It is proved that two di erent common generalizations of orthocomplete and
lattice e ect algebras coincide within the class of separable Archimedean e ect algebras

Tkadlec, Josef

International Journal of Theoretical Physics

Digital Library of the Czech Technical University in Prague

International Journal of Theoretical Physics (2011) 50:3915-3918DOI 10.1007/s10773-011-0781-3Note on generalizations of orthocomplete and latticeeffect algebrasJosef TkadlecReceived: 31 December 2010 / Accepted: 30 March 2011 / Published online: 14 April 2011Abstract It is proved that two different common generalizations of orthocomplete andlattice effect algebras coincide within the class of separable Archimedean effect algebras.Keywords Effect algebra, orthocomplete, lattice, separable, ArchimedeanOvchinnikov [4] introduced weakly orthocomplete orthomodular posets (he called themalternative) as a common generalization of orthocomplete orthomodular posets and ortho-modular lattices and showed that they are disjunctive. Weak orthocompleteness is usefulin the study of orthoatomisticity and disjunctivity might be used to characterize atomistic-ity [4, 8]. Weak orthocompleteness was generalized by De Simone and Navara [1] to theso-called property (W+), which was generalized for effect algebras by Tkadlec [10].Tkadlec [5] introduced the class of orthomodular posets with the maximality propertyas another common generalization of orthocomplete orthomodular posets and orthomodularlattices. He showed various consequences of this property and generalized it to the so-calledproperty (CU) [5, 6, 7, 9].Both properties (W+) and (CU) are generalizations of orthocomplete effect algebras andlattice effect algebras [9, 10]. We show that in the class of separable Archimedean effectalgebras these properties are equivalent.1 Basic notions and propertiesDefinition 1.1 An effect algebra is an algebraic structure (E,⊕,0,1) such that E is a set,0 and 1 are different elements of E and ⊕ is a partial binary operation on E such that forevery a, b, c ∈ E the following conditions hold:J. TkadlecDepartment of Mathematics, Faculty of Electrical Engineering, Czech Technical University, 166 27 Prague,Czech Republice-mail: tkadlec@fel.cvut.cz1(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 if a⊕ 1 is defined.For simplicity, we use the notation E for an effect algebra. A partial ordering on an effectalgebra E is defined by a ≤ b if there is a c ∈ E such that b = a ⊕ c. Such an element cis unique (if it exists) and is denoted by b 	 a. 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. An orthogonality relation on E is defined by a ⊥ b if a⊕ b exists. See, e.g.,Dvurecˇenskij and Pulmannova´ [2], Foulis and Bennett [3].Definition 1.2 Let E be an effect algebra. The isotropic index i(a) of an element a ∈ Eis sup{n ∈ N : na is defined}, where na = ⊕ni=1 a is the sum of n copies of a.An effect algebra E is Archimedean if every nonzero element has a finite isotropic index.The isotropic index of 0 is ∞ in every effect algebra.Definition 1.3 Let E be an effect algebra.A nonempty system (ai)i∈I of (not necessarily distinct) elements of E is called orthogonal,if sums of all finite subsystems are defined.An element a ∈ E is a majorant of an orthogonal system O if it is an upper bound of allsums of finite subsystems of O.Definition 1.4 An effect algebra is separable if every orthogonal system of its distinctelements is countable.It is easy to see that an Archimedean effect algebra is separable if and only if everyorthogonal system of its nonzero elements is countable. On the other hand, there is an un-countable orthogonal system of nonzero elements in every non-Archimedean effect algebra—for a nonzero non-Archimedean element a and an uncountable index set I we can take theorthogonal system (a)i∈I .2 ResultsFirst, let us define our main notions.Definition 2.1 Let E be a partially ordered set.A chain in E is a nonempty linearly (totally) ordered subset of E.An element a ∈ E is an upper bound of a set S ⊆ E if s ≤ a for every s ∈ S.A set S ⊆ E is downward directed if for every a, b ∈ S there is a c ∈ S such that c ≤ a, b.Definition 2.2 An effect algebra E fulfills the condition (CU) if the set of upper boundsof every chain in E is downward directed.An effect algebra E fulfills the condition (W+) if the set of majorants of every orthogonalsystem in E is downward directed.2Let us remark that these properties are not equivalent [10, Example 2.7]. The next the-orem was proved in [10]:Theorem 2.3 Every separable effect algebra fulfilling the condition (CU) fulfills the condi-tion (W+).Let us present auxiliary results concerning general properties of chains. These resultsseem to be known but we did not find a proper reference to them.Let us say that a chain {cβ : β ∈ α} for some ordinal α is ordered if cβ < cγ for everyβ, γ ∈ α with β < γ.Lemma 2.4 Let E be a partially ordered set and C ⊆ E be a chain. Then there is anordinal α and an ordered subchain {cβ : β ∈ α} of C such that for every c ∈ C there is aβ ∈ α with c ≤ cβ.Proof We will construct the desired ordered subchain by the transfinite induction. Let ustake an arbitrary c0 ∈ C. Let α be an ordinal such that cα has not been defined yet and cβhas been already defined for every β ∈ α. If the set Cα = {cβ : β ∈ α} has an upper boundc ∈ C \ Cα then we put cα = c. If there is no such c, the construction is complete. Thisconstruction stops before we reach an ordinal with cardinality greater then the cardinalityof C. The main meaning of the previous lemma is that we are able to find an ordered subchainindexed by an ordinal with the same set of upper bounds. Now let us show that in thecountable case (i.e. finite or with the cardinality of the set of natural numbers N = {0, 1, . . . })the indexing ordinal might be the least infinite ordinal ω or a finite ordinal (natural number).In the latter case it can be even the number 1.Lemma 2.5 Let E be a partially ordered set and C ⊆ E be a countable chain. Then thereis an ordinal α ≤ ω and an ordered subchain {cβ : β ∈ α} of C such that for every c ∈ Cthere is a β ∈ α with c ≤ cβ.Proof If C has a maximal element c then we can take α = 1 and c0 = c. Let us supposethat C does not have a maximal element.C is infinite and therefore there is a bijection f : N → C. For every k ∈ N, we will putck = f(nk) for an increasing sequence (nk)k∈N of natural numbers defined by the followinginduction. Let us put n0 = 0. Let k be a natural number such that ck+1 has not been definedyet and ck has been already defined. Since the set C does not have a maximal element, the set{c ∈ C : c > f(nk)} is infinite and therefore the set Nk = {n ∈ N : n > nk, f(n) > f(nk)}is nonempty. We put nk+1 = minNk.The chain {ck : k ∈ N} is ordered. Let us check the desired property. For every c ∈ Cthere is an n ∈ N such that c = f(n) and, since (nk)k∈N is increasing, a k ∈ N such thatnk ≤ n < nk+1. Due to our construction, c = f(n) ≤ f(nk) = ck. Theorem 2.6 Every separable Archimedean effect algebra fulfilling the condition (W+)fulfills the condition (CU).Proof Let E be a separable Archimedean effect algebra and let C be a chain in E. Accordingto Lemma 2.4 there is an ordinal α and an ordered subchain {cβ : β ∈ α} with the same3set of upper bounds. The system O =(cβ+1 	 cβ : β, (β + 1) ∈ α)is orthogonal. Since E isseparable and Archimedean, O is countable and therefore α is countable. Hence, accordingto Lemma 2.5, there is an ordinal α′ ≤ ω and an ordered subchain {c′β : β ∈ α′} with thesame set of upper bounds. Let a, b be upper bounds of C. Then a, b are majorants of theorthogonal system O′ = (c′0)∪(c′β+1	c′β : β, (β+1) ∈ α′). According to the condition (W+),there is a majorant d ≤ a, b of O′. Hence d is an upper bound of {c′β : β ∈ α′} and thereforeof C. Corollary 2.7 The conditions (CU) and (W+) are equivalent in separable Archimedeaneffect algebras.AcknowledgmentsThe 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 CzechRepublic no. 6840770010.References1. De Simone, A., Navara, M.: Yosida-Hewitt and Lebesgue decompositions of states on orthomod-ular posets. Tech. Report no. 40, Universita` “Federico II” Napoli, Italy (1997)2. Dvurecˇenskij, A., Pulmannova´, S.: New Trends in Quantum Structures. Kluwer Academic,Bratislava (2000)3. Foulis, D. J., Bennett, M. K.: Effect algebras and unsharp quantum logics. Found. Phys. 24,1331–1352 (1994). doi: 10.1007/BF022830364. Ovchinnikov, P. G.: On alternative orthomodular posets. Demonstr. Math. 27, 89–93 (1994)5. Tkadlec, J.: Conditions that force an orthomodular poset to be a Boolean algebra. Tatra Mt.Math. Publ. 10, 55–62 (1997)6. Tkadlec, J.: Central elements of effect algebras. Int. J. Theor. Phys. 43, 1363–1369 (2004). doi:10.1023/B:IJTP.0000048621.17418.bb7. Tkadlec, J.: Atomic sequential effect algebras. Int. J. Theor. Phys. 47, 185–192 (2008). doi:10.1007/s10773-007-9492-18. Tkadlec, J.: Atomistic and orthoatomistic effect algebras. J. Math. Phys. 49, 053505 (2008). doi:10.1063/1.29122289. Tkadlec, J.: Effect algebras with the maximality property. Algebra Universalis 61, 187–194 (2009).doi: 10.1007/s00012-009-0013-310. Tkadlec, J.: Common generalizations of orthocomplete and lattice effect algebras. Int. J. Theor.Phys. 49, 3279–3285 (2010). doi: 10.1007/s10773-009-0108-94

Note on generalizations of orthocomplete and lattice effect algebras

Josef Tkadlec

Crossref

Note on Generalizations of Orthocomplete and Lattice Effect Algebras

Connections between the weak orthocompleteness and the maximality property
in e ect algebras are presented. It is proved that an orthomodular poset with the maximality

property is disjunctive. A characterization of Archimedean weakly orthocomplete e ect
algebras is given

Int. J. Theor. Phys. (2010), 49: 3279–3285DOI 10.1007/s10773-009-0108-9The original publication is available at www.springerlink.com.Common generalizations of orthocomplete and lat-tice effect algebrasJosef TkadlecReceived: 31 October 2008 / Accepted: 24 July 2009 / Published online: 1 August 2009Abstract Connections between the weak orthocompleteness and the maximality propertyin effect algebras are presented. It is proved that an orthomodular poset with the maximal-ity property is disjunctive. A characterization of Archimedean weakly orthocomplete effectalgebras is given.Keywords Effect algebra · weakly orthocomplete · maximality property · disjunctive ·Archimedean · separableOvchinnikov [7] introduced weakly orthocomplete orthomodular posets (he called them alter-native) as a common generalization of orthocomplete orthomodular posets and orthomodularlattices and showed that they are disjunctive. Weak orthocompleteness is useful in the studyof orthoatomisticity and disjunctivity might be used to characterize atomisticity [7, 11]. Weakorthocompleteness was generalized by De Simone and Navara [1].Tkadlec [8] introduced the class of orthomodular posets with the maximality propertyas another common generalization of orthocomplete orthomodular posets and orthomodularlattices. He showed various consequences of this property and generalize it [8, 9, 10, 12].We show that these two notions are incomparable, that maximality property also impliesdisjunctivity in orthomodular posets and present a characterization of Archimedean weaklyorthocomplete effect algebras. We show also some other relations within the class of effectalgebras.J. TkadlecDepartment of Mathematics, Faculty of Electrical Engineering, Czech Technical University, 166 27 Prague,Czech Republice-mail: tkadlec@fel.cvut.cz11 Basic notions and propertiesDefinition 1.1 An effect algebra is an algebraic structure (E,⊕,0,1) such that E is a set,0 and 1 are different elements of E and ⊕ is a partial binary operation on E such that forevery 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 if a⊕ 1 is defined.For simplicity, we use the notation E for an effect algebra. A partial ordering on an effectalgebra E is defined by a ≤ b if there is a c ∈ E such that b = a ⊕ c. Such an element cis unique (if it exists) and is denoted by b 	 a. 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 that a⊕ 0 = a for every a ∈ E and that a cancellation lawis valid: for every a, b, c ∈ E with a⊕ b ≤ a⊕ c we have b ≤ c. An orthogonality relation onE is defined by a ⊥ b if a⊕ b exists. See, e.g., Dvurecˇenskij and Pulmannova´ [2], Foulis andBennett [3].Let E be a partially ordered set. For every a, b ∈ E with a ≤ b we denote [a, b] = {c ∈ E :a ≤ c ≤ b}. A chain in E is a nonempty linearly (totally) ordered subset of E.Obviously, if a ⊥ b and a ∨ b exist in an effect algebra, then a ∨ b ≤ a ⊕ b. The reverseinequality need not be true (it holds in orthomodular posets).Definition 1.2 Let E be an effect algebra. An element a ∈ E is principal if b ⊕ c ≤ a forevery b, c ∈ E such that b, c ≤ a and b ⊥ c.Definition 1.3 An orthoalgebra is an effect algebra E in which, for every a ∈ E, a = 0 ifa⊕ a is defined.An orthomodular poset is an effect algebra in which every element is principal.Every orthomodular poset is an orthoalgebra. Indeed, if a⊕a is defined then a⊕a ≤ a =a⊕ 0 and, according to the cancellation law, a ≤ 0 and therefore a = 0.Orthomodular posets are characterized as effect algebras such that a ⊕ b = a ∨ b forevery orthogonal pair a, b. Let us remark that an orthomodular poset is usually defined asa bounded partially ordered set with an orthocomplementation in which the orthomodularlaw is valid. (See [3, 4])Definition 1.4 Let E be an effect algebra. The isotropic index i(a) of an element a ∈ Eis sup{n ∈ N : na is defined}, where na = ⊕ni=1 a is the sum of n copies of a.An effect algebra E is Archimedean if every nonzero element has a finite isotropic index.The isotropic index of 0 is∞. In an orthoalgebra, a⊕a is defined only for a = 0, hence theisotropic index of every nonzero element is 1. Therefore every orthoalgebra is Archimedean.Definition 1.5 Let E be an effect algebra.A nonempty system (ai)i∈I of (not necessarily distinct) elements of E is called orthogonal,if sums of all finite subsystems are defined.An element a ∈ E is a majorant of an orthogonal system O if it is an upper bound of allsums of finite subsystems of O.2The sum⊕O of an orthogonal system O is the least majorant of O (if it exists).E is orthocomplete if every orthogonal system has the sum.E is weakly orthocomplete if every orthogonal system either has the sum or has no minimalmajorant.Every pair of elements of an orthogonal system is orthogonal. On the other hand, there aremutually orthogonal elements that do not form an orthogonal system if the effect algebrais not an orthomodular poset. Since only 0 is orthogonal to itself in an orthoalgebra andsince the multiplicity of 0 and the order of elements in an orthogonal system do not play animportant role, we may consider sets instead of systems in orthoalgebras. Every majorantof an orthogonal system is its upper bound, these notions coincide just in orthomodularposets. It is easy to see that an effect algebra is weakly orthocomplete if and only if for everyorthogonal system every its minimal majorant is its least majorant (the sum).Let us denote by O(E) the family of all orthogonal systems in an effect algebra E andlet us define an equivalence ∼ on O(E) as follows: for every O1, O2 ∈ O(E), O1 ∼ O2 if forevery a ∈ E \ {0} the multiplicities of a in O1, O2 are the same or both infinite. (We ignorethe difference in the order of elements, in the multiplicity of 0, in infinite multiplicities.) Letus denote by [O] the class of orthogonal systems equivalent to the orthogonal system O andby O(E)|∼ = {[O] : O ∈ O(E)} the set of equivalence classes. We define a partial ordering in O(E)|∼ as follows: for every O1, O2 ∈ O(E), [O1]  [O2] if for every a ∈ E \ {0} themultiplicity of a in O1 is less then or equal to the multiplicity of a in O2 or both thesemultiplicities are infinite.Definition 1.6 Let E be an effect algebra and S ⊆ E. An orthogonal system O in E is amaximal orthogonal system majorated by S, if O is majorated by S (every element of S is amajorant of O) and there is no orthogonal system O′ majorated by S such that [O] ≺ [O′].Let us remark that the “maximality” refers to the partial ordering of the equivalenceclasses. If a maximal orthogonal system O majorated by a set S contains some nonzeroelement with an infinite multiplicity (this element has an infinite isotropic index, the effectalgebra is not Archimedean) then we can add this element to O and the resulting orthogonalsystem will be majorated by S, too. This is impossible in Archimedean effect algebras.Lemma 1.7 Let E be an effect algebra and S ⊆ E. For every orthogonal system O majo-rated by S there is a maximal orthogonal system M majorated by S such that [O]  [M ].Proof If an orthogonal system O′ is majorated by S then every orthogonal system from theclass [O′] is majorated by S. Let us consider the family of classes [O′] of orthogonal systemsmajorated by S such that [O]  [O′]. This is a nonempty family such that every chain in ithas an upper bound. According to Zorn’s lemma, there is a maximal class in this family andwe can take an arbitrary its element to get the desired M . Definition 1.8 An effect algebra E has the maximality property if {a, b} has a maximallower bound for every a, b ∈ E.It is easy to see (going to orthosupplements) that an effect algebra E has the maximalityproperty if and only if {a, b} has a minimal upper bound for every a, b ∈ E.32 General relationsWe will study connections between various properties. First, let us present a useful notion.Definition 2.1 A set S in an effect algebra is downward directed if for every a, b ∈ S thereis a c ∈ S such that c ≤ a, b.It is easy to see that every minimal element in a downward directed set is its least elementand that every set with the least element is downward directed.Theorem 2.2 Let E be an effect algebra. Consider the following properties:(L) E is a lattice.(OC) E is orthocomplete.(W+) For every orthogonal system in E, the set of its majorants is downward directed.(WOC) E is weakly orthocomplete.(CU) For every chain in E, the set of its upper bounds is downward directed.(M) E has the maximality property.Then the following implications hold: (L),(OC)⇒ (W+)⇒ (WOC); (L),(OC)⇒ (CU)⇒(M).Proof (L)⇒ (W+): If a, b are majorants of an orthogonal system O then a ∧ b ≤ a, b is amajorant of O.(OC)⇒ (W+): Obvious.(W+)⇒ (WOC): Every minimal majorant of an orthogonal system is its least majorant.(L)⇒ (CU) [12]: If a, b ∈ E are upper bounds of a chain C ⊆ E, then a ∧ b ≤ a, b is anupper bound of C.(OC)⇒ (CU) [12]: According to [6, Theorem 3.2], every chain in an orthocomplete effectalgebra has the least upper bound.(CU)⇒ (M) [12]: 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 our assumption, there is an upper bound c ≤ a, b of C. Since the chain C ismaximal, c ∈ C is a maximal element of [0, a] ∩ [0, b]. Let us remark that the condition (W+) was introduced by De Simone and Navara [1],the condition (WOC) was introduced by Ovchinnikov [7], the conditions (M) and (CU) wereintroduced by Tkadlec [8, 12].It seems to be an open problem whether the condition (W+) implies the maximalityproperty (it is true in Archimedean effect algebras—see Theorem 3.1) or the condition (CU).Let us present examples showing that no other implication between properties from Theo-rem 2.2 holds except those just mentioned, stated in Theorem 2.2 and direct consequencesof the transitivity.Example 2.3 Let X be a countable infinite set. Let E be the family of finite and cofinitesubsets of X with the ⊕ operation defined as the union of disjoint sets. Then (E,⊕, ∅, X) isan orthomodular lattice (it forms a Boolean algebra) that is not orthocomplete.Example 2.4 Let X be a 6-element set. Let E be the family of even-element subsets of Xwith the ⊕ operation defined as the union of disjoint sets from E. Then (E,⊕, ∅, X) is afinite (hence orthocomplete) orthomodular poset that is not a lattice.4Example 2.5 Let X1, X2, X3, X4 be mutually disjoint 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 weakly orthocomplete ortho-modular poset. Indeed, since singletons are elements of E then every element of E is theunion and therefore the sum of every maximal orthogonal system it majorates (see Theo-rem 3.8). (E,⊕, ∅, X) does not fulfill the condition (W+) from Theorem 2.2: the orthogonalset{{x} : x ∈ X1} has majorants X1 ∪ X2 and X1 ∪ X4 but no majorant less then orequal to both of them. (E,⊕, ∅, X) does not fulfill the maximality property: lower bounds of{X1 ∪X2, X4 ∪X1} are finite subsets of X1, hence there is no maximal lower bound.Example 2.6 ([12]) 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 an orthomodular poset withthe maximality property. Let X = {xn : n ∈ N}, y0 ∈ Y , f : X → Y \ {y0} be a bijection.Then the chain{{x2, . . . , xn, f(x2), . . . , f(xn)} : n ∈ N\{1}} has two minimal upper bounds(X ∪ Y ) \ {x1, f(x1)} and (X ∪ Y ) \ {x1, y0}, hence the condition (CU) from Theorem 2.2 isnot fulfilled.Example 2.7 Let X,Y be disjoint uncountable sets of the same cardinality,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 an orthomodular posetfulfilling the condition (CU) from Theorem 2.2. Indeed, let C be a chain in E and A,B ∈ Eits upper bounds. If C contains a cofinite subset of X ∪Y then C has a maximal element. If Cdoes not contain an infinite set then the set⋃ C is countable and there is an element C ∈ Esuch that⋃ C ⊆ C ⊆ A ∩B.The orthomodular poset is not weakly orthocomplete, because for x0 ∈ X, y0 ∈ Y thereis a bijection f : X → Y \ {y0} and the orthogonal set{{x, f(x)} : x ∈ X \ {x0}} hasdifferent minimal upper bounds (X ∪ Y ) \ {x0, f(x0)} and (X ∪ Y ) \ {x0, y0}.3 Relations in special casesWe will show connection concerning conditions from Theorem 2.2 in some cases.Theorem 3.1 Every Archimedean effect algebra fulfilling the condition (W+) from Theo-rem 2.2 has the maximality property.Proof Let E be an Archimedean effect algebra fulfilling the condition (W+) from Theo-rem 2.2 and let a, b ∈ E. The orthogonal system (0) is majorated by {a, b}. According toLemma 1.7, there is a maximal orthogonal system M majorated by {a, b}. According to thecondition (W+), there is a majorant c ≤ a, b of M . Since E is Archimedean, c is a maximallower bound of {a, b} (otherwise there is a d ∈ E such that c < d ≤ a, b and we can addnonzero d	 c to M). 5Definition 3.2 An effect algebra is separable if every orthogonal system of its distinctelements is countable.It is easy to see that an Archimedean effect algebra is separable if and only if every orthog-onal system of its nonzero elements is countable. On the other hand, there is an uncountableorthogonal system of nonzero elements in every non-Archimedean effect algebras.Theorem 3.3 Every separable effect algebra fulfilling the condition (CU) from Theorem 2.2fulfills the condition (W+) from Theorem 2.2.Proof Let O be an orthogonal system in a separable effect algebra E fulfilling the condi-tion (CU) from Theorem 2.2. Since it is not important to distinguish infinite multiplicitiesof elements in orthogonal systems, we may suppose that they are countable. Since the effectalgebra is separable, the number of elements in O is countable and we can put O = (ai)i∈I(I = N or I = {1, . . . , n} for some n ∈ N). Let a, b be majorants of O. Then the chain{⊕ki=1 ai : k ∈ I} ⊆ [0, a] ∩ [0, b] has an upper bound c ≤ a, b which is a majorant of O. Let us remark that the above theorem cannot be strengthened by replacing the condi-tion (CU) by the maximality property: the orthomodular poset with the maximality prop-erty in Example 2.6 is separable but not even weakly orthocomplete—the orthogonal set{{xn, f(xn)} : n ∈ N \ {1}} has two minimal upper bounds (X ∪ Y ) \ {x1, f(x1)} and(X ∪ Y ) \ {x1, y0}.Both weak orthocompleteness and maximality property implies disjunctivity (see, e.g.,[5]) in orthomodular posets.Definition 3.4 An effect algebra E is disjunctive if for every a, b ∈ E with a 6≤ b there isa nonzero c ∈ E such that c ≤ a and c ∧ b = 0.Theorem 3.5 Every weakly orthocomplete orthomodular poset and every orthomodularposet with the maximality property is disjunctive.Proof For the case of weakly orthocomplete orthomodular posets see [7].Let E be an orthomodular poset with the maximality property and let a, b ∈ E such thata 6≤ b. Since E has the maximality property, there is a maximal element c in [0, a] ∩ [0, b].For the element d = a 	 c we obtain d ≤ a and, since a 6≤ b, d 6= 0. It suffice to prove thatd ∧ b = 0. Indeed, if it is not true then there is a nonzero element e ∈ E such that e ≤ b, d.Hence e ⊥ c and c < c⊕ e ≤ a, b (we use the principality of both a and b)—this contradictsto the maximality of c. Let us remark that Examples 2.5 and 2.7 show that a disjunctive orthomodular posetneed not be weakly orthocomplete nor have the maximality property. Let us show examplesthat the previous theorem cannot be strengthened to orthoalgebras or lattice effect algebras.Example 3.6 The so called Wright triangle [4, Example 2.13] is an orthoalgebra that isfinite (hence orthocomplete and therefore weakly orthocomplete and with the maximalityproperty) and not disjunctive.Example 3.7 C3 = {0, a,1} with a ⊕ a = 1 and x ⊕ 0 = x for every x ∈ C3 is a latticeeffect algebra (hence weakly orthocomplete and with the maximality property) that is notdisjunctive: 1 6≤ a but there is no nonzero b ∈ C3 such that b ∧ a = 0.6Let us present a characterization of Archimedean weakly orthocomplete effect algebras.Theorem 3.8 Let E be an effect algebra.(1) If every element of E is the sum of every maximal orthogonal system it majorates,then E is weakly orthocomplete.(2) If E is weakly orthocomplete and Archimedean, then every element of E is the sumof every maximal orthogonal system it majorates.Proof (1) Let O be an orthogonal system in E and let a be a minimal majorant of O. ThenO is a maximal orthogonal system majorated by a (otherwise we can add a nonzero elementb ≤ a and therefore the element a	 b < a majorates O) and therefore ⊕O = a exists.(2) Let a ∈ E and let O be a maximal orthogonal system majorated by a. Let us supposethat a is not the sum of O and seek a contradiction. Either⊕M does not exist or⊕M < a.In both cases a is not a minimal majorant of O, hence there is a majorant b < a of O. Thena	 b might be added to O and, since E is Archimedean, this contradicts to the maximalityof O. The following example shows that the assumption of Archimedeanity in the part (2) ofthe previous theorem cannot be omitted.Example 3.9 Let E = {0, 1, 2, . . . , n, . . . , n′, . . . , 2′, 1′, 0′} with m ⊕ n = m + n for everym,n ∈ N ∪ {0} and m ⊕ n′ = (n − m)′ for every m,n ∈ N ∪ {0} with m ≤ n. Then(E,⊕, 0, 0′) is a lattice effect algebra (it forms a chain) and therefore weakly orthocomplete.Neither element n′, n ∈ N∪ {0}, is the sum of the maximal orthogonal system ∏n∈N{n}N itmajorates.AcknowledgmentsThe 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 CzechRepublic no. 6840770010.References1. De Simone, A., Navara, M.: Yosida-Hewitt and Lebesgue decompositions of states on orthomod-ular posets. Tech. Report no. 40, Universita` “Federico II” Napoli, Italy (1997)2. Dvurecˇenskij, A., Pulmannova´, S.: New Trends in Quantum Structures. Kluwer Academic,Bratislava (2000)3. Foulis, D. J., Bennett, M. K.: Effect algebras and unsharp quantum logics. Found. Phys. 24,1331–1352 (1994)4. Foulis, D., Greechie, R., Ru¨ttimann, G.: Filters and supports in orthoalgebras. Int. J. Theor.Phys. 31, 789–807 (1992)5. Godowski, R.: Disjunctivity and orthodisjunctivity in orthomodular posets. Demonstr. Math. 12,1043–1049 (1979)6. Jencˇa, G., Pulmannova´, S.: Orthocomplete effect algebras. Proc. Am. Math. Soc. 131, 2663–2671(2003)7. Ovchinnikov, P. G.: On alternative orthomodular posets. Demonstr. Math. 27, 89–93 (1994)8. Tkadlec, J.: Conditions that force an orthomodular poset to be a Boolean algebra. Tatra Mt.Math. Publ. 10, 55–62 (1997)79. Tkadlec, J.: Central elements of effect algebras. Int. J. Theor. Phys. 43, 1363–1369 (2004)10. Tkadlec, J.: Atomic sequential effect algebras. Int. J. Theor. Phys. 47, 185–192 (2008)11. Tkadlec, J.: Atomistic and orthoatomistic effect algebras. J. Math. Phys. 49, 053505 (2008)12. Tkadlec, J.: Effect algebras with the maximality property. Algebra Universalis, to appear.8

Note on generalizations of orthocomplete and lattice effect algebras

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