ADVANCES IN APPLIED MATHEMATICS 19, 200᎐215Ž. 1997 ARTICLE NO. AM970535
Interval and Scale Effect Algebras
M. K. Bennett† and D. J. Foulis
Department of Mathematics and Statistics, Uni¨ersity of Massachusetts, Amherst, Massachusetts
Received November 27, 1996
1. EFFECT ALGEBRAS
Inwx 6 , we introduced the following definition. 1.1. DEFINITION.Aneffect algebra is a system Ž.E, [ ,0,u consisting of a set E with two special elements 0, u g E, called the zero and the unit, and with a partially defined binary operation [ satisfying the following conditions for all p, q, r g E: Ž.i Commutati¨e Law:If p[qis defined, then q [ p is defined and p [ q s q [ p. Ž.ii Associati¨e Law:If q[rand p [ Žq [ r .are defined, then p[qand Ž.p [ q [ r are defined and p [ Ž.Ž.q [ r s p [ q [ r. Ž.iii Orthosupplement Law: For every p g E there exists a unique q g E such that p [ q is defined and p [ q s u. Ž.iv Zero-One Law:If p[uis defined, then p s 0. The archetypal example of an effect algebra is the standard effect algebra Ž.E,[,z,|for a Hilbert space H, where E consists of all self-adjoint operators A on H such that z F A F |. For A, B g E, A [ B is defined iff A q B g E, in which case A [ B [ A q B.Ž We use the notation [ to mean ‘‘equals by definition.’’. If H is the Hilbert space affiliated with a quantum-mechanical system, operators A g E are of significance in repre- senting unsharp measurements or observations on the systemwx 3, 19, 20 . Effect algebras are mathematically equivalent to weak orthoalgebraswx 9 or D-posetswx 14 . Unless confusion threatens, we say that E is an effect algebra when we really mean that Ž.E, [ ,0,u is an effect algebra. Also, if we write an
† Deceased.
200
0196-8858r97 $25.00 Copyright ᮊ 1997 by Academic Press All rights of reproduction in any form reserved. INTERVAL AND SCALE EFFECT ALGEBRAS 201 equation such as p [ q s r, we are asserting both that p [ q is defined and that p [ q s r. In what follows, we assume that E is an effect algebra with unit u. If p, q g E, we say that p is orthogonal to q Ž.in symbols, p H q iff p[qis defined and that p is less than or equal to q Ž.in symbols, p F q iff p [ r s q for some r g E. The unique element q such that p [ q s u, X called the orthosupplement of p, is written as p [ q. Let a12, a ,...,an Ž. be a finite sequence in E.If ns1, we say that [ a1 exists in E and Ž. define [ a11[ a . By recursion, we say that the orthogonal sum Ž. Ž . [a12,a,...,anexists in E iff s [ [ a12, a ,...,any1and s [ anexist Ž. in E, in which case we define [ a12, a ,...,ann[s[a . Alternative notations for the orthogonal sum are a a иии a , na , andŽ if n 12[ [ [ ni[is1 . is understood [iiaii.If[a exists in E, we say that a12, a ,...,anis an orthogonal sequence in E.
If a g E, n is a positive integer and ai [ a for i s 1, 2, . . . , n, we say that na is defined iff [iiaiiexists in E, in which case we define na [ [ a . If n s 0, we define na [ 0. The element a is isotropic iff a H a, that is, iff 2ais defined. If na is defined, but Ž.n q 1 a is not defined, we say that n is the isotropic index of a.If na is defined for all nonnegative integers n,we say that a has infinite isotropic index. A subset S of E is called a subeffect algebra of E iff 0, u g S, S is X closed under p ¬ p , and p, q g S with p H q « p [ q g S. A subeffect algebra S of E is an effect algebra in its own right under the restriction to S of [. Also, for p, q g S, p F q in S iff p F q in E. Indeed, if p, q g S X X and p [ r s q in E, then r s Ž p [ q . g S,so pFqin S. Note that a subeffect algebra of E is closed under the formation of finite orthogonal sums. The basic properties of effect algebras, such as the fact that F is a partial order, are developed inwx 6 and will be assumed in the present paper.
2. INTERVAL ALGEBRAS
In what follows, abelian groups are understood to be additively written and the positive cone of a partially ordered abelian group G is denoted by Gq[Ä4ggGN0Fg. In particular, ޒqqand ޚ denote the standard positive cones in the additive group ޒ of real numbers and the additive group ޚ of integers. If G is a partially ordered abelian group and u g Gq, we define the inter¨al Gqwx0, u [ Ä4p g G N 0 F p F u . We omit the straightforward proof of the following theorem. 202 BENNETT AND FOULIS
2.1. THEOREM. If G is a partially ordered abelian group and u g Gq, then the inter¨al Gqwx0, u can be organized into an effect algebraŽ Gqwx0, u , [ , 0, u. such that p [ q is defined iff p q q F u, in which case p [ q s p q q. Furthermore, the effect-algebra partial order on Gqwx0, u coincides with the restriction to Gqwx0, u of the partial order on G.
2.2. DEFINITION. An effect algebra of the form Gqwx0, u as in Theorem 2.1Ž. or isomorphic to such an effect algebra is called an inter¨al effect algebra Ž.or for short, simply an inter¨al algebra with unit u in the group G.
If H is a Hilbert space and E is the standard effect algebra for H, then E is an interval algebra with unit | in the partially ordered abelian group V of all self-adjoint operators on H. The proof of the following lemma is a straightforward mathematical induction.
2.3. LEMMA. Let G be a partially ordered abelian group, u g Gq, and let q p12,p,..., pn be a finite sequence of elements in the inter¨al algebra G wx0, u . q Then the orthogonal sum [ipii exists in G wx0, u iff Ý piF uinG,in which case [ipiiis Ý p .
If G is an abelian group and M : G, we denote by ²:M the subgroup of G generated by M and by ssgŽ.M the subsemigroup of G consisting of 0 and all finite sums of sequences of elements of G.IfGq is the positive cone in a partially ordered abelian group G, then ²Gq: s Gqqy G and, as is well knownwx 4, 10 , G s ²Gq:iff G is directed in the sense that, for a, b g G, ᭚c g G with a, b F c.
2.4. THEOREM. Let K be a partially ordered abelian group, suppose ugKq,and let S be a subeffect algebra of the inter¨al algebra Kqwx0, u . Let G[²:S and let Gq[ ssgŽ.S . Then G is a partially ordered abelian group with generating cone Gqq and S s G wx0, u .
Proof. Evidently 0, u g S : Gq: G l Kq, Gqq Gq: Gq, Gqly Gqq:Kly K qs Ä40 , and G is a partially ordered abelian group with generating cone Gq.If pgS, then pЈ s u y p g S,so p,uypgGq, and it follows that p g Gqwx0, u . Conversely, suppose that p g Gqwx0, u . qq q Then p, u y p g G : K and ᭚p12, p ,..., pngS:K wx0, u such that q psÝiipand p F u in K. By Lemma 2.3, p s [ipiin K wx0, u and, since Sis a subeffect algebra of Kqwx0, u , it follows that p g S.
2.5. COROLLARY. A subeffect algebra of an inter¨al algebra is an inter¨al algebra. INTERVAL AND SCALE EFFECT ALGEBRAS 203
If E is the standard effect algebra for a Hilbert space H, then the orthomodular lattice ސ of all projection operatorsŽ self-adjoint idempo- tents. on H is a subeffect algebra of E. Elements of ސ are called sharp effects Žor decision effects wx15, 16. and ސ is the prototype for the so-called quantum logics wx1, 3, 5, 12, 17, 18 . Since E is an interval algebra, it follows from Corollary 2.5 that ސ is also an interval algebra.
3. ORDER UNITS AND AMBIENT GROUPS
If G is a partially ordered abelian group, an element u g Gq is called an order unit for G iff, for every g g G, there exists n g ޚq such that g F nu. Note that if G admits an order unit u, then Gq is a generating cone, i.e., G is directed. Indeed, if x, y g G, ᭚n, m g ޚq with x F nu and yFmu, whence, x, y F Ž.n q mu.
3.1. LEMMA. If G is a partially ordered abelian group with a generating cone Gqqq, u g G , and G s ssgŽGqwx0, u ., then u is an order unit in G. Proof. Let g g G. Since Gq generates G, there are elements a, b g Gqqwith g s a y b. Since G s ssgŽGqwx0, u ., there are elements q p12,p,..., pniiiigG wx0, u with a s Ý p . Thus, a s Ý p F nu F nu q b and it follows that g s a y b F nu.
3.2. DEFINITION.IfGis a partially ordered abelian group, then an element u g Gqqsuch that G s ssgŽGqwx0, u .²and G s Gq:is called a generati¨e order unit for G.If uis a generative order unit for G and E is isomorphic to Gqwx0, u , we say that G is an ambient group with order unit u for E. In a lattice-ordered group, every order unit is generative. However, if Gsޚwith the nonstandard cone Gqq[ ޚ _Ä41 , then 2 is an order unit in G, but it is not generative.
3.3. THEOREM. E¨ery inter¨al algebra E has an ambient group G.
Proof. Theorem 2.4 with E s Kqwx0, u . An ambient group for an interval algebra E is not necessarily uniquely determined by E. For instance, let G [ ޚ as an additive group, but let the positive cone in G be Gq[ Ä3n q 4m N n, m g ޚq4Äs 0, 3, 4 4j Ž6 q ޚq.. Then G is an ambient group with order unit 7 for E [ Gqwx0, 7 s Ä40, 3, 4, 7 . The interval algebra E, which is isomorphic to the Boolean algebra 2 2, also has ޚ = ޚ with positive cone ޚq= ޚq and order unit Ž.1, 1 as an ambient group. 204 BENNETT AND FOULIS
4. THE UNIVERSAL AMBIENT GROUP OF AN INTERVAL ALGEBRA
If G is an abelian group, then a G-¨alued measure on the effect algebra Eis a mapping : E ª G such that, for all p, q g E, p H q « Ž.p [ q sŽ.pq Ž.q. Inwx 6 we showed that, associated with each effect algebra E is a so-called uni¨ersal group; that is, an abelian group G and a G-valued measure ␥ : E ª G such that G s ²Ž␥ E .:and, if : E ª G is a G-valued U measure, there is a group homomorphism : G ª G such that s U (␥. We refer to ␥ as the canonical G-valued measure for E. 4.1. THEOREM. Let G be the uni¨ersal group, let ␥ : E ª G be the canonical G-¨alued measure for the effect algebra E, and let Gq[ ssgŽŽ␥ E ... Then a necessary and sufficient condition that E is an inter¨al algebra is that Gqqly G : Ä40,so that G is partially ordered with Gqas its positi¨e cone, and ␥ : E ª Gqw0, ␥ Ž.ux is a surjection such that, for p, q g E, ␥ Ž.p F ␥ Ž.q implies p F q. Proof. The sufficiency of the condition is evident. Conversely, suppose that E is an interval algebra. By Theorem 3.3, there is an ambient group G for E and we can identify E with Gqwx0, u . The injection mapping : E ª G is a G-valued measure on E, so there is a group homomorphism UU :GªGsuch that ␥ŽŽ..pspfor all p g E. q Ž. Let x g G . Then ᭚p12, p ,..., pniigE with x s Ý ␥ p and it follows UŽ. q UŽ q. qq that x s Ýiip g G . Thus, G : G . Also, if x, yx g G , then q q Ýiip,yÝ iipgG,soÝiips0; hence, since p12, p ,..., pngG , p1s q p2 sиии s pn s 0, so x s 0. Thus, G is partially ordered with G as its positive cone. X X X Let p g E. Then p s u y p g E, ␥ Ž.u s ␥ Žp [ p .Ž.Žs ␥ p q ␥ p ., X and it follows that ␥ Ž.p , ␥ Ž.u y␥ Ž.p s␥ Žp .Ž.g␥ E :Gq; hence, ␥ Ž.p g Gqw0, ␥ Ž.u x. Consequently, ␥ Ž.E : Gqw0, ␥ Ž.u x. Conversely, suppose that U U xgGqw0, ␥ Ž.u x, so that x, ␥ Ž.u y x g Gq. Thus, Ž.x , ␥ŽŽ.uyx .s U U U uyŽ.xgŽGq.:Gqand it follows that Ž.x g Gqwx0, u s E. Since q Ž. UŽ. xgG,᭚p12,p,..., pniiiigE with x s Ý ␥ p ,soÝps xgE; UŽ. Ž. hence, x s [ipiig E by Lemma 2.3 and it follows that x s Ý ␥ pi U s␥ŽŽ..x g␥ ŽE .. Therefore, Gqw0, ␥ Ž.u x: ␥ Ž.E and we have ␥ Ž.E s Gqw0, ␥ Ž.u x. Suppose that p, q g E with ␥ Ž.p F ␥ Ž.q in Gqw0, ␥ Ž.u x. Then ␥ Ž.q y U U ␥Ž.pgGq,soqyps␥ŽŽ.qy␥ Ž..pgŽGq.sGqand it follows that p F q in E. 4.2. COROLLARY. If E is an inter¨al algebra with unit u, there exists an ambient group G with order unit u for E such that E s Gqwx0, u and e¨ery group-¨alued measure : E ª H can be extended to a group homomorphism U : G ª H. INTERVAL AND SCALE EFFECT ALGEBRAS 205
Proof. In Theorem 4.1, let G s G and identify E with Gqw0, ␥ Ž.u xby identifying p g E with ␥ Ž.p g Gqw0, ␥ Ž.u x. The group G in Corollary 4.2 is the universal group for the interval algebra E, with the inclusion mapping E ª Gqwx0, u : G as the canonical G-valued measure. Thus, we refer to the group G in Corollary 4.2 as the uni¨ersal ambient group for the interval algebra E. 4.3. DEFINITION. Let G be a partially ordered abelian group. An order unit u in G is said to be uni¨ersal iff G is the universal ambient group for the interval effect algebra Gqwx0, u .
4.4. LEMMA. Let G be a partially ordered abelian group. Then the follow- ing conditions are equi¨alent: Ž.i E¨ery generati¨e order unit in G is uni¨ersal. Ž.ii If u is a generati¨e order unit for G and : Gqwx0, u ª Hisa group-¨alued measure, then can be extended to a group-¨alued measure ࠻ :Gqwx0, 2u ª H. Proof. To prove thatŽ. i « Žii . , assume Ž. i and the hypotheses of Ž ii . . By U Ž.i, can be extended to a group homomorphism : G ª H and the restriction ࠻ of U to Gqwx0, 2u is an H-valued measure. To prove thatŽ. ii « Ž.i , assume Ž. ii , let u be a generative order unit for G, and let : Gqwx0, u ª H be a group-valued measure. Let 1 F n g ޚ. By mathematical induction and the fact that ssgŽGqwx0, u .s Gq, there is a q n unique H-valued measure n: G w0, 2 ux ª H that extends . Since u is an order unit, Gqqis the set-theoretic union of the intervals G w0, 2 nux,so there is a uniquely determined mapping q: Gqª H that extends each of qŽ.qŽ. qŽ. the H-valued measures k . Evidently, x q y s x q y for all x, y g Gq. Since G s Gqy Gq, q can be extended uniquely to a group U homomorphism : G ª H. 4.5. THEOREM. Let G be partially ordered abelian group with no nonzero elements of order 2 such that, for e¨ery x g Gq, there is a y g Gq with 2 y s x. Then e¨ery order unit in G is uni¨ersal. Proof. Let u be an order unit in G. Then Gq is a generating cone. If g g G, there are elements a, b g Gq with g s a y b and there are elements c, d g Gq with a s 2c, b s 2 d; hence there is an element h [ c y d g G with g s 2h. Furthermore, since G contains no nonzero 1 elements of order 2, h is uniquely determined by g and we define 2 g [ h. 1 q Evidently, x ¬ 2x is an order-preserving automorphism of G.If agG , n n there is a positive integer n with a F 2 u and, by 2 applications of the 1 order-preserving automorphism x ¬ 2 x to the latter inequality, we can n find b g Gq with b F u such that 2 b s a. Consequently, Gqs ssgŽGqwx0, u .,souis generative. 206 BENNETT AND FOULIS
࠻ Now let : Gqwx0, ¨ ª H be an H-valued measure. Then : q ࠻Ž. Ž1 . q Gwx0, 2u ª H defined by x s 2 2 x for all x g G wx0, 2u is an H-valued measure that extends ,souis universal by Lemma 4.4.
4.6. COROLLARY. Let F be a subfield of ޒ, let V be a ¨ector space o¨er F, and suppose that the additi¨e abelian group V is partially ordered by the positi¨e cone Vqq in such a way that F Vq: ViqŽ.e., Vq partially orders V as a ¨ector space.. Then e¨ery order unit in the partially ordered abelian group V is uni¨ersal.
4.7. COROLLARY. If V is the additi¨e abelian group of all self-adjoint operators on a Hilbert space, ordered in the usual way, then Ž.V , | is the uni¨ersal group for the standard effect algebra E [ Vqwxz, | .
5. PROBABILITY MEASURES
Probability measures are defined on effect algebras just as they are for orthoalgebraswx 12, 13 .
5.1. DEFINITION.Aprobability measure Ž.or state on the effect algebra Eis an ޒ-valued measure : E ª ޒ such that Ž.E : ޒqand Ž.u s 1. We denote by ⍀Ž.E the set of all probability measures on E.
Let G be the universal group for E and let ␥ : E ª be the canonical U G-valued measure. Then there is a one-to-one correspondence l , U determined by the condition s (␥, between probability measures UU g⍀Ž.Eand group homomorphisms : G ª ޒ such that ␥ŽŽE ..: U ޒqand ␥ŽŽ..us1. In particular, if G is the universal group for an interval algebra E s Gqwx0, u , then the probability measures on E are precisely the restrictions to E of order-preserving group homomorphisms U U :Gªޒthat are normalized in the sense that Ž.u s 1. 5.2. DEFINITION. Let ⌬ : ⍀Ž.E . Ž.i ⌬ is positi¨e iff, for every p g E with p / 0, ᭚ g ⌬ with Ž.p)0. Ž.ii ⌬ is order determining iff, for all p, q g E, Žp .F Ž.q for all g ⌬ « p F q. 5.3. LEMMA. Suppose the effect algebra E admits a positi¨e set of proba- bility measures, let G be the uni¨ersal group for E, and let ␥ : E ª G be the canonical G-¨alued measure. Then G is a partially ordered abelian group with positi¨e cone Gq[ ssgŽŽ␥ E .. and ⍀ ŽE . is the set of mappings of the form U U (␥, where : G ª ޒ is an order-preser¨ing group homomorphism such U that ␥ŽŽ..us1. Furthermore, ␥ Ž.E : Gqw0, ␥ Ž.u x. INTERVAL AND SCALE EFFECT ALGEBRAS 207
Proof. Suppose that g, yg g Gq. Then there are finite sequences Ž. Ž. p12,p,..., pnand q12, q ,...,qmiijjin E with g s Ý ␥ p and yg s Ý ␥ q , Ž. Ž. and it follows that 0 s Ýii␥ p q Ý jj␥ q . Suppose 1 F k F n and pk/ 0. Then there exists a probability measure on E with Ž.pk ) 0 and there U U is a corresponding group homomorphism : G ª ޒ with s (␥. Therefore,
U U U Ž.0 s ÝÝÝÝ Ž.␥ Ž.pijijq Ž.␥ Ž.q s Ž.p q Ž.q ijij
GŽ.pk)0, UŽ. contradicting 0 s 0. Consequently, pi s 0 for 1 F i F n and it follows Ž. q that g s Ýii␥ p s 0, so G is partially ordered by the cone G . Further- U more, for every g ⍀Ž.E , the group homomorphism : Gqª ޒ satisfies
U U U ŽGq .s Ž.Ž.ssgŽ.␥ Ž.E : ssg Ž.␥ Ž.E : ssg Žޒq .: ޒq, UU so is order preserving and ␥ŽŽ..us Ž.us1. Conversely, if U : G ª ޒ is an order-preserving group homomorphism such that U ␥ŽŽ..us1, it is clear that [ (␥ g ⍀Ž.E . X If p g E and p is the orthosupplement of p in E, then ␥ Ž.u s X X X ␥Ž p[p.Ž.Žs␥pq␥p.Žand, since ␥ p., ␥ Žp .g Gq, we have 0 F X ␥Žp .s␥ Ž.uy␥ Žp .F␥ Ž.u. Therefore, ␥ Ž.E : Gqw0, ␥ Ž.u x. 5.4. THEOREM. If E admits an order-determining set of probability mea- sures, then it is an inter¨al algebra. Proof. Evidently, an order-determining set of probability measures is positive. Therefore, by Lemma 5.3, we can organize the universal group G of E into a partially ordered abelian group with positive cone Gqs ssgŽŽ␥ E .., where ␥ : E ª G is the canonical G-valued measure. If g U ⍀Ž.E, denote by : G ª ޒ the corresponding order-preserving group U homomorphism such that s (␥. Suppose p, q g E with ␥ Ž.p F ␥ Ž.q . Then, for every probability mea- U U sure on E, we have ␥ŽŽ..pF␥ŽŽ..q,so Ž.p F Ž.q; hence, by our hypothesis, p F q. Consequently ␥ : E ª Gqwx0, u satisfies p F q m ␥Ž.pF␥ Ž.q,so␥: Eª␥ Ž.E is an order isomorphism. Obviously, ␥ Ž.E is closed under the orthosupplementation in X Gqw0, ␥ Ž.u x. Suppose p, q g E, ␥ Ž.p H ␥ Ž.q in Gqwx0, u , and q [ q s u X X in E. Then ␥ Ž.p F ␥ Ž.u y ␥ Ž.q s ␥ Žq .,so pFq, pHqin E, and ␥Žp[q .s␥ Ž.pq␥ Ž.qs␥ Ž.p[␥ Ž.qin Gqwx0, u . This shows that ␥Ž.Eis a subeffect algebra of Gqwx0, u ; hence, by Corollary 2.5, ␥ Ž.E is an interval algebra. Since ␥ : E ª ␥ Ž.E is an effect-algebra isomorphism, it follows that E is an interval algebra. 208 BENNETT AND FOULIS
Virtually every logico-algebraic system E that has been seriously pro- posed as theŽ. sharp or unsharp logic of a real physical system is at least an effect algebra that carries an order determining set of probability mea- sures. By Theorem 5.4, every such logic E is an interval algebra.
5.5. THEOREM. E¨ery inter¨al algebra admits at least one probability measure.
Proof. If E is an interval algebra, we may assume by Theorem 3.3 that EsGqwx0, u for the universal group G with generative order unit u.By wx11, Corollary 3.3 , there is a group homomorphism : G ª ޒ such that ŽGq. :ޒq and Ž.u s 1. The restriction of to E is a probability measure on L.
6. SCALE ALGEBRAS
Inwx 6 , we defined a scale effect algebra, or just a scale algebra for short, to be a totally ordered effect algebra. The scale algebra ޒqwx0, 1 is called qq the standard scale algebra and, if n g ޚ , the scale algebra Cn [ ޚ wx0, n is called the n-chain wx7. If G is a totally ordered abelian group and u g Gq, it is clear that Gqwx0, u is a scale algebra. In this section, we show that, conversely, every scale algebra is an interval algebra, that it has a totally ordered universal group, and that every order unit in a totally ordered abelian group is universal. We begin by showing that any scale algebra can be ‘‘doubled.’’
X 6.1. LEMMA. LetŽ. A, [ ,0,u be a scale algebra with a ¬ a as the orthosupplement mapping and F as its total order. Then there exists a scale Ž. U algebra D, [D ,0,¨ with d ¬ d as the orthosupplement mapping and FD as its total order such that A : D and, for all a, b g A: Ž. U i aFD b Ž. X U ii a F b « a FDDb and a [ b s a [ b Ž. X U iii a F b m a FDDb and a [ b g A Ž. iv a F b m a FD b Ž. v dgD_A«aFD d Ž. vi d g D « ᭚a, b g A with d s a [D b. Sketch of the proof. The proof, although straightforward, is a bit te- dious, so we content ourselves with a sketch, leaving the details to the UU interested reader. Let I [ Ä4a g A N a - u , let I s Äa N a g I4be a copy INTERVAL AND SCALE EFFECT ALGEBRAS 209 of I that is disjoint from A, and let
UU D[IjÄ4ujIsAjI.
U U U U Extend the bijection : I ª I given by a ¬ a to a bijection : D ª D UU U UUU by defining a s a for a g I and Ž.u s u, noting that d s d for all d g D. Ž. Define a partial binary operation x, y ¬ x [D y for x, y g D only in the following threeŽ. mutually exclusive cases: X Case 1. If x, y g A with x F y , then x [D y [ x [ y. X Ž XX.U Case 2. If x, y g A with y - x, then x [D y [ x [ y . U Case 3. If x g A, a g I, y s a , and x F a, then x [DDy and y [ x are both defined to be bU, where b is the unique element in A for which x [ b s a. Ž. U That D,0,¨,[D is an effect algebra with unit ¨ s 0 and orthosupple- U mentation x ¬ x follows from a routine verification of the four laws in Definition 1.1.Ž The confirmation of the associative law involves an exhaus- tive case analysis.. Ž.i follows from Case 1, Case 2, and the fact that A is totally ordered, so Ž.Ž . Ž . that a [D b is defined for all a, b g A. ii , iii , and iv follow immedi- ately from Case 1. To proveŽ. v , note that if a g A and d g D _ A, then U Ž. Ž . dsb for some b g I : A,soaFD dby ii . To prove vi , suppose d g D.If dgA, then d s d [ 0 s d [D 0 with d,0gA.If dfA, then U U d s c for some c g I : A, and it follows from Case 2 that d s c s XX c[D uwith c , u g A. 6.2. THEOREM. An effect algebra E is a scale algebra iff it admits a totally ordered ambient group. Proof. If E admits a totally ordered ambient group, then it is clearly a scale algebra. Conversely, let E be a scale algebra with unit u.We recursively construct a nested infinite sequence of scale algebras E s иии E012: E : E : such that the partially defined orthogonal sum [iq1 on Eiq1 extends the corresponding partially defined orthogonal sum [i on EiDin the same sense that [ on D extends [ on A in Lemma 6.1. Ž. By part vi of Lemma 6.1 and induction, for each element q g En, there exists a finite sequence of elements p12, p ,..., pkgEsE0:Ensuch that q s p1 [n p2 [nnkиии [ p . ϱ Let C [ D is0 Ei and define the binary operation q on C by x q y s x [nq1 y, where n is any nonnegative integer such that both x and y belong to En. Evidently, q is well defined, E : C, and C has the following 210 BENNETT AND FOULIS properties:
Ž.i C is a commutative cancellation semigroup with zero under q. Ž.ii If x, y g C with x q y s 0, then x s y s 0. Ž.iii If x, y g C, either ᭚z g C with x s y q z or ᭚w g C with y s x q w. X Ž.iv For x, y g E, x F y in E iff x q y g E, in which case x [ y s x q y. Ž.v Egenerates the semigroup C. By a theorem of Birkhoffwx 2, 8, Theorem 4, p. 14 ,Ž. i and Ž ii . imply that there is a partially ordered abelian group G with generating cone Gqs C. Ž. qq Ž. By iii , G is totally ordered by G , E s G wx0, 1L by iv , and E generates CsGq byŽ. v . As a consequence of Theorem 6.2, every scale algebra E is an interval algebra. We remark that the effect algebra D constructed in Lemma 6.1 is actually the tensor product E m C22of E with the 2-chain C wx7. 6.3. LEMMA. If G is a totally ordered abelian group u g Gq and 0 / ngޚqq,then e¨ery element in G wx0, nu is a finite sum of at most n elements in Gqwx0, u . Proof. The proof is by induction on n. The statement is obvious for ns1. Suppose every element in Gqwx0, nu is a finite sum of at most n elements in Gqwx0, u and let x g Gqw0, Ž.n q 1 ux.If xgGqwx0, u , there is nothing to prove, so we can assume that u F x. Therefore, since 0 F x F q nu q u, we have 0 F x y u F nu,so᭚p12,p,..., pngG wx0, u with xyusÝiipand it follows that x s u q Ýiip . 6.4. THEOREM. If G is a totally ordered abelian group, then e¨ery order unit in G is uni¨ersal. Proof. By Lemma 6.3, every order unit in G is generative. Thus, to prove the lemma, we need only verify partŽ. ii of Lemma 4.4. Thus, let ugGqq, let : G wx0, u ª H be a group-valued measure, and x g Gqwx0, 2u . Since G is totally ordered, one of the conditions x F u or u F x holds. If u F x, then u F x F 2u and 0 F x y u F u. Thus, the mapping ࠻ :Gqwx0, 2u ª H, defined by
࠻ Ž.x ,ifxFu, Ž.xs ½Ž.Ž.xyuqu,ifuFx, for all x g Gqwx0, 2u , is a well defined extension of .Ž Note that if has an extension to an H-valued measure on Gqwx0, 2u , it must be given by this formula.. INTERVAL AND SCALE EFFECT ALGEBRAS 211
࠻ Now let x, y, x q y g Gqwx0, 2u . We have to show that Ž.x q y s ࠻ ࠻ Ž.xqŽ.y. This is clear if x q y F u, for then x, y, x q y g Gqwx0, u , so we can assume that u F x q y F 2u. It is also clear if x s y s u, when both sides reduce to 2Ž.u . Thus, we can also assume that one of the two conditions x F u, y F u holds, for otherwise u F x, u F y,so2uFxq yF2u,xqys2u,Ž.Ž.xyuqyyus0, and, owing to the fact that 0 F x y u and 0 F y y u, we have x s y s u. Hence, by symmetry, we can assume that x F u. If u - y, then we have 0 F x,0Fyyu, and x q y y u F u,soŽ xq yyu.ŽŽsx[yyu ..Ž.Ž.sxqyyu, and it follows that
࠻ Ž.Žx q y s x q y y u .Ž.Ž.Ž.Ž.q u s x q y y u q u ࠻ ࠻ sŽ.xq Ž.y.
Thus, we can assume that x F u, y F u, u F x q y F 2u. Hence, x q y y u,uyy,xgGqwx0, u ;soŽ.Ž.xqyyu[uyysxin Gqwx0, u and it follows that Ž.Ž.Ž.x q y y u q u y y s x ,so
Ž.Ž.Ž.xqyyu s x y uyy .
Also, u s y [ Ž.u y y in Gqwx0, u ,so
Ž.u s Ž.y q Žuyy . and again we have
࠻ Žx q y .s Žx q y y u .q Ž.u s Ž.x q Ž.y ࠻࠻ sŽ.xq Ž.y.
6.5. COROLLARY. If E is a scale algebra, then E is an inter¨al algebra, the uni¨ersal ambient group G for E is totally ordered, and if G is any totally ordered ambient group for E, then G is isomorphic to G. Proof. By Theorem 6.2, there is a totally ordered group G with a generative order unit u such that E s Gqwx0, u and, by Theorem 6.4, G is the universal group for E.
6.6. THEOREM. A scale algebra admits a unique probability measure. Proof. Let G be the universal ambient group for the scale algebra E. By Corollary 6.5, G is totally ordered and bywx 10, Corollary 4.17 , there is a UU unique order-preserving homomorphism : G ª ޒ with Ž.u s 1. U Thus, the restriction of to E s Gqwx0, u is the unique element in ⍀Ž.E. 212 BENNETT AND FOULIS
6.7. THEOREM. Let E be a scale algebra with uni¨ersal ambient group G, U let be the unique probability measure on E, and let : G ª ޒ be the unique extension of to a group homomorphism. Then the following condi- tions are mutually equi¨alent: Ž.i E contains no nonzero elements with infinite isotropic index. U Ž.ii Gqs Äg g G N 0 F Ž.g 4 Ž.iii There is a subgroup H of ޒ and an isomorphism : G ª HofG onto H such that Ž.u s 1 and ŽGq.s H l ޒq. Ž.iv E is isomorphic to a subeffect algebra of the standard scale algebra ޒqwx0, u . Ž.v ⍀ ŽE .sÄ4is order determining. Ž.vi ⍀ ŽE .s Ä4 is positi¨e.
U Proof. We note that G is totally ordered and : G ª ޒ is order U preserving with Ž.u s 1. U U Ž.i« Žii . Since is order preserving, Gq: Äg g G N 0 F Ž.g 4. U AssumeŽ. i and let g g G with 0 F Ž.g , but suppose that g f Gq. Then, since G is totally ordered, g / 0 and yg g Gq, hence, 0 F U U U U Ž.ygsy Ž.g and it follows that Ž.g s Ž.yg s 0. Byw 10, Proposition 1.22 and Corollary 4.11x , nŽ.yg - u for all n g ޚq, so the nonzero element yg has infinite isotropic index, contradictingŽ. i . U Ž.ii « Žiii . Assume Ž. ii , let H [ Ž.G , and define : G ª H by U Ž.g[Ž.g for all g g G.If ggkerŽ. , then g, yg g Gq byŽ. ii , hence g s 0. Therefore, : G ª H is an isomorphism such that Ž.u s U U Ž.us1gH. Obviously, ŽGq.s ŽGq.: H l ޒqq.If hgHlޒ , U U then h s Ž.g for some g g G and g g GqbyŽ. ii , so h g ŽGq.s ŽGq.. Ž.iii « Ž.iv Assume Ž. iii and let F [ Ž.ŽE . Since Gq.: ޒq, it fol- lows that : G ª ޒ is order preserving and, because Ž.u s 1, we have Ž. Žq . q FsEsGwx0, u : ޒ wx0, 1 . Let E be the restriction of to E, Ž. Ž. so that E: E ª F is a bijection. Evidently 0 s 0 g F and 1 s u g F.If sgF, then s s Ž.x for some x g E, whence 1 y s s Ž.u y x and F is closed under the orthosupplementation s ¬ 1 y s. Suppose s, t g F with s q t F 1, s s Ž.x , and t s Ž.y with x, y g E. Then Žx q y .F 1sŽ.u,so Žuyxyy .gHlޒqqand therefore u y x y y g G , q Ž. whence x q y g G wx0, u s E. Thus, x H y in E and E x [ y s Ž. Ž. q EExqyss[tin ޒ wx0, 1 . Consequently, F is a subeffect algebra q of ޒ wx0, 1 and Ž.vi « Ž.i Assume Ž. vi and suppose that g g E has infinite isotropic index. Then 0 F Ž.g and n Ž.g s Žng .F 1 for every n g ޚq,so Ž.gs0 and therefore g s 0. Theorem 6.7 suggests the following definition. 6.8. DEFINITION. A scale algebra E is archimedean iff it contains no nonzero element with infinite isotropic index. 6.9. COROLLARY. An archimedean scale algebra is either a chain Cn or it is isomorphic to a topologically dense subeffect algebra of the standard scale algebra ޒqwx0, 1 . Proof. An additive subgroup H of ޒ is either cyclic or dense in ޒ wx10, Lemma 4.21 . 6.10. COROLLARY. The standard scale algebra may be characterized as the only archimedean scale algebra that is infinite and complete as a lattice. 7. NONARCHIMEDEAN SCALE ALGEBRAS In this section, we show how to construct examples of nonarchimedean scale algebras; in fact, we sketch a general construction that provides many such examples as a special case. Following Wilcewx 21 , we define a partial abelian semigroup Ž.PAS to be a set P equipped with a partially defined binary operation [ that is commutative and associative in the sense of Definition 1.1. A zero for a PAS P is an element 0 g P such that p [ 0 s p for all p g P and P is said to be positi¨e iff it has a zero and satisfies p [ q s 0 « p s q s 0 for all p, q g P. The PAS P satisfies the cancellation law iff, for all p, q, r g P, p [ q s p [ r « q s r. Note that every effect algebra is a positive PAS satisfying the cancellation law. In what follows, we assume that P is a positi¨e PAS satisfying the cancellation law. For p, q g P, define p F q iff ᭚r g P with p [ r s q. Evidently, P is partially ordered by F . For p, q g P, we define q y p g P iff p F q, in which case q y p is the unique element in P such that p[Ž.qypsq. In what follows, we denote by PU the order-theoretic dual of P. In other U U UUU words, P s Ä pp< gP4, the mapping p ¬ p is a bijection, and p F q U iff q F p for all p, q g P. We assume that P l P s л and define U U U U : P ª P by Ž p . s p for all p g P. U In what follows, let E [ P j˙ P . We introduce a partially defined binary operation $ on E as follows: If p, q g P, then p $ q is defined iff 214 BENNETT AND FOULIS U U p [ q is defined, in which case p $ q [ p [ q. Also, p $ q s q $ p is UU U defined iff p F q, in which case p $ q s q $ p [ Ž.q y p . Note that U U p $ q is never defined. U 7.1. THEOREM. Ž E,0,0 ,$. is an effect algebra and the partial order on P and on its dual PU is the restriction to P and to PU of the partial order on E. Proof. The commutative law is obvious and the associative law follows U U from a straightforward case analysis. Evidently, x $ x s 0 for all x g E. U U Also, if x, y g E and x $ y s 0 , then either x g P and y s x or y g P U U U U and x s y ; hence, x $ y s 0 implies that y s x .If xgEand x $ 0 is defined, then x F 0, so x s 0 and therefore E is an effect algebra. If p, q g P, x g E, and p $ x s q, then x g P and p [ x s q. There- fore, the partial order on P is the restriction to P of the partial order on U U E. Likewise, p $ r s q for some r g P iff r F p and q s p y r, and U such an r exists iff q F p, so the partial order on P is the restriction to PU of the partial order on E. By Theorem 7.1, any positive PAS with cancellation P can be enlarged to an effect algebra E.If P is already an effect algebra, then E is isomorphic to the cartesian product P = C2 wx7. 7.2. THEOREM. Let K be any totally ordered abelian group and let P s Kq. U Then E s P j˙ P is a scale algebra and e¨ery element in P has infinite isotropic index in E. U Proof. Evidently, both P and P are totally ordered. Also, if p, q g P, U U U then r [ p q q s p [ q is defined in P and p $ r s q ,so pFq in E and it follows that E is a scale algebra. If p g P, then np g P for all n g ޚq,so phas infinite isotropic index. Taking K in Theorem 7.2 to be any nonzero totally ordered abelian group, we obtain a nonarchimedean scale algebra E. REFERENCES 1. E. Beltrametti and G. Cassinelli, The logic of quantum mechanics, in ‘‘Encyclopaedia of Mathematics and its Applications,’’ Vol. 15Ž. G.-C. Rota, Ed. , Addison-Wesley, Reading, MA, 1981. 2. G. Birkhoff, Lattice ordered groups, Ann. Math. 43 Ž.1942 , 298᎐331. 3. P. Busch, P. Lahti, and P. Mittelstaedt, ‘‘The Quantum Theory of Measurement,’’ Lecture Notes in Physics, New Series M2, Springer-Verlag, Berlin, 1991. 4. M. R. Darnel, ‘‘Theory of Lattice-Ordered Groups,’’ Dekker, New York, 1995. 5. D. J. Foulis and C. H. Randall, What are quantum logics and what ought they to be?, in ‘‘Current Issues in Quantum Logic’’Ž. E. Beltrametti and B. van Fraassen, Eds. , Ettore Majorana International Science Series, Vol. 8, pp. 35᎐52, Plenum, New York, 1981. INTERVAL AND SCALE EFFECT ALGEBRAS 215 6. D. J. Foulis and M. K. Bennett, Effect algebras and unsharp quantum logics, Found. Phys. 24 Ž.1994 , 1331᎐1352. 7. D. J. Foulis, R. J. Greechie, and M. K. Bennett, Sums and products of interval algebras, Internat. J. Theoret. Phys. 33 Ž.1994 , 2119᎐2136. 8. L. Fuchs, ‘‘Partially Ordered Algebraic Systems,’’ International Series of Monographs on Pure and Applied Mathematics, Vol. 28, Pergamon, Oxford, 1963. 9. R. Giuntini and H. Greuling, Toward a formal language for unsharp properties, Found. Phys. 19 Ž.1989 , 931᎐945. 10. K. Goodearl, ‘‘Partially Ordered Abelian Groups with Interpolation,’’ Mathematical Surveys and Monographs, Vol. 20, Am. Math. Soc., Providence, RI. 11. K. Goodearl and D. Handleman, Rank functions and K0 of regular rings, J. Pure Appl. Algebra 7 Ž.1976 , 195᎐216. 12. S. Gudder, ‘‘Quantum Probability,’’ Academic Press, Boston, 1988. 13. J. Hamhalter, M. Navara, and P. Ptak,´ States on orthoalgebras, Internat. J. Theoret. Phys. 34 Ž.1995 , 1439᎐1465. 14. F. Kopkaˆ and F. Chovanec, D-posets, Math. Slo¨aca 44 Ž.1994 , 21᎐34. 15. G. Ludwig, ‘‘Foundation of Quantum Mechanics,’’ Vol. 2, Springer-Verlag, New York, 1983r1985. 16. G. Ludwig, ‘‘An Axiomatic Basis for Quantum Mechanics,’’ Vol. 2, Springer-Verlag, New York, 1986r1987. 17. P. Ptak´´ and S. Pulmannova, in ‘‘Orthomodular Structures as Quantum Logics’’Ž A. van der Merwe, Ed.. , Fundamental Theories of Physics, Vol. 44, Kluwer Academic, Dor- drecht, 1991. 18. C. Piron, in ‘‘Foundations of Quantum Physics’’Ž. A. Wightman, Ed. , Mathematical Physics Monograph Series, Benjamin, Reading, MA, 1976. 19. E. Prugovecki, ‘‘Stochastic Quantum Mechanics and Quantum Space Time,’’ 2nd ed., Reidel, Dordrecht, 1986. 20. F. Schroeck and D. Foulis, Stochastic quantum mechanics viewed from the language of manuals, Found. Phys. 20 Ž.1990 , 823᎐858. 21. A. Wilce, Partial abelian semigroups, Internat. J. Theoret. Phys. 34 Ž.1955 , 1807᎐1812.