The maximality property was introduced in [9] in orthomodular posets as
a common generalization of orthomodular lattices and orthocomplete orthomodular
posets. We show that various conditions used in the theory of e ect
algebras are stronger than the maximality property, clear up the connections
between them and show some consequences of these conditions. In particular,
we prove that a Jauch{Piron e ect algebra with a countable unital set of states
is an orthomodular lattice and that a unital set of Jauch{Piron states on an
e ect algebra with the maximality property is strongly order determining

Tkadlec, Josef

Algebra Universalis

Digital Library of the Czech Technical University in Prague

Algebra Universalis 61 (2009), 187–194.DOI 10.1007/s00012-009-0013-3The original publication is available at www.springerlink.com.Effect algebras with the maximality propertyJosef TkadlecDepartment of Mathematics, Faculty of Electrical Engineering, Czech TechnicalUniversity, 166 27 Praha, Czech Republic, tkadlec@fel.cvut.czAbstractThe maximality property was introduced in [9] in orthomodular posets asa common generalization of orthomodular lattices and orthocomplete ortho-modular posets. We show that various conditions used in the theory of effectalgebras are stronger than the maximality property, clear up the connectionsbetween them and show some consequences of these conditions. In particular,we prove that a Jauch–Piron effect algebra with a countable unital set of statesis an orthomodular lattice and that a unital set of Jauch–Piron states on aneffect algebra with the maximality property is strongly order determining.1 Basic notionsEffect algebras as generalizations of orthomodular posets (quantum logics) are stud-ied in the axiomatics of quantum systems—see, e.g., [1, 2].1.1. Definition. An effect algebra is an algebraic structure (E,⊕,0,1) such that Eis a set, 0 and 1 are distinct elements of E and ⊕ is a partial binary operation onE such that for every a, b, c ∈ E the following conditions hold:(1) a⊕ b = b⊕ a if a⊕ b exists,(2) (a⊕ b)⊕ c = a⊕ (b⊕ c) if (a⊕ b)⊕ c exists,(3) there is a unique a′ ∈ E such that a⊕ a′ = 1 (orthosupplement),(4) a = 0 whenever a⊕ 1 is defined.For simplicity, we use the notation E for an effect algebra. A partial ordering onan effect algebra E is defined by a ≤ b iff there is a c ∈ E such that b = a⊕ c. Suchan element c is unique (if it exists) and is denoted by b	a. The element 0 (1, resp.)is the least (the greatest, resp.) element of E with respect to this partial ordering.For every a, b ∈ E, a′′ = a and b′ ≤ a′ whenever a ≤ b. It can be shown thata⊕ 0 = a for every a ∈ E and that a cancellation law is valid: for every a, b, c ∈ Ewith a⊕ b ≤ a⊕ c we have b ≤ c. An orthogonality relation on E is defined by a ⊥ biff a⊕ b exists (iff a ≤ b′). (See, e.g., [1, 2].)For a ≤ b we denote [a, b] = {c ∈ E : a ≤ c ≤ b}. A chain in E is a nonemptylinearly (totally) ordered subset of E.The work was supported by the grant of the Grant Agency of the Czech Republicno. 201/07/1051 and by the research plan of the Ministry of Education of the Czech Republicno. 6840770010.Keywords: effect algebra, Jauch–Piron state, maximality property.AMS Subject Classification (2000): 03G12; 06C15, 81P10.1Obviously, if a ⊥ b and a ∨ b exist in an effect algebra, then a ∨ b ≤ a⊕ b. Thereverse inequality need not be true (it holds in orthomodular posets).1.2. Definition. Let E be an effect algebra. An element a ∈ E is principal ifb⊕ c ≤ a for every b, c ∈ E such that b, c ≤ a and b ⊥ c.1.3. Definition. An orthoalgebra is an effect algebra E in which, for every a ∈ E,a = 0 whenever a⊕ a is defined.An orthomodular poset is an effect algebra in which every element is principal.An orthomodular lattice is an orthomodular poset that is a lattice.Every orthomodular poset is an orthoalgebra. Indeed, if a ⊕ a is defined thena⊕a ≤ a = a⊕0 and, according to the cancellation law, a ≤ 0 and therefore a = 0.Orthoalgebras are characterized by the following conditions: the orthosupple-mentation is an orthocomplementation (i.e., a ∨ a′ = 1 for every a) or a ⊕ b is aminimal upper bound of a, b for every a, b. Orthomodular posets are characterizedas effect algebras such that a⊕ b = a ∨ b for every orthogonal pair a, b. (See [2, 3].)Let us remark that an orthomodular poset is usually defined as a bounded partiallyordered set with an orthocomplementation in which the orthomodular law is valid.Let us present a special construction of orthomodular posets that we will use insome examples.1.4. Proposition. Let X 6= ∅, E ⊆ expX be nonempty such that the followingconditions are fulfilled:(1) X \A ∈ E whenever A ∈ E,(2) A ∪B ∈ E whenever A,B ∈ E are disjoint.Then (E,⊕, ∅, X) with A ⊕ B = A ∪ B for disjoint A,B ∈ E is an orthomodularposet such that the orthosupplement is the set-theoretic complement and the partialordering is the inclusion.Proof. Obvious.1.5. Definition. An orthomodular poset of the form of Proposition 1.4 is calledconcrete.It is well-known (see, e.g., [8]) that an orthomodular poset has a concrete rep-resentation iff it has a strongly order determining set of two-valued states (see thenext section for these notions).Let us present two important notions we will use in the sequel.1.6. Definition. A system (ai)i∈I of (not necessarilly distinct) elements of an effectalgebra E is orthogonal if⊕i∈F ai is defined for every finite set F ⊆ I.An effect algebra E is orthocomplete if for every orthogonal system (ai)i∈I ofelements of E the supremum∨{⊕i∈F ai : F ⊆ I is finite} exists.1.7. Definition. An effect algebra E has the maximality property if [0, a] ∩ [0, b]has a maximal element for every a, b ∈ E.Obviously, every finite effect algebra has the maximality property and everylattice effect algebra has the maximality property: a ∧ b is a maximal (even thegreatest) element of [0, a] ∩ [0, b] for every a, b.22 States2.1. Definition. Let E be an effect algebra. A state s on E is a mapping s :E → [0, 1] such that:(1) s(1) = 1,(2) s(a⊕ b) = s(a) + s(b) whenever a⊕ b is defined.A set S of states on E is unital if, for every a ∈ E \ {0}, there is a state s ∈ Ssuch that s(a) = 1.A set S of states on E is strongly order determining if, for every a, b ∈ E witha 6≤ b, there is a state s ∈ S such that s(a) = 1 > s(b).Obviously, for every state s we have s(0) = 0, s(a′) = 1− s(a) for every a ∈ E,s(a) ≤ s(b) for every a, b ∈ E with a ≤ b. It is easy to see that every strongly orderdetermining set of states on an effect algebra is unital.There are orthomodular lattices without any state [4]. On the other hand, thereis a strongly order determining set of special two-valued states on concrete ortho-modular posets (it is easy to verify that they are indeed states):2.2. Definition. Let E ⊆ expX be a a concrete orthomodular poset, x ∈ X. Thestate sx on E defined bysx(A) ={0 , x /∈ A ,1 , x ∈ A , A ∈ E ,is called carried by the point x.Let us present two observations describing the impact of a sufficiently large statespaces to the properties of the algebraic structure.2.3. Proposition. Every effect algebra with a unital set of states is an orthoalgebra.Proof. Let E be an effect algebra with a unital set S of states. Let a ∈ E be suchthat a ⊕ a is defined. Then 1 ≥ s(a ⊕ a) = 2 s(a) and therefore s(a) ≤ 12 for everystate s ∈ S. Since S is unital, we obtain that a = 0.2.4. Proposition. Every effect algebra with a strongly order determining set ofstates is an orthomodular poset.Proof. Let E be an effect algebra with a strongly order determining set S of states.Let us prove that every element of E is principal. Let a, b, c ∈ E be such thatb, c ≤ a and b ⊥ c. Then for every state s ∈ S with s(a′) = 1 we consecutivelyobtain: 0 = s(a) = s(b) = s(c) = s(b⊕ c), s((b⊕ c)′) = 1. Since the set S is stronglyorder determining, we obtain that a′ ≤ (b⊕ c)′ and therefore b⊕ c ≤ a.33 Jauch–PironnessAn important role in the axiomatics of quantum structures play the so-called Jauch–Piron states [5, 8].3.1. Definition. Let E be an effect algebra. A state s on E is Jauch–Piron if forevery a, b ∈ E with s(a) = s(b) = 1 there is a c ∈ E such that c ≤ a, b and s(c) = 1.An effect algebra is Jauch–Piron if every state on it is Jauch–Piron.It is easy to see that every Boolean algebra is Jauch–Piron. There are Jauch–Piron effect algebras that are not orthoalgebras (e.g., the 3-element chain C3 ={0, a,1} with a⊕ a = 1 and x⊕ 0 = x for every x ∈ C3) and orthomodular latticesthat are not Jauch–Piron (e.g., MO2 = {0, a, a′, b, b′,1} with a⊕ a′ = b⊕ b′ = 1 andx⊕ 0 = x for every x ∈ E; the two-valued states are not Jauch–Piron).The following statement was proved in [10, Proposition 2.6], we will generalizeit later (Theorem 3.7).3.2. Lemma. Every Jauch–Piron effect algebra with a countable unital set of stateshas the maximality property.3.3. Proposition. Every effect algebra with the maximality property and with aunital set of Jauch–Piron states is an orthomodular poset.Proof. Let E be an effect algebra with the maximality property and with a unitalset S of Jauch–Piron states. Let us suppose that E is not an orthomodular posetand seek a contradiction. There are elements a, b, c ∈ E such that b, c ≤ a, b ⊥ cand b⊕ c 6≤ a. Let us denote d = b⊕ c. Since E has the maximality property, thereis a maximal element e in [0, a′] ∩ [0, d′]. Since d 6≤ a, we obtain that a′ 6≤ d′ andtherefore e < a′ and a′	 e 6= 0. Since the set S is unital, there is a state s ∈ S suchthat s(a′ 	 e) = 1. Hence s(a′) = 1, 0 = s(e) = s(a) = s(b) = s(c) = s(d), s(d′) = 1,s(d′ 	 e) = 1. Since the state s is Jauch–Piron, there is an element f ∈ E suchthat f ≤ (a′ 	 e), (d′ 	 e) and s(f) = 1. Hence f 6= 0 and e < e ⊕ f ≤ a′, d′—thiscontradicts to the maximality of e.Let us remark that there are effect algebras with the maximality property thatare not orthoalgebras—e.g., the 3-element chain C3 = {0, a,1} with a ⊕ a = 1and x ⊕ 0 = x for every x ∈ C3. It seems to be an open question whether theassumption of the maximality property in Proposition 3.3 might be omitted (it isnot a consequence of the existence of a countable unital set of Jauch–Piron states—see Example 3.8). Proposition 3.3 cannot be improved to orthomodular lattices—seeExample 3.5 ({12 (sx + sy) : x, y ∈ X, x 6= y} is a unital set of Jauch–Piron states).It is well-known and easy to see that every state on a Boolean algebra is Jauch–Piron and that a unital set of states on a Boolean algebra is strongly order deter-mining. Let us generalize the latter statement.3.4. Theorem. A set of Jauch–Piron states on an effect algebra with the maximalityproperty is unital if and only if it is strongly order determining.4Proof. ⇐: Obvious.⇒: Let E be an effect algebra with the maximality property and with a unitalset S of Jauch–Piron states. Let a, b ∈ E such that a 6≤ b. Let c ∈ E be a maximalelement of [0, a]∩ [0, b]. Then c < a and therefore a	c 6= 0. Since the set S is unital,there is a state s ∈ S such that s(a	 c) = 1 and therefore s(a) = 1. Let us supposethat s(b) = 1 and seek a contradiction. Since s is Jauch–Piron, there is an elementd ∈ E such that d ≤ a 	 c, d ≤ b and s(d) = 1. Hence d 6= 0 and c < c ⊕ d ≤ a.According to Proposition 3.3, b is principal and therefore c⊕d ≤ b—this contradictsto the maximality of c.Let us remark that we proved Theorem 3.4 using Proposition 3.3. On the otherhand, Proposition 3.3 is a direct consequence of Theorem 3.4 and Proposition 2.4,hence we might prove it by this way if we prove Theorem 3.4 by another way.Let us present examples showing that the assumptions in Theorem 3.4 cannotbe omitted.3.5. Example. Let X = {a, b, c, d}, E be the family of even-element subsets of Xwith the ⊕ operation defined as the union of disjoint sets. Then (E,⊕, ∅, X) is afinite (hence with the maximality property) concrete orthomodular poset and theset S = {sa, sb, sc} of states carried by points a, b, c is a unital set of (two-valued)states on E that is not strongly order determining: {a, d} 6≤ {a, b} but there is nostate s ∈ S such that s({a, d}) = 1 > s({a, b}). (States in S are not Jauch–Piron.)3.6. Example. Let X1, X2, X3, X4 be nonempty mutually disjoint sets, X1, X3 beinfinite, X =⋃4i=1Xi,E0 = {∅, X1 ∪X2, X2 ∪X3, X3 ∪X4, X4 ∪X1, X} ,E = {(A \ F ) ∪ (F \A) : F ⊆ X1 ∪X3 is finite, A ∈ E0} ,A⊕B = A∪B for disjoint A,B ∈ E. Then (E,⊕, ∅, X) is a concrete orthomodularposet and the set S = {sx : x ∈ X1∪X3} of states carried by points from X1∪X3 isa unital set of (two-valued) Jauch–Piron states on E. The set S is not strongly orderdetermining because X1 ∪X4 6≤ X1 ∪X2 and for every s ∈ S with s(X1 ∪X4) = 1there is an x ∈ X1 such that s = sx and therefore s(X1∪X2) = 1. (E does not havethe maximality property.)3.7. Theorem. Every Jauch–Piron effect algebra with a countable unital set ofstates is an orthomodular lattice.Proof. Let E be a Jauch–Piron effect algebra with a countable unital set S of states.According to Lemma 3.2, E has the maximality property. According to Theorem 3.4,the set S is strongly order determining. According to Proposition 2.4, E is anorthomodular poset. Let us show that a ∧ b exists for every a, b ∈ E. (Then alsoa ∨ b = (a′ ∧ b′)′ exists for every a, b ∈ E.) If [0, a] ∩ [0, b] = {0} then 0 = a ∧ b.Let us suppose that there is a nonzero element c ∈ E such that c ≤ a, b. Thenthere is a state s ∈ S such that s(c) = 1. Hence s(a) = s(b) = 1 and the setSa,b = {s ∈ S : s(a) = s(b) = 1} is nonempty and countable. Let s0 be a σ-convexcombination with nonzero coefficients of all (possibly infinitely many) states fromSa,b, i.e., there are real numbers αs > 0 for s ∈ Sa,b such that∑s∈Sa,b αs = 1 and5s0 =∑s∈Sa,b αss. Then s0 is a state on E such that s0(a) = s0(b) = 1. Sincethe state s0 is Jauch–Piron, there is an element c0 ∈ E such that c0 ≤ a, b and1 = s0(c0) =∑s∈Sa,b αs s(c0). Hence, for every s ∈ Sa,b, αs s(c0) = αs and therefores(c0) = 1. For every c ∈ [0, a]∩ [0, b] and every s ∈ S with s(c) = 1 we have s ∈ Sa,band therefore s(c0) = 1. Since S is strongly order determining, c ≤ c0 for everyc ∈ [0, a] ∩ [0, b]. Hence c0 = a ∧ b.Let us present showing examples that the conditions in Theorem 3.7 cannot beomitted. There is a concrete (hence with a strongly order determining set of two-valued states) Jauch–Piron orthomodular poset that is not a lattice—see [7] (everyunital set of states on it is uncountable). As the following example shows there is anorthomodular poset with a countable strongly order determining set of (two-valued)Jauch–Piron states that does not have the maximality property and therefore it isnot a lattice (there are non-Jauch–Piron states).3.8. Example. Let X1, X2, X3, X4 be mutually disjoint countable infinite sets, X =⋃4i=1Xi,E0 = {∅, X1 ∪X2, X2 ∪X3, X3 ∪X4, X4 ∪X1, X} ,E = {(A \ F ) ∪ (F \A) : F ⊆ X is finite, A ∈ E0} ,A⊕B = A∪B for disjoint A,B ∈ E. Then (E,⊕, ∅, X) is a concrete orthomodularposet and the set S = {sx : x ∈ X} of states carried by points is a countablestrongly order determining set of two-valued Jauch–Piron states on E. The set[∅, X1 ∪ X2] ∩ [∅, X4 ∪ X1] consists of finite subsets of X1, hence E does not havethe maximality property. As an example of a non-Jauch–Piron state we can take aσ-convex combination with nonzero coefficients of states from {sx : x ∈ X1}.4 Relationship of various conditions4.1. Theorem. Let E be an effect algebra. Consider the following properties:(F) E is finite.(CF) E is chain-finite.(OC) E is orthocomplete.(JPCU) E is Jauch–Piron with a countable unital set of states.(L) E is a lattice.(CU) For every a, b ∈ E, every chain in [0, a] ∩ [0, b] has an upper bound in[0, a] ∩ [0, b].(M) E has the maximality property.Then the following implications hold: (F)⇒ (CF)⇒ (OC)⇒ (CU)⇒ (M), (JPCU)⇒(L)⇒ (CU).Proof. (F)⇒ (CF): Obvious.(CF)⇒ (OC): Every orthogonal system in a chain-finite effect algebra is finite.Hence E is orthocomplete.(OC)⇒ (CU): Let C be a chain in [0, a] ∩ [0, b]. According to [6, Theorem 3.2],every chain in an orthocomplete effect algebra has a supremum. This supremumobviously belongs to [0, a] ∩ [0, b].6(CU)⇒ (M): Let a, b ∈ E. Since [0, a] ∩ [0, b] ⊇ {0}, the family of chains in[0, a]∩ [0, b] is nonempty. According to Zorn’s lemma, there is a maximal chain C in[0, a]∩ [0, b]. According to the assumption, there is an upper bound c ∈ [0, a]∩ [0, b]of C. Since the chain C is maximal, c ∈ C is a maximal element of [0, a] ∩ [0, b].(JPCU)⇒ (L): See Theorem 3.7.(L)⇒ (CU): Let a, b ∈ E. The element a ∧ b is an upper bound for every chainin [0, a] ∩ [0, b].Let us present examples that show that the scheme of implications in the previoustheorem cannot be improved in the sense that no other implications (except directconsequences of the transitivity) between given conditions hold.4.2. Example. Let X be an infinite set, y /∈ X, E = {∅} ∪ {{x, y} : x ∈ X} ∪{X \ {x} : x ∈ X} ∪ {X ∪ {y}}, A ⊕ B = A ∪ B for disjoint A,B ∈ E. Then(E,⊕, ∅, X ∪ {y}) is an infinite chain-finite concrete orthomodular lattice.4.3. Example. Let X be an uncountable set, E = expX with A ⊕ B = A ∪ Bfor disjoint A,B ∈ E. Then (E,⊕, ∅, X) is an orthocomplete concrete orthomodularlattice (it forms a Boolean algebra) such that there is an uncountable set of mutuallyorthogonal elements. Hence it is not chain-finite and every unital set of states on Eis uncountable.4.4. Example. Let X be a countable infinite set. Let E be a family of finite andcofinite subsets of X with the ⊕ operation defined as the union of disjoint sets. Then(E,⊕, ∅, X) is a concrete orthomodular lattice (it forms a Boolean algebra) fulfillingthe condition (JPCU) (every state on a Boolean algebra is Jauch–Piron, there is acountable unital set of states carried by points) that is not orthocomplete.4.5. Example. Let X be a 6-element set. Let E be the family of even-elementsubsets of X with the ⊕ operation defined as the union of disjoint sets from E.Then (E,⊕, ∅, X) is a finite concrete orthomodular poset that is not a latice.4.6. Example. Let X,Y be disjoint infinite countable sets,E0 = {A ⊆ (X ∪ Y ) : card(A ∩X) = card(A ∩ Y ) is finite} ,E = E0 ∪ {(X ∪ Y ) \A : A ∈ E0} ,A ⊕ B = A ∪ B for disjoint A,B ∈ E. Then (E,⊕, ∅, X ∪ Y ) is a concrete ortho-modular poset with the maximality property. Let X = {xn : n ∈ N}, y0 ∈ Y , f :X → Y \{y0} be a bijection, A = (X∪Y )\{x1, f(x1)}, B = (X∪Y )\{x1, y0}. Thenthe chain{{x2, . . . , xn, f(x2), . . . , f(xn)} : n ∈ N \ {1}} in [∅, A] ∩ [∅, B] does nothave an upper bound in [∅, A] ∩ [∅, B], hence the condition (CU) from Theorem 4.1is not fulfilled.Let us remark that not all effect algebras have the maximality property (seeExample 3.8).7References[1] Dvurecˇenskij, A., Pulmannova´, S.: New Trends in Quantum Structures. KluwerAcademic Publishers, Bratislava, 2000.[2] Foulis, D. J., Bennett, M. K.: Effect algebras and unsharp quantum logics,Found. Phys. 24 (1994), 1331–1352.[3] Foulis, D., Greechie, R., Ru¨ttimann, G.: Filters and supports in orthoalgebras.Internat. J. Theoret. Phys. 31 (1992), 789–807.[4] Greechie, R. J.: Orthomodular lattices admitting no states. J. Combin. TheorySer. A 10 (1971), 119–132.[5] Jauch, J.: Foundations of Quantum Mechanics. Addison-Wesley, Reading,Mass., 1968.[6] Jencˇa, G., Pulmannova´, S.: Orthocomplete effect algebras. Proc. Amer. Math.Soc. 131 (2003), 2663–2671.[7] Mu¨ller, V.: Jauch–Piron states on concrete quantum logics. Internat. J. Theo-ret. Phys. 32 (1993), 433–442.[8] Pta´k, P., Pulmannova´, S.: Orthomodular Structures as Quantum Logics. KluwerAcademic Publishers, Dordrecht, 1991.[9] Tkadlec, J.: Conditions that force an orthomodular poset to be a Boolean alge-bra. Tatra Mt. Math. Publ. 10 (1997), 55–62.[10] Tkadlec, J.: Central elements of effect algebras. Internat. J. Theoret. Phys. 43(2004), 1363–1369.8

Effect algebras with the maximality property

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