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The Center of an Effect Algebra OnJer 12: 91-J()6, J995 91 。 1995 Kluwer AClulemic Pllhlishers. /'rillletl ÎI/ I/re Nelfwrlnllds. The Center of an Effect Algebra R. J. GREECHIE Deparlmelll of Mathemaliιzα Il d Slatisli c.~， μluisia l/ a Tedl University, RUSIOI/, LA 71272, U. S.A. D. FO ULIS Departmel/l of MalhemalÎCs and SlalÎslics, UllíversÎly of Massachusetts, Amhersl, MA 01003, U. S.A. and S. P U L M A N N 0 V A Malhemalicai Insli/Ule qf lhe Siovak Academy 01 Scie l/ ce.~. Bralísiava, Siov(/kia Comm附 11ι (Received: 5 October 1994; accepted: 30 Oc t恰在 r 1994) Abstr.lct. An effcct algebra is a partial al 革命 ra modeled on the 51在 ndard effcct algebra of positive sclf­ adjoint operaωrs dominated 均 the identity on a Hilbcrt space. 巴 very effect algc七ra is parli器lI y ordered in a na lU ral 曹町，豁然辜gested by the partial order on the standard e 汀ect algcbra. An effect algebra is said to be distri 七日 live if, as a posel, it forms a distributivc lattice. We dcfine and slUdy lhe ccnler of an effccl algebra, relate it to cartesian-product factorizatiolls, detenninc the cenler of lhe stand 辑rd effect algebra, and characterize all fìnite distributive effecl algebras as products of chains and diamoll必. Mathematics Subject Classilicatiolls (1赞 91). 81810，的 GI2 ， 06C 15, 06泣。， 06625. Key words. Effect algebra, orlbosupplement, isolropic elemenl, principal el 巳 ment ， center, íntcrval algebra, effcct ring, scale algebra, distribulive effccl algebra. 1. lnlroduction Motiγ 在ted by the use of positive operator valued measures in stochastíc (or phase space) quantum mechanics 口， 3, 22, 25, 26] and by the study of fuzzy or unsharp quantum logics 抖， 5, 15, 20] there has been a recent surge of interest ìn a new c1 ass of orthostructures called weak (or generalized) orthoalgebras [1 5, 1 坷 ， difference (or D) poset:s [7-10, 17 ，凹， 21 ， 24] ， orξffect algebra;￥ [2 ，泣， 13]. By and large, these structures are mathematical1y equivalent, and it is mostJy a matter of taste which to select. ln what follows, we choose to focus on effect algebras because lhe additíve structure of such algebras 在 rticulates wel1 with that、 of the partially ordered abelian groups which often figure in their representation {2]. Thus, we assume that the reader is somewhat familiar with 丘， 12, 13], although 毫 for convenience, we reproduce most of the pertinent basìc defìnitions and results. 92 R. J. GREECHIE ET AL. In this paper we define and stlldy the notion of the cεnter of an effect algebra, relate central elements with factorization into cartesian prodllcts, determine the center of a standard e仔"e ct algebra on a Hilbert space, and llSe central elements to help characterÎze all finite dÎstributive effect algebras. 2. Effect AJgebras In [12] an 吃ffect algebra is defined to be an algebraic system 仰， 0 ，包，盼 consÎsting of a set two speci 在 I elements 0, u E E called the zero 在nd the unit, and a partÎally defined binary operatÎon φon E that satÎsfies the following conditions for all p ， q ， r 巳 E: 。 ) [Commutative La w] If pæq is de白 ned ， then qæp is defined and pæq 之二 qæp. (ii) fAssociatìve Law] If q æ r is de白 ned and p æ (q æ r) is defined, then p æ q is defined , (p æ 的① r is defined ，在 nd pæ(qæ 付出 (p æq)ær. (iii) [Orthosupplementation Law] For every p ε E there exists a llnique q E E sllch th 低 pæ q is d巳 fined and p æ q 口 u. (iv) [Zero 唰 Unit Law] If u æ p Îs defined, then p = O. For simplicity, we often refer to E , rather than to (E ， O ， u ， æ) ， 挠 s beÎng an effect algebra. For the remainder 01 this paper， 附 assume that E 必 an effect algebra with 川 it 1L. If p, q ε E ， we say that p and q are orthogonal and wrÎ te p .1 q Îff pæ q is defined in E. If we write p æ q = r , we mean that p 1. q and p æ q rε E. If p ， q εE and p æ q = 1L , we call q the orthosupplement of p and wrÎte pl := q. (We use the notatÎon := to mean "equals by definition.") If el. 向尸·叮已白 is a finite seqllence of elements of E, then, by recursion, we say tÌlat the orthogonal sum e I æ 句 æ.. .æen exÎsts in E Îff s : 之二 e( 6)句@… φ e n … t exists in E and s .1 e", in which c部e we clefine et æ 句 æ..ω Cn S ( e". Alternative 则往往 on for the orthogonal Sllm Îs EÐj=1 句， or simply EÐj ej ifπis llnderstood. If EÐj ej exists in E , we s 在y that et. e2 ， 币. .， εn is an orthogonal sequence in E. The basic properties of orthogonal seqllences, which can be fOllnd În f12, Theorem 4. 坷， will be llsed withollt comment in what follows. If pε E and PhP2,'" ,Pn is an orthogonal seqllence, where Pl = P2 … =P托 Z p, 飞时 define 叩 EÐjPj ‘ If p 并 o and 2p is defined (i.e., if P 1. p), we say that p is an ìsotropic element of E. If np is defined, bllt (n + 1)p is not, we refer to n as the isotropic index of p and we call smt(p) :=ηp the isotropic summit of p. If np is defined for all positive integers n , we say that p has infinite isotropic index. An orthoalgebriα[6 ， 10 ，口. 14, 16, 17, 21 , 24] may be defined as an effect algebra with no isotropic elements. A sllbset F of E is called a s始吃ffect algebra of E iff 哟。，1L ξ F ， (ii) p ε F=争 pl E F , and (iii) p, q E F with p .l q 冲 pæq ε 丑在vidently ， sllch an F is an eff配 t algebra in ils own right under the restriction of æ to 贝 TIIU CENTERυF 八 N EFI'ECT ^LliEIJI之八 1)3 Ir B and F arc elTcct algcbras, a mapping cþ: B • F is callcd a l11 orl'hislII iff it maps the unit 01' E to the unit 01‘ F and preserves orthogonal sums În the sense that, if p, q εE with p -L守， then 功 (P) 土版的 and qy(p (Ð q) = 圳的 6 科 q). 巳vidently， if qy: E • F is a morphism, then 版时= U and 份怡') = qy(p)'. An isomorpl出mqy: E • F is defìned to be a bijective 1lI0rphism such that, for p, q E E , qy (P)l 动 (q) 咋 p -L q. Evídently, a bijective 1I1 0rphism 功 :E 叫 F is an isol11orphism iff ç奋斗 :F • E is also a morphism. 3. The Order Structure of an Effect Algebra lt is shown in 112] that the relation ::s; defined for p, q 巴 E by p ::s; q iff 3r 巳 E with pEÐr 口 q is a partial order on E and 0 ::s; p ::s; u holds for all 1J E E. lt is also shown that the orthosupplementation mapping p f-+ p' Ís an order嘟 rev 巳rsing involution and that q -L 1J iff q ::s; p'. Furthermore, E satisfies the foIlowing ca/lcellatioll law: If pEÐq ::s; rEÐq , then 1J::S; r.1辑、lote that, if F is a sub-effect al 忍ebra of E , then the partial order on F is the restriction to F 0 1' the partial order on E. lf F is an effect algebra and φ:E • F is a morphism, then, for all p, q E E , l' ::s; q 吟 qy (P) ::s; qy(q). 趴Ilthermol毡， a surjective morphism 在 E → F is an isolllorphism iff, for all p, q ε E ， qy (P) 运 qy(q) 吨 p ~豆 q. lf we say that the effect algebra E has a certain orde川heoretic property, we mean that the poset (1言，::S;) has that property. For inslance, if we say that E is lattice ordered ，附 mean that (瓦::S;) is a 1 在 ttice. We use the usual notation 1) 八 q and p V q for the infimum (greatest lower bound) and supremum (Ieast upper bound) of elements p, q in E whell theyexìst. If we write, for instance, p 八 q = T , we mean tl1 at the infimum p 八 q exists and equals T. τwo elements p, q E E are said to be disjoinf iff p 八 q 批 O. If e E E , we defìne the Înterval E[O, e] {p 毛 Elp ::S; 叶. As usual, a 1 附lzero elementαε E is caIled an atom iff E[O, a] 刊，叶. We say that E is atol1时 i 纪 for 0 并 pε E ， there is an atomαε E with a ::s; p. lf E is lìnite, or 1l10re generally, if E satisfies chain conditìons, then E is atomic and every nonzero element in E is a fìnite orthogonal sum of atoms [12, Theorem 4.11]. 3. t. DEFlNITION. A nonempty subset 1 of E is called an ideal in E iff for all p, q {二 E with p -L q, pEÐq ε 1 iff p, q E 1. Suppose 1 is an ideal in E , p, r E E, p ::s; r and T {二 1. Since p ::s; γ ， 3q E E with pEf) q = 1"，在 nd it follows that p E 1.
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