ARCHIMEDEAN UNITAL GROUPS WITH FINITE UNIT INTERVALS

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ARCHIMEDEAN UNITAL GROUPS WITH FINITE UNIT INTERVALS

DAVID J. FOULIS Received 23 October 2002

LetGbe a unital group with a finite unit intervalE, letnbe the number of atoms in E, and letκbe the number of extreme points of the state spaceΩ(G). We introduce canonical order-preserving group homomorphisms ξ:Zⁿ →G andρ:G→Z^κ linkingGwith the simplicial groupsZⁿandZ^κ. We show thatξis a surjection and ρis an injection if and only ifGis torsion-free. We give an explicit construction of the universal group (unigroup) forEusing the canonical surjectionξ. IfGis torsion-free, then the canonical injectionρis used to show thatGis Archimedean if and only if its positive cone is determined by a finite number of homogeneous linear inequalities with integer coefficients.

2000 Mathematics Subject Classification: 06F20.

1. Introduction and basic definitions. In this paper, we continue the study of unital groups with finite unit intervals begun in [2,3]. Motivation for this study can be found in [2]. Although we will attempt to keep this paper some- what self-contained, we make free use of the notation, nomenclature, and re- sults of [2,3].

We begin by setting forth notation and recalling some basic definitions. We write a partially ordered abelian groupGadditively, and denote the positive cone inGbyG⁺:= {g∈G|0≤g}[7]. IfG⁺generatesGas an abelian group, that is, ifG=G⁺−G⁺, thenGis said to bedirected. A subsetFofG⁺iscone gen- eratingif and only if every element ofG⁺is a sum of a finite sequence of (not necessarily distinct) elements ofF. Various definitions of “Archimedean” can be found in the literature. We use the following [7, page 20]:GisArchimedean if and only if, fora,b∈G, the conditionna≤bfor every positive integern implies thata≤0.

IfGis a partially ordered abelian group andu∈G⁺, we define theinterval E:=G⁺[0,u]:= {g∈G|0≤g≤u}. ThusEforms a bounded partially ordered set under the restriction of the partial order≤onGtoE. The intervalEcan be organized into aneffect algebraunder the partial binary operation⊕obtained by restriction of+toE. For the details see [1,5].

An elementu∈G⁺is called anorder unitif and only if each element ofGis dominated by a positive integer multiple ofu[7, page 4]. Aunital group[2] is a partially ordered abelian groupGwith a specified order unitu, called theunit, such that the intervalE:=G⁺[0,u], called theunit interval, is cone generating.

IfGis a unital group with unitu, thenGis directed, andG= {0}u=0. IfG is directed,u∈G⁺, andG⁺[0,u]is cone generating, thenGis a unital group with unitu.

If G is a unital group with unit interval E, and ifK is an abelian group, then a mapping φ:E→Kis called a K-valued measure onE if and only if p,q,p+q∈E⇒φ(p+q)=φ(p)+φ(q). For instance, ifΦ:G→Kis a group homomorphism, then the restrictionφ:=Φ|EofΦtoEis aK-valued measure onE. If every K-valued measure onE is the restriction toE of a group ho- momorphism fromGtoK, thenGis called aK-unital group. Aunigroupis a unital group that isK-unital for every abelian groupK[5,6].

LetGandHbe unital groups with unit intervalsEandL, respectively, and unitsuandv, respectively. A mappingφ:E→Lis aneffect-algebra morphism [5, Definition 6.1] if and only ifφ(u)=vand, regarded as a mappingφ:E→ H, it is anH-valued measure. Letφ:E→L be an effect-algebra morphism.

If p_i∈E, k_i are nonnegative integers for i=1,2,...,n, and n

i=1k_ip_i ∈E, thenφ(n

i=1k_ip_i)=n

i=1k_iφ(p_i)∈L. Ifφ:E→Lis a bijective effect-algebra morphism andφ⁻¹:L→Eis also an effect-algebra morphism, thenφ:E→L is aneffect-algebra isomorphism.

We use the usual notationR, Q, andZ for the ordered field of real num- bers, the ordered field of rational numbers, and the ordered ring of integers, respectively. Thus, thestandard positive coneinRisR⁺:= {x²|x∈R}and thestandard positive conesinQandZareQ⁺:=Q∩R⁺andZ⁺:=Z∩Q⁺. We often disregard the multiplicative structures ofR,Q, andZand regard them as partially ordered additive abelian groups. As such, and with 1 as the unit, R,Q, andZprovide examples of unigroups.

LetG= {0}be a unital group with unituand unit intervalE. Then astatefor Gis a group homomorphismω:G→Rsuch thatω(G⁺)⊆R⁺andω(u)=1.

The set of all states forG, called thestate spaceofG, is denoted byΩ(G). By [7, Corollary 4.4, Proposition 6.2],Ω(G)is a nonempty compact convex subset of the locally convex Hausdorff linear topological spaceR^Gwith the topology of pointwise convergence. Ifω∈Ω(G)andω(G)⊆Q, thenωis called aQ-valued state. Evidently,ω∈Ω(G)isQ-valued if and only ifω(E)⊆Q⁺. Aprobability measureonEis anR-valued measureπonEsuch thatπ(E)⊆[0,1]⊆Rand π(u)=1. The restriction toEof a stateω∈Ω(G)is a probability measure on E. IfGisR-unital, then each probability measureπ onEis the restriction to Eof a uniquely determined stateω∈Ω(G).

Let∆⊆Ω(G). Then∆is said to bestrictly positive if and only if, for each 0=p ∈G⁺, there exists ω∈ ∆ with 0< ω(p). If ω∈Ω(G), and {ω} is strictly positive, we say that ωis a strictly positive state. IfG⁺ = {p ∈G| 0≤ω(p)for allω∈∆}, then∆is said to becone determining. By [7, Theorem 4.14], G is Archimedean if and only ifΩ(G) is cone determining. By defini- tion,∆isseparating if and only if, for allg=0∈G, there existsω∈∆with ω(g)=0.

Ifr is a positive integer, we understand thatR^r,Q^r, andZ^r are organized into additive abelian groups with coordinatewise operations. Thestandard par- tial order for each of these groups is the coordinatewise partial order deter- mined by the standard total orders on R, Q, and Z, and the corresponding standard positive conesare(R⁺)^r,(Q⁺)^r, and(Z⁺)^r.

With the standard partial order, Z^r forms the so-called simplicial group [7, page 47]. As a simplicial group, Z^r is an Archimedean lattice-ordered group with the smallest order unit, namely(1,1,...,1). An elementv∈(Z⁺)^r is an order unit if and only if all of its coordinates are strictly positive. If v is an order unit in the simplicial groupZ^r, then the unit interval(Z⁺)^r[0,v]

forms a finite MV-algebra [4]. Conversely, every finite MV-algebra has this form. Ifv=(1,1,...,1), then(Z⁺)^r[0,v] is isomorphic to the finite Boolean algebra2^r.

This paper is focused on the study of a unital group G with a finite unit intervalEand on the question of just whenGis Archimedean (i.e., carries a cone-determining set of states). IfGcarries a cone-determining set of states, then it is clear that Gis torsion-free, that is, 0 is the only element of finite order inG, so in Sections4and5we will be paying special attention to the torsion-free case. IfG= {0}is a unital group with a finite unit intervalE, then there are atoms (minimal nonzero elements) inE, and every nonzero element inEdominates at least one atom. Thus, we will be working with the following data.

1.1. Standing assumptions and notation. For the remainder of this paper, Gis a unital group with unitu=0 and with a finite unit intervalE=G⁺[0,u].

The distinct atoms inEare denoted bya1,a2,...,an.

By [2, Lemma 5.1], the finite set{a1,a2,...,a_n} ⊆E⊆G⁺is cone generating, and since G is directed, {a1,a2,...,an}is a finite set of generators for the abelian group G. By the fundamental theorem for finitely generated abelian groups, the torsion subgroupG_τofGis finite andGis a direct sum ofG_τand a torsion-free subgroupH⊆Gof finite rankr >0. Ifη:G→His the natural projection homomorphism with ker(η)=Gτ, then by [2, Theorem 4.1]Hcan be organized into a unital group with unitη(u), with positive coneH⁺=η(G⁺), and with a finite unit intervalL. Moreover, there is an affine isomorphismω

ωfrom Ω(G)onto Ω(H)such that ω=ω◦ηfor all ω∈Ω(G). As H is a torsion-free group of finite rankr, there is a group isomorphismφ:H→Z^r and, by [3, Lemma 3.2],φcan be chosen in such a way thatφ(H⁺)⊆(Z⁺)^r. By [2, Lemma 3.5], the set ofQ-valued states onHis separating.

2. The canonical surjectionξ. In this section, we introduce a surjective order-preserving group homomorphismξfrom the simplicial groupZⁿonto the unital groupG. Recall that ifZis a free abelian group,B⊂Zis a free basis forZ,Kis an abelian group, andf:B→Kis a function, then there is a unique group homomorphismφ:Z→Kthat agrees withf onB.

Definition2.1. Letd1=(1,0,...,0),d2=(0,1,...,0),...,dn=(0,0,...,1) be the standard free basis for the additive abelian groupZⁿ. Define thecanoni- cal surjectionξ:Zⁿ→Gto be the uniquely determined group homomorphism ξ:Zⁿ→Gsuch thatξ(di)=aifori=1,2,...,n.

Lemma2.2. (i)Ifx=(x1,x2,...,xn)∈Zⁿ, thenξ(x)=_n

i=1xiai. (ii)The mappingξ:Zⁿ→GsatisfiesG⁺=ξ((Z⁺)ⁿ).

(iii)The mappingξ:Zⁿ→GsatisfiesG=ξ(Zⁿ).

(iv)The mappingξ:Zⁿ→Gis a surjective order-preserving group homomor- phism and its kernel satisfies the conditionker(ξ)∩(Z⁺)ⁿ= {0}.

Proof. Part (i) follows immediately fromDefinition 2.1. Part (ii) follows from (i) and the fact that {a1,a2,...,a_n} is a cone-generating set. Part (iii) follows from (ii) and the fact thatG is directed. To prove (iv), we begin by observing that the group homomorphismξ:Zⁿ→G is order preserving by (ii) and surjective by (iii). Supposex=(x1,x2,...,x_n)∈ker(ξ)∩(Z⁺)ⁿ. Then n

i=1x_ia_i=0 and 0≤x_ifori=1,2,...,n. Sincex_ia_i∈G⁺fori=1,2,...,n, it follows thatxiai=0 fori=1,2,...,n. Therefore, since 0=ai∈G⁺, we have x_i=0 fori=1,2,...,n.

Definition2.3. DefineT:=ξ⁻¹(u)∩(Z⁺)ⁿ. Vectorst∈T are calledmulti- plicity vectorsforG(cf. [2, Definition 5.2]). DefineDto be the subgroup ofZⁿ generated by the set of all differencest−sfort,s∈T, letG^∗ be the quotient groupZⁿ/D, and letξ^∗:Zⁿ→G^∗be the natural surjective group homomor- phism with ker(ξ^∗)=D.

Lemma2.4. (i)Ifs,t∈T, thens≤t⇒s=t.

(ii)The setT is finite and nonempty.

(iii)Ifx∈(Z⁺)ⁿ, thenξ(x)∈E∃t∈T,x≤t.

Proof. (i) Ifs,t∈T and s≤t, then t−s∈ker(ξ)∩(Z⁺)ⁿ, so s=t by Lemma 2.2(iv).

(ii) By (i),T forms an antichain in the positive cone(Z⁺)ⁿof the simplicial groupZⁿ, henceT is a finite set [8].

(iii) Letx∈(Z⁺)ⁿ. Ift∈T andx≤t, then 0≤ξ(x)≤ξ(t)=u, soξ(x)∈E.

Conversely, supposeξ(x)∈E. Thenu−ξ(x)∈G⁺, so there existsy∈(Z⁺)ⁿ withξ(y)=u−ξ(x). Therefore,x+y∈(Z⁺)ⁿwithξ(x+y)=u, and it follows thatt:=x+y∈T withx≤t.

Definition2.5. LetT = {t1,t2,...,tm}. Then them×nmatrix[t_ij] with t1,t2,...,t_mas its successive row vectors is called the multiplicity matrixfor G. The m×(n+1) matrixM obtained by appending a final column to the matrix[t_ij]consisting entirely of−1’s is called therelation matrix forG(cf.

[2, Definition 5.3]).

The relation matrixMencodesmfundamental relations_n

j=1tijaj−u=0 fori=1,2,...,msatisfied by the generatorsa_jforGand the unitu(see [2, Theorem 5.4]).

Theorem2.6. (i) rank(T )=rank(M).

(ii) rank(G)+rank(T )≤n+1.

(iii)IfGis torsion-free, thenGisR-unital if and only ifrank(G)+rank(T )= n+1.

Proof. We will prove that the last column ofM is a rational linear com- bination of its first ncolumns, from which (i) follows. There is a Q-valued state ω∈Ω(G). Let q_j := −ω(a_j) for j =1,2,...,n. As n

i=1t_ija_j =u for i=1,2,...,m, we haven

i=1tijqj= −ω(u)= −1 fori=1,2,...,m. Parts (ii) and (iii) now follow from [2, Theorem 5.6].

Theorem2.7. The abelian groupG^∗can be organized into a unigroup with unitu^∗ :=ξ^∗(t), independent of the choice of t∈T, and with positive cone (G^∗)⁺:=ξ^∗((Z⁺)ⁿ). Then there is an effect-algebra isomorphismpp^∗from Eonto the unit intervalE^∗:=(G^∗)⁺[0,u^∗], andξ^∗is the canonical surjection forG^∗. Moreover, there is a surjective order-preserving group homomorphism β:G^∗→Gsuch thatβ((G^∗)⁺)=G⁺,β(p^∗)=pfor allp∈E, andξ=β◦ξ^∗.

Proof. Evidently,D⊆ker(ξ), whenceD∩(Z⁺)ⁿ= {0}byLemma 2.2(iv), so G^∗can be organized into a partially ordered abelian group with positive cone (G^∗)⁺:=ξ^∗((Z⁺)ⁿ). Since the simplicial groupZⁿis directed andξ^∗:Zⁿ→G^∗ is a surjective order-preserving group homomorphism, it follows thatG^∗ is directed. Clearly,u^∗:=ξ^∗(t)is independent of the choice oft∈T, whence u^∗∈(G^∗)⁺.

We claim that ifx∈(Z⁺)ⁿ, then

ξ(x)∈E⇐⇒ξ^∗(x)∈E^∗. (2.1)

To prove (2.1), supposex∈(Z⁺)ⁿ. Ifξ(x)∈E, then byLemma 2.4(iii) there ex- istst∈Twithx≤t, whence 0≤ξ^∗(x)≤ξ^∗(t)=u^∗, soξ^∗(x)∈(G^∗)⁺[0,u^∗]= E^∗. Conversely, ifξ^∗(x)∈E^∗, there existsy∈(Z⁺)ⁿ such thatu^∗−ξ^∗(x)= ξ^∗(y), whenceξ^∗(x+y)=ξ^∗(t)for any choice oft∈T,x+y−t∈D⊆ker(ξ), andξ(x)+ξ(y)=ξ(t)=u. Therefore,ξ(x)∈E, and (2.1) follows.

For eachi=1,2,...,n, we havedi∈(Z⁺)ⁿwithξ(di)=ai∈E, whence by (2.1),ξ^∗(di)∈E^∗. Since(Z⁺)ⁿis the set of all linear combinations ofd1,d2,..., d_nwith nonnegative integer coefficients, it follows that{ξ^∗(d_i)|i=1,2,...,n} is a finite cone-generating subset ofE^∗inG^∗. Therefore,u^∗is an order unit in(G^∗)⁺andG^∗is a unital group with unitu^∗and unit intervalE^∗.

We claim that ifx,y∈(Z⁺)ⁿ, then

ξ(x)=ξ(y)∈E⇐⇒ξ^∗(x)=ξ^∗(y)∈E^∗. (2.2)

To prove (2.2), supposex,y∈(Z⁺)ⁿ. Ifξ(x)=ξ(y)∈E, then there existsz∈ (Z⁺)ⁿ such that u−ξ(x)=ξ(z), whencex+z∈T and likewise y+z∈T, sox−y=(x+z)−(y+z)∈D=ker(ξ^∗), andξ^∗(x)=ξ^∗(y)∈E^∗ by (2.1).

Conversely, ifξ^∗(x)=ξ^∗(y)∈E^∗, thenx−y∈ker(ξ^∗)=D⊆kerξ, soξ(x)= ξ(y)∈Eby (2.1), proving (2.2).

Ifp∈E, we definep^∗∈E^∗as follows: choosex∈(Z⁺)ⁿwithp=ξ(x)and letp^∗:=ξ^∗(x). By (2.2),p^∗is well defined and the mapping^∗:E→E^∗ given bypp^∗is a bijection. Evidently, this notation is consistent withu^∗=ξ^∗(t), t∈T, as defined previously. Because ker(ξ^∗)=D⊆ker(ξ), there is a uniquely determined surjective group homomorphismβ:G^∗→Gsuch thatβ◦ξ^∗=ξ.

Clearly,β((G^∗)⁺)=ξ((Z⁺)ⁿ)=G⁺, soβis order preserving. Ifh∈E^∗, there existsx∈(Z⁺)ⁿwithh=ξ^∗(x), whenceβ(h)=β(ξ^∗(x))=ξ(x)∈Eby (2.1) and(β(h))^∗=ξ^∗(x)=h. Ifp∈E, there existsx∈(Z⁺)ⁿ withp=ξ(x)∈E, whencep^∗=ξ^∗(x)∈E^∗ with β(p^∗)=β(ξ^∗(x))=ξ(x)=p. Therefore, the restrictionβ|E^∗ ofβtoE^∗is a bijective effect-algebra morphism ofE^∗ontoE with^∗:E→E^∗as its inverse.

Supposep,q,p+q∈Eand choosex,y∈(Z⁺)ⁿwithp=ξ(x)andq=ξ(y).

Thenx+y∈(Z⁺)ⁿ withξ(x+y)=p+q∈E. Therefore,(p+q)^∗ =ξ^∗(x+ y)=ξ^∗(x)+ξ^∗(y)=p^∗+q^∗, so^∗ :E→E^∗ is an effect-algebra morphism.

Consequently, both^∗:E→E^∗ and its inverseβ|E^∗:E^∗→Eare effect-algebra isomorphisms.

Because^∗:E→E^∗is an effect-algebra isomorphism, the atoms inE^∗area^∗_i fori=1,2,...,n. Also,ξ^∗(di)=a^∗_i fori=1,2,...,n, and it follows thatξ^∗ is the canonical surjection forG^∗.

To prove thatG^∗is a unigroup, suppose thatKis an abelian group andφ: E^∗→Kis aK-valued measure. DefineΦ:Zⁿ→Kto be the unique group homo- morphism such thatΦ(d_i)=φ(a^∗_i)fori=1,2,...,n. Supposet=(t₁,t₂,...,t_n)

∈T. Thus, u=n

i=1t_ia_i, whence u^∗=n

i=1t_ia^∗_i, and sinceφis aK-valued measure,φ(u^∗)=_n

i=1tiφ(a^∗_i)=_n

i=1tiΦ(di)=Φ(_n

i=1tidi)=Φ(t). There- fore, ift,s∈T, we haveΦ(t−s)=φ(u^∗)−φ(u^∗)=0, so ker(ξ^∗)=D⊆ker(Φ), and it follows that there exists a group homomorphismφ^∗:G^∗ →K such that φ^∗◦ξ^∗ =Φ. Consequently,φ^∗(a^∗_i)=φ^∗(ξ^∗(di))=Φ(di)=φ(a^∗_i)for i=1,2,...,n. Since every element inE^∗is a linear combination of the atoms a^∗_i with nonnegative integer coefficients andφis aK-valued measure, it fol- lows that the group homomorphismφ^∗agrees withφonE^∗.

Corollary2.8. The unital groupGis a unigroup if and only if ker(ξ)⊆D. The unigroupG^∗inTheorem 2.7is uniquely determined (up to an isomor- phism of unital groups) by the structure of the effect algebraE, and it is called theunigroup for the effect algebraE [1]. As can be seen from the proof of Theorem 2.7, the structure ofG^∗ is encoded in the setT of multiplicity vec- tors, hence it is implicit in the canonical surjectionξ.

3. Q-valued states. We maintain the assumptions and notation ofSection 1.1andDefinition 2.3. In this section, we establish a bijective correspondence ω↔ω¯ between Q-valued states ω∈Ω(G) and surjective order-preserving

group homomorphisms ¯ω:G→Z, and we use the mappingωω¯ to define an order-preserving group homomorphismρfromGinto a simplicial group Z^κ.

By [3, Theorem 5.3], the state spaceΩ(G)is a polytope and its set of extreme points∂e(Ω(G))is a finite set ofQ-valued states. A stateω∈Ω(G)is said to bedispersion free if and only if it takes on only the values 0 and 1 on the unit intervalE=G⁺[0,u]. Every dispersion-free stateω∈Ω(G)belongs to

∂e(Ω(G)).

Notation 3.1. Let {ω1,ω2,...,ωκ} := ∂e(Ω(G)) be the set of extreme points of the polytopeΩ(G).

Lemma 3.2. Let φ: G→Q be a group homomorphism and assume that φ(G⁺)⊆Q⁺(i.e.,φis order preserving). Then

(i) 0< φ(u)if and only if there is at least onei∈ {1,2,...,n}with 0<

φ(a_i);

(ii) φ(u)=0if and only ifφis the zero homomorphism;

(iii) ifφ(u)=0, thenω:G→Qdefined byω:=(1/φ(u))φis aQ-valued state.

Proof. (i) If 0< φ(a_i), then the fact that a_i ≤ u implies 0< φ(a_i)≤ φ(u). Conversely, suppose 0< φ(u)and lett=(t1,t2,...,t_n)∈T. Thenu= _n

i=1tiai, whence 0< φ(u)=_n

i=1tiφ(ai), and it follows that at least one φ(a_i)must be strictly positive.

(ii) Supposeφ(u)=0. Thenφ(a_i)=0 for i=1,2,...,nby (i) and, since {a1,a2,...,an}is a set of generators forG, it follows thatφis the zero homo- morphism. The converse is obvious.

(iii) Suppose φ(u)=0. Thenω=(1/φ(u))φ is a group homomorphism fromGtoQ,ω(G⁺)⊆Q⁺, andω(u)=1.

Lemma3.3. Letζ:G→Z be a group homomorphism. Then the following conditions are mutually equivalent:

(i) ζ(G)=Z; (ii) ζ⁻¹(1)= ∅;

(iii) the nonzero integers in the list ζ(a1),ζ(a2),...,ζ(an) are relatively prime.

Proof. (i)(ii). Obviously (i)⇒(ii). Conversely, if (ii) holds, there existsg1∈ Gwithζ(g1)=1, whenceζ(kg1)=kζ(g1)=kandζ(G)=Z.

(ii)(iii). Suppose (ii) holds, so there existsg1∈Gwithζ(g1)=1. Because {a1,a2,...,a_n}is a set of generators forG, there are integersx1,x2,...,x_nsuch that g1=n

i=1x_ia_i, whence 1=n

i=1x_iζ(a_i), and (iii) follows. Conversely, if (iii) holds, there are integersx1,x2,...,xn such that 1=_n

i=1xiζ(ai) and g1:=n

i=1x_ia_i∈ζ⁻¹(1).

Definition 3.4. Suppose thatφ:G→Qis a nonzero group homomor- phism such thatφ(G⁺)⊆Q⁺. We define the group homomorphism ¯φ:G→Q

as follows: 0≤φ(ai) fori=1,2,...,nand 0< φ(ai)for at least one iby Lemma 3.2. Write the strictly positive rational numbers in the listφ(a1),φ(a2), ...,φ(a_n)as reduced fractions, letq >0 be the least common multiple of the denominators of the positive fractions in the list, and letp >0 be the greatest common divisor of the positive integers in the listqφ(a1),qφ(a2),...,qφ(an).

Define ¯φ:G→Qby ¯φ(g):=(q/p)φ(g)for allg∈G.

Lemma3.5. Letφ:G→Qbe a nonzero group homomorphism such that φ(G⁺)⊆Q⁺. Then

(i) ¯φ:G→Zis a surjective group homomorphism,φ(G¯ ⁺)⊆Z⁺, and0<

φ(u);¯

(ii) ω:=(1/φ(u))¯ φ¯is aQ-valued state forG;

(iii) ifω=(1/φ(u))¯ φ, then¯ ω¯=φ;¯ (iv) ¯φ=φif and only ifφ(G)=Z.

Proof. (i) By our choices ofqandp inDefinition 3.4, ¯φ(a_i)∈Z⁺fori= 1,2,...,m, at least one of these integers is positive, and the positive integers are relatively prime. Because{a1,a2,...,an}is a cone-generating set inGand φ(a¯ _i)∈Z⁺fori=1,2,...,n, we have ¯φ(G⁺)⊆Z⁺, whence, sinceGis directed, φ(G)¯ ⊆Z. ByLemma 3.3, ¯φ(G)=Z, and byLemma 3.2, 0<φ(u).¯

(ii) Thatω=(1/φ(u))¯ φ¯is aQ-valued state follows fromLemma 3.2(iii).

(iii) We haveω(a_i)=φ(a¯ _i)/φ(u)¯ fori=1,2,...,n, and the nonzero nu- merators of these fractions are relatively prime, whence ¯q := φ(u)¯ is the least common multiple of denominators of the positive fractions in the list ω(a1),ω(a2),...,ω(an), and ¯p:=1 is the greatest common divisor of the positive integers in the list ¯qω(a1),qω(a¯ 2),...,qω(a¯ _n). Therefore, ¯ω(a_i)= (¯q/p)ω(a¯ i)=φ(a¯ i)fori=1,2,...,nand, since{a1,a2,...,an}is a set of gen- erators for the groupG, it follows that ¯ω=φ.¯

(iv) If ¯φ=φ, thenφ(G)=φ(G)¯ =Zby (i). Conversely, supposeφ(G)=Z. Then φ(ai)∈ Z⁺ fori=1,2,...,n and by Lemma 3.3the positive integers in the listφ(a1),φ(a2),...,φ(an)are relatively prime. Hence, p=q=1 in Definition 3.4, and we have ¯φ=φ.

Theorem3.6. The mappingωω¯is a bijection from the set of allQ-valued states onGonto the set of all surjective order-preserving group homomorphisms ζ:G→Z.

Proof. Ifω∈Ω(G)and ω(G)⊆Q, then ¯ω:G→Z is a surjective order- preserving group homomorphism by Lemma 3.5(i). Letζ :G →Z be a sur- jective order-preserving group homomorphism. ByLemma 3.2, 0< ζ(u)and ω:= (1/ζ(u))ζ is aQ-valued state on G. By Lemma 3.5(iv), ¯ζ =ζ, so by Lemma 3.5(iii), ¯ω=ζ. We have only to prove that the mapping ωω¯ is injective on the set ofQ-valued states. Thus, supposeωis aQ-valued state onGandζ=ω. Then, by¯ Definition 3.4, there is a positive rational number λ such thatζ=λω, and it follows that ζ(u)=λω(u)=λ·1=λ, whence ω=(1/ζ(u))ζis uniquely determined byζ.

Evidently, a stateω∈Ω(G)is dispersion free if and only ifω(G)⊆Z. Thus, inTheorem 3.6, the dispersion-free states (if any) onGare exactly theQ-valued statesω∈Ω(G)such that ¯ω=ω.

Definition3.7. Since the extreme points ω1,ω2,...,ω_κ ofΩ(G) areQ- valued states, we can define ¯ωi:=ωifori=1,2,...,κ. The mappingρ:G→Z^κ is defined byρ(g):=(ω¯1(g),ω¯2(g),...,ω¯κ(g))for allg∈G. We also define v:=ρ(u)∈Z^κ.

Theorem3.8. (i)The mappingρ:G→Z^κ is a group homomorphism and ρ(G⁺)⊆(Z⁺)^κ.

(ii)The elementv=(ω¯1(u),ω¯2(u),...,ω¯κ(u))∈(Z⁺)^κ is an order unit in the simplicial groupZ^κ.

(iii)With the standard positive cone(Z⁺)^κand withvas unit, the simplicial groupZ^κis a unigroup with unit interval(Z⁺)^κ[0,v].

(iv) ker(ρ)= {g∈G|ω(g)=0for allω∈Ω(G)}.

(v)As an abelian group,Gis torsion-free if and only ifρ:G→Z^κis an injec- tion.

Proof. (i) Clearlyρis a group homomorphism and, since each ¯ωi maps G⁺intoZ⁺, it follows thatρ(G⁺)⊆(Z⁺)^κ.

(ii) We have 0<ω¯1(u),ω¯2(u),...,ω¯_κ(u), so all coordinates of the vectorv are strictly positive, and it follows thatvis an order unit in(Z⁺)^κ.

(iii) The standard free basis vectors

(1,0,0,...,0),(0,1,0,...,0),(0,0,1,...,0),...,(0,0,0,...,1) (3.1) belong to(Z⁺)^κ[0,v]and they form a set of generators for the positive cone (Z⁺)^κ. Therefore,Z^κ is a unital group with unitv. Since the simplicial group Z^κis lattice ordered andvis an order unit,Z^κis a unigroup with unitv.

(iv) Letg∈G. Thenρ(g)=0if and only if ¯ω_i(g)=0 fori=1,2,...,κ if and only ifωi(g)=0 fori=1,2,...,κ. But, since everyω∈Ω(G)is a convex combination ofω1,ω2,...,ωκ, it follows thatωi(g)=0 fori=1,2,...,κif and only ifω(g)=0 for allω∈Ω.

(v) SupposeGis torsion-free and let 0=g∈G. By [3, Lemma 3.2], there is a group isomorphismφ:G→Z^r such thatφ(G⁺)⊆(Z⁺)^r, hence by [2, Lemma 3.5] there is aQ-valued stateω∈Ω(G)with ω(g)=0, sog∈kerρby (iv).

Therefore, ifG is torsion-free, thenρ is injective. Conversely, supposeρ is injective,kis a positive integer,g∈G, andkg=0. Thenkρ(g)=0∈Z^κ, so ρ(g)=0, and thereforeg=0. Consequently,Gis torsion-free.

In Theorem 3.8, the unit interval(Z⁺)^κ[0,v] is an MV-algebra and the re- strictionρ|EofρtoEis an effect-algebra morphism ofEinto(Z⁺)^κ[0,v].

4. The canonical injection ρ. We now begin to focus on the question of just whenG is Archimedean. IfGis Archimedean, thenGis torsion-free, so in this section and the next one we will adopt as a standing hypothesis the

assumption thatG is torsion-free. Thus, by [3, Lemma 3.3], we can cast our standing hypothesis as follows.

Standing assumptions and notation. In this section and the next one, we assume thatris a positive integer,G=Z^ras an additive abelian group,Gis a unital group with unitu=(u1,u2,...,ur),G⁺⊆(Z⁺)^r, and the unit interval E=G⁺[0,u]is finite. The atoms inEare denoted bya1,a2,...,an.

All of the previous results are applicable to the unital group G, and now we have the advantage of a representation of the elements ofG as vectors inZ^r in such a way that all vectors inG⁺ have nonnegative coordinates. By Theorem 3.8(v),ρ:G→Z^κ is an order-preserving injective group homomor- phism fromGinto the simplicial groupZ^κ, whenceρ(G)is a subgroup ofZ^κ that is isomorphic (as a group) to G. Also, Z^κ is a lattice-ordered unigroup with unitv=ρ(u)and the restrictionρ|EofρtoE=G⁺[0,u]is an injective effect-algebra morphism ofEinto the MV-algebra(Z⁺)^κ[0,v].

Definition4.1. (i) The mappingρ:G→Z^κis called thecanonical injection.

(ii) Then×rmatrix overZ⁺ witha1,a2,...,anas its successive row vectors is denoted byA0=[a_ij].

(iii) Lete1=(1,0,...,0),e2=(0,1,...,0),...,e_r=(0,0,...,1)be the standard free basis vectors for the abelian groupZ^r.

(iv) Becausea1,a2,...,a_n generate the group G, there are (not necessarily uniquely determined) integersc_ijfori=1,2,...,randj=1,2,...,nsuch that ei=_n

j=1cijaj. LetCbe ther×nmatrixC:=[cij].

(v) Forj=1,2,...,r, letπ_j:Z^r→Zbe the projection homomorphism onto thejth coordinate.

The rows of the matrixA0are thenatoms in the unit intervalE=G⁺[0,u].

The canonical surjectionξ:Zⁿ→G=Z^r is given by the formulaξ(x1,x2,..., x_n)=(x1,x2,...,x_n)A0, and (t1,t2,...,t_n)∈(Z⁺)ⁿ is a multiplicity vector if and only if(t1,t2,...,tn)A0=(u1,u2,...,ur). Ther×nmatrixC overZ is a left inverse forA0, that is,CA0=1r=ther×ridentity matrix. Therefore, for eachj=1,2,...,r, we haven

i=1c_jia_ij=1, whence the nonzero integers in the jth row of the matrixCare relatively prime, as are the positive integers in the jth column ofA0.

If 1≤j≤r, the projection homomorphismπ_j:G→Zis surjective and, ow- ing to the fact thatG⁺⊆(Z⁺)^r, we haveπj(G⁺)⊆Z⁺. Therefore, byLemma 3.2, there exists iwith 1≤i≤n such thatπj(ai)=aij >0, that is, there is at least one strictly positive integer in thejth column ofA0. Also, byLemma 3.2, π_j(u)=u_j>0 and, byTheorem 3.6, there is a uniquely determinedQ-valued state γj ∈Ω(G) such that γj =πj. Evidently, γj =(1/uj)πj. The Q-valued statesγ1,γ2,...,γ_r, which correspond to the columns of the matrixA0, form a separating set of states forG.

Lemma4.2. Every stateω∈Ω(G)is a unique affine linear combination of the statesγj:=(1/uj)πjforj=1,2,...,r.

Proof. Fori,j =1,2,...,r, we have γj(ei)= (1/uj)πj(ei)= (1/uj)δij, whereδijis the Kronecker delta. Letω∈Ω(G)and letsi:=uiω(ei)=ω(uiei) for i=1,2,...,r. Then r

i=1s_i =ω(u)= 1. Also φ:=r

j=1s_jγ_j is a group homomorphismφ:G→R, and φ(ei)=_r

i=1(sj/uj)δij =si/ui=ω(ei)for i=1,2,...,r. Since the group homomorphismsφandωagree on the free basis e1,e2,...,e_r forG, it follows thatω=φ, andωis an affine linear combination of the statesγjforj=1,2,...,r. To prove that the coefficients are uniquely determined byω, supposeω=r

j=1hjγjwithhj∈Rforj=1,2,...,r. Then ω(e_i)=r

j=1h_j(1/u_j)δ_ij=h_i/u_i, whenceh_i=s_ifori=1,2,...,r.

Since{ai|i=1,2,...,n}is a finite set of generators forG, it follows that {ρ(ai)|i=1,2,...,n}is a finite set of generators forρ(G)⊆Z^κ. This observa- tion brings us to our next definition.

Definition4.3. (i) Letw_i:=ρ(a_i)=(ω¯1(a_i),ω¯2(a_i),...,ω¯_κ(a_i))∈Z^κ for i=1,2,...,n.

(ii) LetW:=[wij]be then×κmatrix with the vectorsw1,w2,...,wn as its successive rows.

(iii) LetP=[p_ij]be ther×κmatrix overZgiven by the productP:=CW. Since0≤ai≤ufori=1,2,...,n, it follows that0≤ρ(ai)≤ρ(u), whence wi∈(Z⁺)^κ[0,v]fori=1,2,...,n. ByDefinition 4.3(ii),

w_ij=ω¯_j a_i

fori=1,2,...,n, j=1,2,...,κ. (4.1) The columns of the matrixWcorrespond to the extreme pointsω1,ω2,...,ωκ

ofΩ(G)and its row vectorswi,i=1,2,...,n, generate the subgroupρ(G)of Z^κ. By part (ii) ofTheorem 4.4, the canonical injectionρcorresponds to right multiplication of vectors in G=Z^r by ther×κ matrixP. By part (v) of the theorem, the columns of the matrixP=CWare coefficients of homogeneous linear inequalities overZthat must be satisfied by vectors in the positive cone G⁺.

Theorem4.4. Lety=(y1,y2,...,yr)∈G=Z^r. Then (i) ¯ωj(y)=r

i=1yipij forj=1,2,...,κ;

(ii) ρ(y)=yP;

(iii) ρ(y)∈(Z⁺)^κ0≤r

i=1pijyiforj=1,2,...,κ;

(iv) ρ(y)∈(Z⁺)^κ0≤ω(y)for allω∈Ω(G);

(v) y∈G⁺⇒0≤r

i=1p_ijy_iforj=1,2,...,κ.

Proof. (i) We havey=r

i=1y_ie_i=r i=1y_in

k=1c_ika_k, whence, forj=1, 2,...,κ,

ω¯_j(y)= r i=1

n k=1

y_ic_ikω¯_j ak

= r i=1

y_i n k=1

c_ikw_kj= r i=1

y_ip_ij. (4.2)

Parts (ii) and (iii) follow from (i) and the definition ofρ.

(iv) Supposeρ(y)∈(Z⁺)^κ. Then 0≤ω¯i(y), so 0≤ωi(y)fori=1,2,...,κ.

Ifω∈Ω(G), thenωis a convex combination ofω1,ω2,...,ωκ, and it follows that 0≤ω(y). Conversely, if 0≤ω(y)for allω∈Ω(G), then 0≤ω_i(y), so 0≤ω¯i(y)fori=1,2,...κ, whenceρ(y)∈(Z⁺)^κ.

(v) It follows from (iii) and the fact thatρ(G⁺)⊆(Z⁺)^κ.

5. The Archimedean property. We maintain the hypotheses and notation ofSection 4. The canonical surjectionξand the canonical injectionρare order- preserving group homomorphisms

Z^{n ξ}→G ^ρ→Z^κ (5.1)

that link the torsion-free unital groupGwith the simplicial groupsZⁿandZ^κ and that satisfy

di ξ

→ai ρ

→wi fori=1,2,...,n (5.2) as well as

G⁺=ξ

Z⁺n, ρG⁺

⊆ Z⁺κ

∩ρ(G). (5.3)

As attractive as the setup in (5.1), (5.2), and (5.3) may be, there is an obvious asymmetry in (5.3). Indeed, althoughξdetermines the positive cone inGvia G⁺=ξ((Z⁺)ⁿ),ρdoes not necessarily determineG⁺. For symmetry, one would like to have

G⁺=ρ⁻¹ Z⁺κ

. (5.4)

Evidently, (5.4) holds if and only if the inclusion in (5.3) is an equality, that is, if and only if

ρ G⁺

= Z⁺κ

∩ρ(G), (5.5)

and condition (5.5) is equivalent to the requirement that, fora,b∈G,

a≤binG⇐⇒ρ(a)≤ρ(b)in the simplicial groupZ^κ. (5.6) The imageρ(G)ofGunderρis the subgroup ofZ^κgenerated by the vectors w1,w2,...,w_n, and it can be organized into a partially ordered abelian group in either of the following two natural ways.

(i) Use the restriction toρ(G)of the partial order on the simplicial group Z^κ. In this case, the positive cone forρ(G)is the induced positive cone(ρ(G))⁺

=ρ(G)∩(Z⁺)^κ.

(ii) Partially orderρ(G)with the subcone(ρ(G))⁺=ρ(G⁺)of the induced positive coneρ(G)∩(Z⁺)^κ as its positive cone. In this case,ρ(G)is a unital group with unitv=ρ(u)and is isomorphic as a unital group toGunderρ: G→ρ(G).

The equivalent conditions (5.4), (5.5), and (5.6) are themselves equivalent to the requirement that these two partial orders are the same. By the follow- ing theorem, all of these conditions are equivalent to the condition thatGis Archimedean, or equivalently, thatΩ(G)is cone determining.

Theorem5.1. The following conditions are mutually equivalent:

(i) G⁺=ρ⁻¹((Z⁺)^κ);

(ii) Gis Archimedean;

(iii) Ω(G)is cone determining, that is,G⁺= {y∈G|0≤ω(y)for allω∈ Ω(G)};

(iv) G⁺= {(y1,y2,...,y_r)∈Z^r|0≤r

i=1p_ijy_iforj=1,2,...,κ};

(v) the positive cone G⁺ is determined by a finite system of homogeneous linear inequalities overZ, that is, there exists anr×smatrix[bij]overZ such thatG⁺= {(y1,y2,...,y_r)∈Z^r|0≤r

i=1b_ijy_iforj=1,2,...,s}. Proof. (i)⇒(ii). Assume that (i) holds. The abelian groupG is isomorphic underρ:G→ρ(G)to the subgroupρ(G)ofZ^κ. Organizeρ(G)into a partially ordered abelian group under the restriction to ρ(G)of the standard partial order onZ^κ. Thenρ(G)inherits the Archimedean property from the simplicial groupZ^κ. Also, condition (i) is equivalent to the requirement that the isomor- phismρ:G→ρ(G)is an isomorphism of partially ordered abelian groups, whenceGacquires the Archimedean property.

(ii)(iii). Follows from [7, Theorem 4.14].

(iii)(iv). Follows from parts (iii) and (iv) ofTheorem 4.4.

(iv)⇒(i). Follows fromTheorem 4.4(iv).

(iv)⇒(v). This is obvious.

(v)⇒(iii). Assume (v) and forj=1,2,...,sdefine the group homomorphism φj:G→Zbyφj(y):=r

i=1bijyi fory=(y1,y2,...,yr)∈G=Z^r. By (v), we haveφ_j(G⁺)⊆Z⁺forj=1,2,...,s. By dropping all occurrences (if any) of the zero homomorphism from the listφ1,φ2,...,φs, we can and do assume that φj=0 forj=1,2,...,s. Forj=1,2,...,s, define ¯φj:=φjas inDefinition 3.4.

ByLemma 3.5(i), ¯φ_j:G→Zis a surjective order-preserving group homomor- phism forj=1,2,...,s. Furthermore, by (i),G⁺= {y∈G|0≤φ¯j(y)forj= 1,2,...,s}. ByLemma 3.5(ii),νj:=(1/φ¯j(u))φ¯jis aQ-valued state inΩ(G)for j=1,2,...,s. Therefore, by (i),G⁺= {y∈G|0≤ν_j(y)forj=1,2,...,s}, from which (iii) follows.

Corollary5.2. IfGis Archimedean, then the restrictionρ|EofρtoEem- beds the effect algebraEinto the MV-algebra(Z⁺)^κ[0,v]as a subeffect algebra ρ(E).

To help fix ideas, we present a very simple example withn=3,r=2, and κ=2 to illustrate some of the ideas developed in this paper.

Example5.3. LetG=Z²as an additive abelian group and definea_i∈Gfor i=1,2,3 bya1:=(1,0),a2:=(1,1), anda3:=(1,2). Letξ:Z³→Gbe defined by