha4) (j = 1, 2).
Consequently,
xj ∗ (cm1 f) = x ∗ (ejMf(m)cm1 ) (j = 1, 2),
as desired.
21
This completes the proof of the lemma. //
Let A be a two generator Abelian `-group of rank 2. Let y1, y2 ∈ A+
be generators of A. As already noted, we can identify A with a sublatticesubgroup of Aut(R,≤) which is the interpretation of the sublattice subgroupC of L(f) generated by x1, x2. That is, A is an `-homomorphic image ofC under the natural map induced by φ : xj 7→ yj (j = 1, 2). As shown inSection 3, we can realise A as a standard direct system given by a recursivesequence of maps Mf(n) from Z
2 to Z2. Moreover, ym1 yn2 ≥ 1 in A iff for some
` ∈ N we have (m,n)Mf(0)Mf(1) . . .Mf(`) ≥ (0, 0) in the cardinal order on Z2.
But, in L(f) we have x(m,n) ≥ 1 iff x(m,n)(f c1)` ≥ 1 iff x(m,n)Mf(0)...Mf(`) ≥ 1.Since x1, x2 ≥ 1 we see that the map ψ : yj 7→ xj (j = 1, 2) induces an`-homomorphism of A into L(f). Now ψφ is the identity on A, whence ψis an `-embedding. This completes the proof of Theorem A in the case thatthe Abelian `-group has two generators and rank 2 (and hence the proof ofCorollary 1.1). //
By Proposition 4.1 we may regard any finitely generated Abelian `-groupof rank n as a standard direct system with n generators y1, . . . , yn. So theonly modification that is needed for rank n instead of rank 2 is to considerthe n2 − n+ 1 standard matrices of En in place of the three in E2. We adjoinx1, . . . , xn with the same relations as we gave above for xj , with j ∈ 1, . . . , ninstead of j ∈ 1, 2, requiring additionally that [xi, xj ] = 1 (i, j ∈ 1, . . . , n).The proof of the presentability of all recursive functions of one variable isidentical except that for s we need the relations [s, cn
2−n+11 ] = 1 and x
cm1 sj =
xejMs(m)cm1 for j = 1, . . . , n and m = 0, . . . , n2 − n; the deduction afterwards
is as above. This completes the proof of Theorem A.//
Acknowledgements: We wish to thank Jorge Martinez for inviting usboth to the Ordered Algebraic Structures Conference in February/March 2000at the University of Florida, Gainesville; it was instrumental in starting thisresearch.
This work was done in Michaelmas Term 2001 while the second authorwas visiting Cambridge. He is most grateful to DSI, Milano and the ItalianMURST Project on Many-Valued Logic, ex 40%, for the funds that madethis possible, and to the members of DPMMS for their support. He wouldalso like to thank Dr. Rosemary Summers and her family for their wonderfulhospitality.
Finally, during the revision of this paper, we learnt of the death of B.H. Neumann. He was one of the great pioneers of both combinatorial group
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theory and ordered groups. We would like to believe that this combination ofthose ideas would have interested him. Sadly, we can only dedicate this paperto him, in memoriam.
References
[1] L. V. Ahlfors, Complex Analysis, McGraw-Hill, New York, 1953.
[2] A. Baker, A concise introduction to the Theory of Numbers, Cam-bridge University Press, 1984.
[3] K. Baker, Free vector lattices, Canadian J. Math. 20 (1968), 58–66.
[4] L. Berrgren, J. Borwein, and P. Borwein, Pi: a source book, Springer-Verlag, Heidelberg, 2000 (2nd edition).
[5] W. M. Beynon, Applications of duality in the theory of finitely generatedlattice-ordered groups, Canadian J. Math. 29 (1977), 243–254.
[6] G. Birkhoff, Lattice-ordered groups, Annals of Math. 43 (1942), 298–331.
[7] Black Swamp Problem Book, (ed. W. C. Holland), Bowling GreenState University, Ohio, USA.
[8] P. F. Conrad, The structure of a lattice-ordered group with a finite numberof disjoint elements, Michigan Math. J. 7 (1960), 171–180.
[9] H. Davenport, The Higher Arithmetic, Cambridge University Press,1952.
[10] C. De Concini & C. Procesi, Complete symmetric varieties. II. Intersec-tion theory, in Algebraic groups and related topics (Kyoto/Nagoya,1983), North-Holland, Amsterdam, 1985, 481–513.
[11] E. G. Effros, D. E. Handelman & C.-L. Shen, Dimension groups and theiraffine representations, American J. Math. 102 (1980), 385 - 407.
[12] G. Ewald, Combinatorial Convexity and Algebraic Geometry,Springer-Verlag, Heidelberg, 1996.
[13] W. Fulton, An introduction to Toric Varieties, Annals Math. Studies131, Princeton University Press, 1993.
23
[14] A. M. W. Glass, Partially Ordered Groups, Series in Algebra 7, WorldScientific Pub. Co., Singapore, 1999.
[15] A. M. W. Glass and Y. Gurevich, The word problem for lattice-orderedgroups, Trans. American Math. Soc. 280 (1983), 127–138.
[16] G. H. Hardy and E. M. Wright, Introduction to the Theory of Num-bers, Clarendon Press, Oxford, 1979 (5th edition).
[17] G. Higman, Subgroups of finitely presented groups, Proc. Royal Soc. Lon-don, Series A, 262 (1961), 455–475.
[18] W. C. Holland, The lattice-ordered group of automorphisms of an orderedset, Michigan Math. J. 10 (1963), 399–408.
[19] P. Jaffard, Extensions des groupes reticules et applications, Publ. Sci.Universite Algeria, Series A, 1 (1954), 197–222.
[20] Kourovka Note Book, Unsolved Problems in Group Theory.
[21] R. C. Lyndon and P. E. Schupp, Combinatorial Group Theory,Ergebnisse der Math. 89, Springer-Verlag, Heidelberg, 1977.
[22] V. Marra, Every abelian `-group is ultrasimplicial, J. Algebra 225 (2000),872–884.
[23] V. Marra, Non-Boolean partitions: a mathematical investigationthrough lattice-ordered Abelian groups and MV-algebras, Ph.D.Thesis, Universita degli Studi di Milano, 2002.
[24] V. Marra and D. Mundici, MV -algebras and Abelian `-groups: a fruit-ful interaction, in Ordered algebraic structures, (ed. J. Martinez),Kluwer Acad. Pub., Dordrecht, 2002, 57–88.
[25] V. Marra and D. Mundici, Combinatorial fans, lattice-ordered groups,and their neighbours: a short excursion, Sem. Lothar. Comb. 47 (2001),Article B47f.
[26] D. Mundici, Farey stellar subdivisions, ultrasimplicial groups, and K0 ofAF C∗-algebras, Advances in Math. 68 (1988), 23 –39.
[27] D. Mundici and G. Panti, Decidable and undecidable prime theories ininfinite-valued logic, Annals of Pure and Applied Logic 108 (2001), 269–278.
24
[28] J. Robinson, Recursive functions of one variable, Proc. American Math.Soc. 19 (1968), 815–820.
[29] H. Rogers, Jr., Theory of Recursive Functions and Effective Com-putability, McGraw-Hill, New York, 1967.
[30] C. P. Rourke and B. J. Sanderson, Introduction to piecewise-lineartopology, Ergebnisse Math. 69, Springer-Verlag, Heidelberg, 1972.
[31] H. M. Stark, An introduction to Number Theory, MIT Press, 1970.
Authors’ addresses:
Department of Pure Mathematics and Mathematical Statistics,Centre for Mathematical Sciences,Wilberforce Rd.,Cambridge CB3 0WB,England.
DSI,Universita degli Studi di Milano,via Comelico 39,20100 Milano,Italy.
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