Sharply Orthocomplete Effect Algebras

Special types of effect algebras $E$ called sharply dominating and S-dominating were introduced by S. Gudder in \cite{gudder1,gudder2}. We prove statements about connections between sharp orthocompleteness, sharp dominancy and completeness of $E$. Namely we prove that in every sharply orthocomplete S-dominating effect algebra $E$ the set of sharp elements and the center of $E$ are complete lattices bifull in $E$. If an Archimedean atomic lattice effect algebra $E$ is sharply orthocomplete then it is complete.


Introduction
An algebraic structure called an effect algebra has been introduced by D.J. Foulis and M.K. Bennett (1994). The advantage of an effect algebra is that effect algebras provide a mechanism for studying quantum effects, or more general, in non-classical probability theory their elements represent events that may be unsharp or pairwise non-compatibble. Lattice effect algebras are in some sence a nearest common generalization of orthomodular lattices [13] that may include non-compatible pairs of elements, and M V -algebras [3] that may include unsharp elements. More precisely a lattice effect algebra E is an orthomodular lattice iff every element of E is sharp (i.e., x and "non x" are disjoint) and it is an M V -effect algebra iff every pair of elements of E is compatible. Moreover, in every lattice effect algebra E the set of sharp elements is an orthomodular lattice ( [10]), and E is a union of its blocks (i.e., maximal subsets of pairwise compatible elements that are M V -effect algebras (see [21])). Thus a lattice effect algebra E is a Boolean algebra iff every pair of elements are compatible and every element of E is sharp.
However, non-lattice ordered effect algebra E is so general that its set S(E) of sharp elements may form neither an orthomodular lattice nor any regular algebraic structure. S. Gudder (see [7,8]) introduced special types of effect algebras E called sharply dominating, whose set S(E) of sharp elements forms an orthoalgebra and also so called S-dominating ,whose set S(E) of sharp elements forms an orthomodular lattice. In [7], S. Gudder showed that a standard Hilbert space effect algebra E(H) of bounded operators on a Hilbert space H between zero and identity operators (with partially defined usual operation + ) are S-dominating. Hence S-dominating effect algebras may be useful abstract models for sets of quantum effects in physical systems.
We study these two special kinds of effect algebras. We show properties of some remarkable sub-effect algebras of such effect algebras E satisfying the condition that E is sharply orthocomplete. Namely properties of their blocks, sets of sharp elements and their centers. It is worth to note that in [11] it was proved that there are even Archimedean atomic M V -effect algebras which are not sharply dominating hence they are not S-dominating.
2 Basic def initions and some known facts Definition 1 ([4]). A partial algebra (E; ⊕, 0, 1) is called an effect algebra if 0, 1 are two distinct elements and ⊕ is a partially defined binary operation on E which satisfy the following conditions for any x, y, z ∈ E: We often denote the effect algebra (E; ⊕, 0, 1) briefly by E. On every effect algebra E the partial order ≤ and a partial binary operation ⊖ can be introduced as follows: x ≤ y and y ⊖ x = z iff x ⊕ z is defined and x ⊕ z = y.
If E with the defined partial order is a lattice (a complete lattice) then (E; ⊕, 0, 1) is called a lattice effect algebra (a complete lattice effect algebra).

Definition 2.
Let E be an effect algebra. Then Q ⊆ E is called a sub-effect algebra of E if (ii) if out of elements x, y, z ∈ E with x ⊕ y = z two are in Q, then x, y, z ∈ Q.
If E is a lattice effect algebra and Q is a sub-lattice and a sub-effect algebra of E then Q is called a sub-lattice effect algebra of E.
Note that a sub-effect algebra Q (sub-lattice effect algebra Q) of an effect algebra E (of a lattice effect algebra E) with inherited operation ⊕ is an effect algebra (lattice effect algebra) in its own right.
For an element x of an effect algebra E we write ord(x) = ∞ if nx = x ⊕ x ⊕ · · · ⊕ x (n-times) exists for every positive integer n and we write ord(x) = n x if n x is the greatest positive integer such that n x x exists in E. An effect algebra E is Archimedean if ord(x) < ∞ for all x ∈ E.
A minimal nonzero element of an effect algebra E is called an atom and E is called atomic if under every nonzero element of E there is an atom.
For a poset P and its subposet Q ⊆ P we denote, for all X ⊆ Q, by Q X the join of the subset X in the poset Q whenever it exists.
We say that a finite system F = (x k ) n k=1 of not necessarily different elements of an effect algebra (E; ⊕, 0, 1) is orthogonal if x 1 ⊕ x 2 ⊕ · · · ⊕ x n (written n k=1 x k or F ) exists in E. Here we define x 1 ⊕ x 2 ⊕ · · · ⊕ x n = (x 1 ⊕ x 2 ⊕ · · · ⊕ x n−1 ) ⊕ x n supposing that x k is defined and We also define ∅ = 0. An arbitrary system G = (x κ ) κ∈H of not necessarily different elements of E is called orthogonal if K exists for every finite K ⊆ G. We say that for an orthogonal system G = (x κ ) κ∈H the element G (more precisely E G) exists iff { K | K ⊆ G is finite} exists in E and then we put We call an effect algebra E orthocomplete [9] if every orthogonal system G = (x κ ) κ∈H of elements of E has the sum G. It is known that every orthocomplete Archimedean lattice effect algebra E is a complete lattice (see [22,Theorem 2.6]).
Recall that elements x, y of a lattice effect algebra E are called compatible (written x ↔ y) iff x ∨ y = x ⊕ (y ⊖ (x ∧ y)) (see [15]). P ⊆ E is a set of pairwise compatible elements if x ↔ y for all x, y ∈ P . M ⊆ E is called a block of E iff M is a maximal subset of pairwise compatible elements. Every block of a lattice effect algebra E is a sub-effect algebra and a sub-lattice of E and E is a union of its blocks (see [21]). Lattice effect algebra with a unique block is called an M V -effect algebra. Every block of a lattice effect algebra is an M V -effect algebra in its own right.
An element w of an effect algebra E is called sharp (see ( [7,8] Note that clearly E is sharply dominating iff for every x ∈ E there exists x ∈ S(E) such that The well known fact is that in every S-dominating effect algebra E the subset S(E) = {w ∈ E | w ∧ w ′ = 0} of sharp elements of E is a sub-effect algebra of E being an orthomodular lattice (see [8,Theorem 2.6]). Moreover if for D ⊆ S(E) the element E D exists then E D ∈ S(E) hence S(E) D = E D. We say that S(E) is a full sublattice of E (see [10]).
Let G be a sub-effect algebra of an effect algebra E. We say that G is bifull in E, if, for any D ⊆ G the element G D exists iff the element E D exists and they are equal. Clearly, any bifull sub-effect algebra of E is full but not conversely (see [12]).
The notion of a central element of an effect algebra E was introduced by Greechie-Foulis-Pulmannová [6]. An element c ∈ E is called central (see [18] is a Boolean algebra, see [6]. If E is a lattice effect algebra then z ∈ E is central iff z ∧ z ′ = 0 and z ↔ x for all x ∈ E, see [19]. Thus in a lattice effect algebra E, An effect algebra E is called centrally dominating (see also [5] for the notion central cover) if for every x ∈ E there exists c x ∈ C(E) such that An element a of a lattice L is called compact iff, for any D ⊆ L, a ≤ D implies a ≤ F for some finite F ⊆ D. A lattice L is called compactly generated iff every element of L is a join of compact elements.

Sharply orthocomplete ef fect algebras
In an effect algebra E the set S(E) = {x ∈ E | x ∧ x ′ = 0} of sharp elements plays an important role. In some sense we can say that an effect algebra E is a "smeared set S(E)" of its sharp elements, while unsharp effects are important in studies of unsharp measurements ( [4,2]). S.Gudder proved (see [8]) that, in standard Hilbert space effect algebra E(H) of bounded operators A on a Hilbert space H between null operator and identity operator, which are endowed with usual + defined iff A+B is in E(H), the set S(E(H)) of sharp elements forms an orthomodular lattice of projections operators on H. Further in (see [8,Theorem 2.2]) it was shown that in every sharply dominating effect algebra the set S(E) is a sub-effect algebra of E. Moreover, in [7,Theorem 2.6] it is proved that in every S-dominating effect algebra E the set S(E) is an orthomodular lattice. We are going to show that in this case S(E) is bifull in E. Proof . Let S ⊆ S(E).
(1) Assume that z = S(E) S ∈ S(E) exists. Let us show that z is the least upper bound of S in E. Let y ∈ E be an upper bound of S. Then y ∧ z exists and it is an upper bound of S as well. Hence, for any s ∈ S, s ≤ y ∧ z. This yields that s ≤ y ∧ z ≤ y ∧ z, for all s ∈ S, y ∧ z ∈ S(E). Hence z ≤ y ∧ z ≤ y ∧ z ≤ z. Then z = y ∧ z ≤ y i.e., z is really the least upper bound of S in E.
(2) Conversely, let z = E S ∈ E exists. Let y ∈ S(E) be an upper bound of S in S(E). Then y ∧ z exists and it is again an upper bound of S. As E is sharply dominating, there exists a greatest sharp element y ∧ z ≤ y ∧ z and hence s ≤ y ∧ z ≤ y ∧ z. This gives that z = y ∧ z ∈ S(E). Thus z = S(E) S ∈ S(E).

Corollary 1. If E is a sharply dominating lattice effect algebra then S(E) is bifull in E.
Definition 4. An effect algebra E is called sharply orthocomplete (centrally orthocomplete (see [5])) if for any system (x κ ) κ∈H of elements of E such that there exists an orthogonal system (w κ ) κ∈H , w κ ∈ S(E) with x κ ≤ w κ , κ ∈ H (an orthogonal system (c κ ) κ∈H , c κ ∈ C(E) with x κ ≤ c κ , κ ∈ H) there exists Theorem 2. Let E be a sharply orthocomplete S-dominating effect algebra. Then

(i) S(E) is a complete orthomodular lattice bifull in E.
(ii) C(E) is a complete Boolean algebra bifull in E.
(iii) E is centrally dominating and centrally orthocomplete.

Proof . (i): From [8, Theorem 2.6] we know that S(E) is an orthomodular lattice and a sublattice effect algebra of E.
Let us show that S(E) is orthocomplete. Let S ⊆ S(E), S orthogonal. Then for every finite F ⊆ S we have that E F = E F = S(E) F ∈ S(E). Moreover, for any s ∈ S, s ≤ s. Since S(E) is bifull in E by Theorem 1 and E is sharply orthocomplete we have that E S = E S = S(E) S ∈ S(E) exists. Since S(E) is an Archimedean lattice effect algebra we have from [22,Theorem 2.6] that S(E) is complete.

It follows by (i) that, for any
(iv): Since C(E) is an atomic Boolean algebra we have C(E) {p ∈ C(E) | p atom of C(E)} = 1.
4 Sharply orthocomplete lattice ef fect algebras M. Kalina in [12] has shown that even in an Archimedean atomic lattice effect algebra E with atomic center C(E) the join of atoms of C(E) computed in E need not be equal to 1. Next examples and theorems show connections between sharp orthocompleteness, sharp dominancy and completeness of an effect algebra E as well as bifullness of S(E), C(E) and atomic blocks in a lattice effect algebra E.

Example 1. Example of a compactly generated sharply orthocomplete M V -effect algebra that is not complete.
It is enough to take the Chang M V -effect algebra E = {0, a, 2a, 3a, . . . , (3a) ′ , (2a) ′ , a ′ , 1} that is not Archimedean (hence non-complete), it is compactly generated (every x ∈ E is compact) and obviously sharply orthocomplete (the center C(E) = S(E) is trivial) and hence sharply dominating.
Example 2. Example of a sharply dominating Archimedean atomic lattice M V -effect algebra E with complete and bifull S(E) that is not sharply orthocomplete.
Then E 0 is a sub-lattice effect algebra of E (hence it is an M V -effect algebra), evidently sharply dominating and it is not sharply orthocomplete (since it is non-complete). S(E 0 ) = {{0 n , 1 n } | n = 1, 2, . . . } is a complete Boolean algebra and S(E 0 ) = C(E 0 ) is a bifull sub-lattice of E 0 . Lemma 1. Let E be a sharply orthocomplete Archimedean atomic M V -effect algebra. Then E is complete.
Proof . Let A ⊆ E be a set of all atoms of E. Then 1 = E {n a a|a ∈ A} = E {n a a|a ∈ A}, n a a ∈ C(E) = S(E) are atoms of C(E) for all a ∈ A. By [23, Theorem 3.1] we have that E is isomorphic to a subdirect product of the family {[0, n a a] | a ∈ A}. The corresponding lattice effect algebra embedding ϕ : E → {[0, n a a] | a ∈ A} is given by ϕ(x) = (x ∧ n a a) a∈A .
Let us check that E is isomorphic to {[0, n a a] | a ∈ A}. It is enough to check that ϕ is onto. Let (x a ) a∈A ∈ {[0, n a a] | a ∈ A}. Then (x a ) a∈A is an orthogonal system and x a = k a a ≤ n a a ∈ S(E) for all a ∈ A. Hence x = E {x a | a ∈ A} = E {k a a | a ∈ A} ∈ E exists. Evidently, ϕ(x) = (x ∧ n a a) a∈A = (k a a) a∈A = (x a ) a∈A . Example 3. Example of a sharply orthocomplete Archimedean M V -effect algebra that is not complete.
If we omit in Lemma 1 the assumption of atomicity in E it is enough to take the M V -effect algebra E = {f : [0, 1] → [0, 1] | f continuous function} which is a sub-lattice effect algebra of a direct product of copies of the standard M V -effect algebra of real numbers [0, 1] that is Archimedean, sharply orthocomplete (the center C(E) = S(E) = {0, 1} is trivial) and hence sharply dominating. Moreover, E is not complete.
It is well known that an Archimedean lattice effect algebra E is complete if and only if every block of E is complete (see [22,Theorem 2.7]). If moreover E is atomic then E may have atomic as well non-atomic blocks [1]. K. Mosná [16,Theorem 8] has proved that in this case Hence every non-atomic block of E is covered by atomic ones. Moreover, many properties of Archimedean atomic lattice effect algebras as well as their non-atomic blocks depend on properties of their atomic blocks.
For instance, an Archimedean atomic lattice effect algebra E is sharply dominating iff every atomic block of E is sharply dominating (see [11]). Moreover, we can prove the following: Theorem 3. Let E be an Archimedean atomic lattice effect algebra. Then the following conditions are equivalent: (i) E is complete.
(ii) Every atomic block of E is complete.
In this case every block of E is complete.
Proof . (i) =⇒ (ii): This is trivial, as every block M of E is a full sub-lattice effect algebra of E.
(ii) =⇒ (i): It is enough to show that E is orthocomplete. From [22, Theorem 2.6] we then get that E is complete.
Let G ⊆ E be a -orthogonal system. Then, for every x ∈ G, there is a set A x of atoms of E and positive integers k a , a ∈ A x such that such that x = E {k a a | a ∈ A x }. Moreover, for any F ⊆ G finite we have that {A x | x ∈ F } is an orthogonal set of atoms. Hence A G = {A x | x ∈ G} is an orthogonal set of atoms of E and there is a maximal orthogonal set A of atoms of E such that A G ⊆ A. Therefore there is an atomic block M of E with A ⊆ M . By assumption M G exists and M G = E G, as M is bifull in E because E is Archimedean and atomic (see [17]).  (ii) C(E) is a complete Boolean algebra bifull in E.
(iii) E is sharply dominating, centrally dominating and S-dominating.
(iv) If moreover E is Archimedean and atomic then E is a complete lattice effect algebra.
Proof . (i), (iii): Let S ⊆ S(E), S orthogonal. Then, for any s ∈ S, s ≤ s. Hence (since S(E) is full in E) E S = E S = S(E) S ∈ S(E) exists. Since S(E) is an Archimedean lattice effect algebra we have from [22,Theorem 2.6] that S(E) is complete. Moreover, let x ∈ E and let G = (w κ ) κ∈H , w κ ∈ S(E), κ ∈ H be a maximal orthogonal system of mutually different elements such that w x = E {w κ | κ ∈ H} ≤ x. Let us show that y ∈ S(E), y ≤ x =⇒ y ≤ w x ∈ S(E). Clearly, w x ∈ S(E). Assume that y ≤ w x . Then w x < y ∨ w x ≤ x. Hence z = (y ∨ w x ) ⊖ w x = 0 and G ∪ {z} is an orthogonal system of mutually different elements such that y ∨ w x = w x ⊕ z = E {w κ | κ ∈ H} ⊕ z ≤ x, a contradiction with the maximality of G. Therefore y ≤ w x and E is sharply dominating, hence S-dominating and from Theorem 2 we get that E is centrally dominating. From Theorem 1, we get that S(E) is bifull in E.
(iv): Assume now that E is a sharply orthocomplete Archimedean atomic lattice effect algebra. Then every atomic block M of E is sharply orthocomplete Archimedean atomic M V -effect algebra and hence it is a complete M V -effect algebra by Lemma 1. By Theorem 3, E is a complete lattice effect algebra.
Theorem 5. Let E be an atomic lattice effect algebra. Then the following conditions are equivalent: (i) E is complete.
(ii) E is Archimedean and sharply orthocomplete.