A Cantor-Bernstein type theorem for effect algebras

Algebra univers. 48 (2002) 399–411 0002-5240/02/040399 – 13 c Birkh¨
auser Verlag, Basel, 2002

Algebra Universalis

A Cantor-Bernstein type theorem for effect algebras ˇa Gejza Jenc Abstract. We prove that if E1 and E2 are σ-complete effect algebras such that E1 is a factor of E2 and E2 is a factor of E1 , then E1 and E2 are isomorphic.

1. Introduction An effect algebra (see [6], [18] and [7]) is a partial algebra (E; ⊕, 0, 1) with a binary partial operation ⊕ and two nullary operations 0, 1 satisfying the following conditions. (E1) If a ⊕ b is defined, then b ⊕ a is defined and a ⊕ b = b ⊕ a. (E2) If a ⊕ b and (a ⊕ b) ⊕ c are defined, then b ⊕ c and a ⊕ (b ⊕ c) are defined and (a ⊕ b) ⊕ c = a ⊕ (b ⊕ c). (E3) For every a ∈ E there is a unique a
∈ E such that a ⊕ a
= 1. (E4) If a ⊕ 1 exists, then a = 0. Effect algebras were introduced by Foulis and Bennett in their paper [6]. Independently, Chovanec and Kˆ opka introduced an essentially equivalent structure called D-poset (see [18]). Another equivalent structure was introduced by Giuntini and Greuling in [7]. One can construct examples of effect algebras from any partially ordered abelian group (G, ≤) in the following way: Choose any positive u ∈ G; then, for 0 ≤ a, b ≤ u, define a ⊕ b iff a + b ≤ u and then put a ⊕ b = a + b. With such a partial operation ⊕, the interval [0, u] becomes an effect algebra ([0, u], ⊕, 0, u). Effect algebras which arise from partially ordered abelian groups in this way are called interval effect algebras, see [2]. Example 1.1. Let (R, ≤) be the partially ordered additive group of real numbers, where ≤ is the usual partial order. Restrict + to the interval [0, 1] in the way indicated in the above paragraph; then ([0, 1], ⊕, 0, 1) is an effect algebra. Presented by J. Mycielski. Received May 23, 2001; accepted in final form April 1, 2002. 2000 Mathematics Subject Classification: 06C15, 03G12, 81P10. Key words and phrases: Cantor-Bernstein theorem, effect algebra, weak congruence. ˇ SR, Slovakia. This research is supported by grant G-1/7625/20 of MS 399

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Another prominent example of an interval effect algebra is the standard effect algebra consisting of all bounded self-adjoint operators on a Hilbert space between 0 and the identity operator. Example 1.2. Let E be the four-element set {0, a, b, 1}. Define a⊕a = b⊕b = 0⊕1 and ∀x ∈ E : 0 ⊕ x = x ⊕ 0 = x; in all other cases the partial sum is undefined. Then (E, ⊕, 0, 1) is an effect algebra. In an effect algebra E, we write a ≤ b iff there is c ∈ E such that a ⊕ c = b. Since every effect algebra is cancellative, ≤ is a partial order on E. In this partial order, 0 is the least and 1 is the greatest element of E. Moreover, it is possible to introduce a new partial operation ; b  a is defined iff a ≤ b and then a ⊕ (b  a) = b. It can be proved that a ⊕ b is defined iff a ≤ b
iff b ≤ a
. Therefore, it is usual to denote the domain of ⊕ by ⊥. If E is an effect algebra such that (E, ≤) is a lattice, we say that E is lattice ordered. An element a of an effect algebra is called an atom iff a is minimal with respect to the property 0 < a. Example 1.3. The smallest example of a non-lattice ordered effect algebra is a six-element effect algebra with two atoms a, b satisfying a ⊕ b ⊕ b = a ⊕ a ⊕ a = 1. Note that this equality defines E up to isomorphism. Among lattice ordered effect algebras, there are two important subclasses, which arise from quantum and fuzzy logic, respectively: orthomodular lattices and MValgebras. Example 1.4. Let (L; ∧, ∨,
, 0, 1) be an orthomodular lattice. Write a ⊕ b = a ∨ b iff a ≤ b
, otherwise let a ⊕ b be undefined. Then (L, ⊕, 0, 1) is an effect algebra. Effect algebras which are associated with orthomodular lattices in this way can be characterized as lattice ordered effect algebras satisfying the implication a ⊥ b =⇒ a ∧ b = 0. Example 1.5. An MV-algebra (cf. [4], [19]), (M ; ⊕, ¬, 0), is a commutative semigroup satisfying the identities x ⊕ 0 = x, ¬¬x = x, x ⊕ ¬0 = ¬0 and x ⊕ ¬(x ⊕ ¬y) = y ⊕ ¬(y ⊕ ¬x). There is a natural partial order in an MV-algebra, given by y ≤ x iff x = x ⊕ ¬(x ⊕ ¬y). For every MV-algebra (M ; ⊕, ¬, 0) can be considered as a effect algebra (M ; ⊕, 0, ¬0), when we restrict the operation ⊕ to the domain ⊥= {(x, y) : x ≤ ¬y}. Effect algebras which are associated with MV-algebras can be characterized as lattice ordered effect algebras satisfying the implication a ∧ b = 0 =⇒ a ⊥ b. (Cf. [1])

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As proved by Rieˇcanov´a in [21], every lattice ordered effect algebra is a union of maximal sub-effect algebras which are MV-algebras. These are called blocks. For example, the lattice-ordered effect algebra in Example 1.2 contains two blocks; each of them is a three-element MV-algebra. In the case of of orthomodular lattices, the blocks are maximal Boolean subalgebras. This result shows that the lattice ordered effect algebras are a very natural generalization of orthomodular lattices. In [15], some of the Rieˇcanov´a’s results were generalized for a particular non-lattice ordered class of effect algebras, called homogeneous algebras. There is another natural connection with the class of orthomodular lattices. Call an element of an effect algebra sharp iff a ∧ a
= 0. The set of all sharp elements in a lattice-ordered effect algebra such that a ∧ a
= 0 is an orthomodular lattice. This was proved in [16]. A finite family of elements A = (a1 , . . . , an ) of an effect algebra is called orthogonal iff ⊕A = a1 ⊕ · · · ⊕ an is defined. An infinite family A = (ak )k∈M is called orthogonal iff all finite subfamilies of A are orthogonal. An orthogonal family A = (ak )k∈M is called summable iff
 A = {ai1 ⊕ · · · ⊕ ain : {i1 , . . . , in } ⊆ M } exists. An effect algebra E is called σ-complete iff every countable orthogonal family of elements of E is summable. Of course, every finite effect algebra is σ-complete. It is easy to see that 1.1 is a σ-complete effect algebra. The interval effect algebra of polynomial functions [0, 1] → [0, 1] is not σ-complete. Later we shall prove that a lattice ordered effect algebra E is σ-complete iff (E, ≤) is a σ-complete lattice. Let E1 , E2 be effect algebras. Define the ⊥ and ⊕ on E1 × E2 as follows: for a1 , b1 ∈ E1 and a2 , b2 ∈ E2 we have (a1 , a2 ) ⊥ (b1 , b2 ) ⇔ a1 ⊥ b1 and a2 ⊥ b2 , and then (a1 , a2 )⊕(b1 , b2 ) = (a1 ⊕b1 , a2 ⊕b2 ). Then we say that (E1 ×E2 , ⊕, (0, 0), (1, 1)) is the direct product of E1 , E2 . It is easy to see that the direct product of effect algebras is an effect algebra. Let us now state the main result of the present paper. Theorem 1.6. Let E1 , E2 be σ-complete effect algebras. Suppose that there exist effect algebras F, G such that E1 ∼ = E2 × F E2 ∼ = E1 × G. Then E1 ∼ = E2 . The assumption that E1 and E2 are σ-complete cannot be dropped, since (as proved by Hanf in [12]) there is a Boolean algebra A such that A × A × A ∼ = A, but A×A ∼  A. Then we may put E1 = A and E2 = A × A to obtain a counterexample. =

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Recently, two more special versions of the main Theorem of [13] appeared. In [23], it was proved that A1 and A2 can be σ-complete orthomodular lattices. In [22], Theorem 4.1, it was proved that A1 and A2 can be σ-complete M V -algebras. On the International Conference on Fuzzy Set Theory an its Application (Liptovk´ y J´ an 2000), A. De Simone, D. Mundici, P. Pt´ ak and M. Navara raised the question, whether Theorem 1.6 holds in a subclass of effect algebras. Section 2 contains basic definitions and relationships concerning effect algebras and some more general structures. We characterize the class of partial abelian monoids which are partially ordered in their algebraic preorder. In section 3, we focus on the class of σ-complete effect algebras and weak congruences on them. We prove that, for a σ-complete effect algebra E and a countably additive weak congruence ∼ satisfying a certain condition, the algebraic preorder of the quotient partial abelian monoid E/∼ is a partial order. In section 4, we prove that the center (see [9]) of a σ-complete effect algebra is a σ-complete Boolean algebra. We apply a result from section 2 to prove that the center of a σ-complete effect algebra modulo the isomorphism of central ideals forms a partial abelian monoid which is partially ordered. Theorem 1.6 is then a simple consequence of the latter result. 2. Effect algebras and other partial abelian monoids An effect algebra need not be lattice ordered. However, as proved in [9], the following relationship between ∧, ∨ and ⊕ holds: if a ∨ b exists and a ⊥ b, then a ∧ b exists and a ⊕ b = (a ∧ b) ⊕ (a ∨ b). Moreover, it is easy to check that, for every subset B of an effect algebra such  that B exists and for every x ≥ B,   x  ( B) = {x  b : b ∈ B}. Let E1 , E2 be effect algebras. A map φ : E1 → E2 is called a homomorphism iff it satisfies the following condition. (H1) φ(1) = 1 and if a ⊥ b, then φ(a) ⊥ φ(b) and φ(a ⊕ b) = φ(a) ⊕ φ(b). A homomorphism φ : E1 → E2 of effect algebras is called full iff the following condition is satisfied. (H2) If φ(a) ⊥ φ(b), then there exist a1 , b1 ∈ E1 such that a1 ⊥ b1 , φ(a) = φ(a1 ) and φ(b) = φ(b1 ). A bijective, full homomorphism is called an isomorphism. Let E1 be an effect algebra. A subset E2 ⊆ E1 is a subeffect algebra of E1 iff 0, 1 ∈ E2 , E2 is closed under the
operation, and a, b ∈ E2 with a ⊥ b =⇒ a ⊕ b ∈ E2 .

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Another possibility of creating a substructure of an effect algebra E is to restrict ⊕ to an interval [0, a] = {x ∈ E : 0 ≤ x ≤ a}, where a ∈ E, so that ([0, a], ⊕) becomes a relative subalgebra (cf. e.g. [8]) of the partial algebra (E, ⊕). We can then consider [0, a] as an effect algebra, letting a act as the unit element. In what follows, we denote such effect algebras by [0, a]E . Let E be an effect algebra. A subset I of E is called an ideal of E iff the following condition is satisfied. x, y ∈ I and x ⊥ y ⇔ x ⊕ y ∈ I A partial abelian monoid is a partial algebra (P, ⊕, 0) satisfying conditions (E1) and (E2), with a neutral element 0. A partial abelian monoid P is said to be positive iff P satisfies the following condition. If a ⊕ b = 0, then a = 0 It is possible to define ≤ for a general partial abelian monoid exactly the same way as for effect algebras. However, the relation ≤ need not be a partial order. Proposition 1. Let P be a partial abelian monoid. Then ≤ is a partial order on P iff for all a, b, c ∈ P the following condition is satisfied. a = a ⊕ b ⊕ c =⇒ a = a ⊕ b Proof. For every partial abelian monoid, ≤ is a preorder, i.e., a reflexive and transitive relation. Thus, ≤ is a partial order iff ≤ is antisymmetric. Assume that ≤ is a partial order and let a = a ⊕ b ⊕ c. Since a ≤ a ⊕ b and a ⊕ b ≤ a, a = a ⊕ b. For the ‘if’ part, let a ≤ d and d ≤ a. There are b, c ∈ P such that a ⊕ b = d and d ⊕ c = a. Thus, a ⊕ b ⊕ c = a and, by assumption, this implies that a ⊕ b = a. Therefore, d = a and we see that ≤ is a partial order on P .
Corollary 2.1. Let P be a partial abelian monoid such that ≤ is a partial order on P . Then P is positive. Proof. Let x, y ∈ P , x ⊕ y = 0. Then x ⊕ y ⊕ x = x. By Proposition 1, this implies that x ⊕ y = x. Thus, by assumption, x = 0.
Obviously, every cancellative and positive partial abelian monoid satisfies the conditions of Proposition 1. 3. Weak congruences and σ-complete effect algebras Let E be an effect algebra. A relation ∼ on E is a weak congruence iff the following conditions are satisfied.

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(C1) ∼ is an equivalence relation. (C2) If a1 ∼ a2 , b1 ∼ b2 , a1 ⊥ b1 , a2 ⊥ b2 , then a1 ⊕ b1 ∼ a2 ⊕ b2 . If E is an effect algebra and ∼ is a weak congruence on E, the quotient E/∼ (⊕ is defined on E/∼ in an obvious way) need not to be a partial abelian monoid, since the associativity condition may fail (cf. [11]). This fact motivates the study of sufficient conditions for a weak congruence to preserve associativity. The following condition was considered in [5]. (C5) If b ⊥ c and a ∼ b ⊕ c, then there are b1 , c1 such that b1 ∼ b, c1 ∼ c, b1 ⊥ c1 and a = b1 ⊕ c1 . In [5], it was proved that for a partial abelian monoid P and a weak congruence ∼, satisfying (C5), the quotient P/∼ is again a partial abelian monoid. Moreover, it is easy to prove that the eventual positivity of P is preserved for such ∼. However, for an effect algebra E, the (C5) property of ∼ does not guarantee that the
operation is preserved by ∼ or, equivalently, that E/∼ is an effect algebra. We refer the interested reader to [20] and [11] for further details concerning congruences on effect algebras and partial abelian monoids. Example 3.1. Let 2N be the effect algebra associated with the Boolean algebra of all subsets of N. If a, b are two subsets of N, put a ∼ b iff a and b are of the same cardinality. Then ∼ is a weak congruence satisfying (C5) and 2N /∼ is isomorphic to the partial abelian monoid (N ∪ {∞}; +, 0). Example 3.2. Let A be an involutive ring with unit, in which x∗ x + y ∗ y = 0 implies x = y = 0. Let P (A) be the set of all projections in A. For e, f ∈ P (A), write e ⊕ f = e + f iff ef = 0, otherwise let e ⊕ f be undefined. Then (P (A); ⊕, 0, 1) is an effect algebra. For e, f in P (A), write e ∼ f iff there is w ∈ A such that e = w∗ w and f = ww∗ . Then ∼ is a weak congruence on P (A) and ∼ satisfies (C5). Proposition 2. Let E be an effect algebra. Let ∼ be a weak congruence satisfying (C5). Then ≤ is a partial order on E/∼ iff for all a, b, c ∈ E the following condition is satisfied. a ∼ a ⊕ b ⊕ c =⇒ a ∼ a ⊕ b (1) Proof. By Proposition 1, ≤ is a partial order on E/∼ iff [a]∼ = [a]∼ ⊕ [b]∼ ⊕ [c]∼ =⇒ [a]∼ = [a]∼ ⊕ [b]∼

(2)

is satisfied. Thus, it suffices to prove that (1) and (2) are equivalent. Assume that (2) is satisfied. Let a, b, c ∈ P be such that a ∼ a⊕ b ⊕ c. Obviously, [a]∼ = [a]∼ ⊕ [b]∼ ⊕ [c]∼ . By (2), this implies [a]∼ = [a]∼ ⊕ [b]∼ . By assumption, a ⊥ b so we can write a ∼ a ⊕ b.

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On the other hand, assume that (1) is satisfied. Let a, b, c ∈ P be such that [a]∼ = [a]∼ ⊕ [b]∼ ⊕ [c]∼ . Since [a]∼ ⊥ [b]∼ , there are a1 ∼ a and b1 ∼ b such that a1 ⊥ b1 . Since [a1 ⊕ b1 ]∼ ⊥ [c]∼ , there are c1 ∼ c and d ∼ a1 ⊕ b1 , such that c1 ⊥ d. Since ∼ satisfies (C5), d ∼ a1 ⊕ b1 implies that there are a2 , b2 ∈ P such that a2 ∼ a1 , b2 ∼ b1 and d = a2 ⊕ b2 . Thus, a2 ⊕ b2 ⊕ c1 ∼ a2 . By (1), this implies that a2 ⊕ b2 ∼ a2 . Since [a]∼ = [a2 ]∼ and [b]∼ = [b2 ]∼ , we see that
[a]∼ = [a]∼ ⊕ [b]∼ . Let E be a σ-complete effect algebra. Let (ak )k∈N be an orthogonal family. It is easy to check that, 
ak = a1 ⊕ · · · ⊕ ak . k∈N

k∈N

The following lemma gives a useful characterization of σ-complete effect algebras. Although it was proved in [10], we include the proof here because we need to refer to it in what follows. Lemma 3.3. [10] Let E be an effect algebra. The following are equivalent: (a) E is σ-complete. (b) For each non-increasing sequence (ak )k∈N , ∧(ak )k∈N exists. (c) For each non-decreasing sequence (ak )k∈N , ∨(ak )k∈N exists. Proof. (c)⇒(a): Let (bk )k∈N be an orthogonal family. For every k ∈ N, put ak = b1 ⊕ · · · ⊕ bk . Evidently, ak is a non-decreasing sequence, so ∨(ak )k∈N exists and  equals k∈N bk . (a)⇒(b): Let (ak )k∈N be a non-increasing sequence. For every k ∈ N, put bk = ak  ak+1 ; (bk )k∈N is the difference sequence of (ak )k∈N . Evidently, (bk )k∈N is   an orthogonal sequence. Denote b = k∈N bk . We claim that a1 b = ai . Indeed, for all k ∈ N we have ak = a1  (b1 ⊕ · · · ⊕ bk−1 ) ≥ a1  b. Thus, a1  b is a lower bound of (ak )k∈N . On the other hand, let c be a lower bound of (ak )k∈N . Then, for all k ∈ N, c ≤ ak = a1  (b1 ⊕ · · · ⊕ bk−1 ). This implies that a1  c ≥ b1 ⊕ · · · ⊕ bk−1 . Therefore, a1  c ≥ b which is equivalent to c ≤ a1  b.
(b)⇒(c): This is a consequence of the equivalence a ≤ b iff b
≤ a
. Recall, that a lattice is called σ-complete iff every countable set of elements has a supremum. Corollary 3.4. Let (E; ⊕, 0, 1) be a lattice-ordered effect algebra. E is σ-complete as an effect algebra iff E is σ-complete as a lattice. Proof. It is obvious that the σ-completeness of E as a lattice is equivalent to the condition (b) of Lemma 3.3.


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Thus, the class of σ-complete effect algebras includes effect algebras arising from σ-complete MV-algebras and effect algebras arising from σ-complete orthomodular lattices. Let E be a σ-complete effect algebra, let ∼ be a weak congruence on E. Then ∼ is said to be countably additive iff the following condition is satisfied: for all orthogonal families (ak )k∈N ⊆ E and (bk )k∈N ⊆ E such that, for all k ∈ N, ak ∼ bk   we have k∈N ak ∼ k∈N bk . Proposition 3. Let E be a σ-complete effect algebra. Let ∼ be a countably additive weak congruence satisfying (C5). Then ≤ is a partial order on E/∼. Proof. Let a0 , b0 , c0 ∈ P be such that a0 ∼ a0 ⊕ b0 ⊕ c0 . By Proposition 2, it suffices to prove that a0 ∼ a0 ⊕ b0 . When we apply the (C5) property of ∼ twice, we immediately obtain that a0 ∼ a0 ⊕ b0 ⊕ c0 implies that there exist a1 , b1 , c1 ∈ P such that a1 ∼ a0 , b1 ∼ b0 , c1 ∼ c and a0 = a1 ⊕ b1 ⊕ c1 . Similarly, since a1 ∼ a1 ⊕ b1 ⊕ c1 , there exist a2 , b2 , c2 ∈ P such that a2 ∼ a1 , b2 ∼ b1 and c2 ∼ c1 . This way, we can construct sequences (ak )k∈N , (bk )k∈N such that a0 ⊕ b 0 ≥ a0 ≥ a1 ⊕ b 1 ≥ a1 ≥ a2 ⊕ b 2 ≥ · · ·

(3)

Moreover, for all k ∈ N, ak ∼ a0 and bk ∼ b0 . Since E is a σ-complete effect algebra, the non-decreasing sequence (3) possesses an infimum, say f . (See Lemma 3.3.) Consider the following sequences (dk )k∈N , (ek )k∈N : d0 = (a0 ⊕ b0 )  a0 ∼ b0

e0 = d2

d1 = a0  (a1 ⊕ b1 ) ∼ c0

e1 = d1

d2 = (a1 ⊕ b1 )  a1 ∼ b0

e2 = d4

d3 = a1  (a2 ⊕ b2 ) ∼ c0

e3 = d3

.. .

.. .

Note that (dk )k∈N is the difference sequence of (3). Similarly, (ek )k∈N arises from the difference sequence of the sequence a0 ≥ a1 ⊕ b 1 ≥ a1 ≥ a2 ⊕ b 2 ≥ · · · by swapping the elements with indices 2k and 2k + 1, for k ∈ N. Consider the proof of Lemma 3.3, part (a)⇒(b). We have
dk ⊕ f a0 ⊕ b 0 = k∈N

a0 =


k∈N

ek ⊕ f

(4)

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Since for all k ∈ N ek ∼ dk and ∼ is countably additive, this implies that a0 ⊕ b0 ∼
a0 . Without going into details, it is worth mentioning that, in view of Example 3.2, Proposition 3 implies that for every Rickart ∗-ring in which every countable family of projections has a supremum, the set of equivalence classes of projections under the relation ∼ is partially ordered. This was proved in [3] under the name Schr¨ oder-Bernstein theorem. (See also [17].) 4. Central elements; main result Let E be an effect algebra. Suppose that there is an isomorphism φ : E → E1 × E2 . For every such φ, the elements φ−1 (1, 0) and φ−1 (0, 1) are called central elements of E. We write C(E) for the set of all central elements of an effect algebra E. It was proved in [9] that the set of all central elements forms a subeffect algebra of E, which is a Boolean algebra. Moreover, the joins and meets of elements of C(E) exist in E and coincide with their joins and meets in C(E). If a, b ∈ C(E) are orthogonal, we have a ∨ b = a ⊕ b and a ∧ b = 0. For all a ∈ C(E), the interval [0, a] is a ⊕-subalgebra and hence an ideal of E. These ideals are called central ideals. By [5], a central ideal in an effect algebra E can be characterized as an ideal I satisfying the following conditions. • I = [0, a] for some a ∈ E. • I is a Riesz ideal, i.e., if i ∈ I and i ≤ a ⊕ b, then there exist i1 , i2 ∈ I, such that i1 ≤ a, i2 ≤ b, i ≤ i1 ⊕ i2 . For every central element a, the map x → a ∧ x is a full homomorphism, which maps E onto [0, a]E (cf. [14]). Lemma 4.1. Let E be a σ-complete effect algebra. Let (ak )k∈N ⊆ E be a countable orthogonal family of central elements. Let (xk )k∈N be a family of elements satisfying   xk ≤ ak , for all k ∈ N. Then k∈N xk exists and equals k∈N xk . Proof. Let M be a finite nonempty subset of N. Obviously, (xk )k∈M is an orthog onal family, so k∈M xk exists. Observe that  [0, ak ]E ) × [0, (a1 ⊕ · · · ⊕ ak )
]E . E∼ =( Therefore,

 k∈M

xk =



⊕(xk )k∈N =

k∈M

xk . This implies that   x1 ⊕ · · · ⊕ xk = x1 ∨ · · · ∨ xk = xk .

k∈M



k∈N

k∈N

k∈N




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Proposition 4. Let E be a σ-complete effect algebra. Let (an )n∈N ⊆ C(E) be a  countable orthogonal family. Denote a = n∈N an . Then  [0, an ]E . [0, a]E ∼ = n∈N

Moreover, a is central.  Proof. Let φ : [0, a]E → n∈N [0, an ]E be a map given by φ(x) = (x ∧ an )n∈N . We will prove that φ is an isomorphism.  To prove that φ is onto, let (xk )k∈N ∈ n∈N [0, an ]E . Observe that (xk ) is an  orthogonal family and put x = n∈N xn . We will prove that φ(x) = (xn )n∈N . We see that
φ(x) = (x ∧ an )n∈N = (( xk ) ∧ an )n∈N . k∈N

Fix any n ∈ N. By associativity of ⊕,

x ∧ an = ( xk ) ∧ an = (xn ⊕ ( k∈N

xk )) ∧ an .

k∈N\{n}

Since the orthogonal family (xk )k∈N\{n} satisfies the conditions of Lemma 4.1,
 xk )) ∧ an = (xn ⊕ ( xk )) ∧ an (xn ⊕ ( k∈N\{n}

k∈N\{n}

Since an is a central element,  (xn ⊕ ( xk )) ∧ an = (xn ∧ an ) ⊕ (( k∈N\{n}

= xn ⊕ ((



xk ) ∧ an )

k∈N\{n}



xk ) ∧ an )

k∈N\{n}

Since, for all k ∈ N \ {n}, xk ∧ an = 0, we see that  an ∧ ( xk ) = 0. k∈N\{n}

Thus, for all n ∈ N, (x ∧ an ) = xn . Therefore, φ is onto. To see that φ is one-to-one it suffices to prove that, for all x ∈ [0, a],
x= x ∧ an . n∈N

Since


n∈N

x ∧ an =



(x ∧ a1 ) ⊕ · · · ⊕ (x ∧ an ) =

n∈N

 n∈N

x ∧ (a1 ⊕ · · · ⊕ an ),

(5)

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 we see that n∈N x ∧ an ≤ x. Moreover,
 x( x ∧ an ) = x  ( (x ∧ a1 ) ⊕ · · · ⊕ (x ∧ an )) n∈N

n∈N

=x(



x ∧ (a1 ⊕ · · · ⊕ an )) =

n∈N



x  (x ∧ (a1 ⊕ · · · ⊕ an )).

n∈N

Since a1 ⊕ · · · ⊕ an is a central element, x  (x ∧ (a1 ⊕ · · · ⊕ an )) = x ∧ a1
∧ · · · ∧ an
. Therefore,



x  (x ∧ (a1 ⊕ · · · ⊕ an )) =

n∈N



x ∧ a1
∧ · · · ∧ an


n∈N

≤



a ∧ a1
∧ · · · ∧ an


n∈N

=



n∈N

=a

a  (a ∧ (a1 ⊕ · · · ⊕ an )) 

a1 ⊕ · · · ⊕ an = 0.

n∈N

 Hence, for all x ∈ [0, a], x = n∈N x ∧ an and this implies that φ is a one-to-one mapping. Let x, y ∈ [0, a]E , x ⊥ y. Obviously, φ(x) ⊥ φ(y), and it is easy to check that φ(x ⊕ y) = φ(x) ⊕ φ(y). Moreover, φ(a) = (an )n∈N , so φ preserves the unit element of the effect algebra [0, a]E . Thus, φ is a homomorphism of effect algebras. Assume φ(x) ⊥ φ(y). This implies that, for all n ∈ N, x ∧ an ⊥ y ∧ an and, consequently, (6) (x ∧ an )n∈N ∪ (y ∧ an )n∈N   is an orthogonal family. By (5), x = n∈N x ∧ an and y = n∈N y ∧ an . Thus, orthogonality of (6) implies that x ⊥ y and φ is a full homomorphism. Let us prove that [0, a] is an ideal in E. Obviously, x ⊕ y ∈ [0, a] =⇒ x ⊥ y and x, y ∈ [0, a]. For the proof of the opposite implication, let x, y ∈ [0, a], x ⊥ y.   By (5), x = n∈N x ∧ an and y = n∈N y ∧ an . Consider the family G = ((x ∧ an ) ⊕ (y ∧ an ))n∈N .  It is obvious that G is an orthogonal family and that G = x ⊕ y. Since each of the an ’s is a central element, (x ∧ an ) ⊕ (y ∧ an ) = (x ⊕ y) ∧ an. Thus, we can apply    Lemma 4.1 to prove that G = G. Since G ⊆ [0, a], G ∈ [0, a]. Therefore,  x ⊕ y = G ∈ [0, a] and we see that [0, a] is an ideal. By [5], an ideal I in an effect algebra is central iff I is an interval and I is a Riesz ideal. Hence, it remains to prove that [0, a] is a Riesz ideal. Let i ≤ x ⊕ y, where

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i ∈ [0, a]. For all n ∈ N, i ∧ an ≤ (x ∧ an ) ⊕ (y ∧ an ).   Since i ∈ [0, a], i = n∈N i ∧ an . Put i1 = n∈N x ∧ an and i2 = n∈N y ∧ an .
Obviously, i ≤ i1 ⊕ i2 where i1 , i2 ∈ [0, a], i1 ≤ x and i2 ≤ y. 

Let E be an effect algebra. For a, b ∈ C(E), we write a ≈ b iff [0, a]E is isomorphic to [0, b]E . Proposition 5. Let E be a σ-complete effect algebra. Then C(E) is a σ-complete Boolean algebra and ≈ is a countably additive weak congruence satisfying (C5). Proof. By Proposition 4, C(E) is σ-complete and ≈ is a countably additive weak congruence. It is easy to check that ≈ satisfies (C5). Indeed, assume a, b, c ∈ C(E) with a ⊕ b ≈ c. Let φ be an isomorphism φ : [0, a ⊕ b]E → [0, c]E . Obviously, a and b are central elements of [0, a ⊕ b]E . Therefore, φ(a) and φ(b) are central in [0, c]. Since c is a central element, φ(a) and φ(b) are central elements. It remains to observe that φ(a) ⊕ φ(b) = c and that φ(a) ≈ a, φ(b) ≈ b.
Finally, let us prove Theorem 1.6. Proof. Since E1 ∼ = E1 × G × F , there exist elements a1 , a2 ∈ C(E1 ), such that [0, a1 ]E1 ∼ = E1 , [0, a2 ]E1 ∼ = E2 , a1 ≤ a2 . Proposition 5 implies that the assumptions of Proposition 3 are satisfied. Hence, the algebraic preorder ≤ on the partial abelian monoid C(E1 )/≈ is a partial order. It is evident that [1]≈ is the greatest element in C(E1 )/≈. Therefore, [a1 ]≈ ≤ [a2 ]≈ and [a1 ]≈ = [1]≈ imply [a2 ]≈ = [1]≈ , i.e.,
[0, a2 ]E1 ∼ = [0, 1]E1 . Thus, E1 ∼ = E2 . References [1] M. K. Bennett and D. J. Foulis, Phi-symmetric effect algebras, Foundations of Physics, 25 (1995), 1699–1722. [2] M. K. Bennett and D. J. Foulis, Interval and scale effect algebras, Advances in Applied Mathematics, 19 (1997), 200–215. [3] S. K. Berberian, Baer ∗-rings, Springer-Verlag, 1972. [4] C. C. Chang, Algebraic analysis of many-valued logics, Trans. Amer. Math. Soc., 89 (1959), 74–80. [5] G. Chevalier and S. Pulmannov´ a, Some ideal lattices in partial abelian monoids, preprint, Mathematical Institute of Slovak Academy of Sciences, Bratislava, 1998. [6] D. J. Foulis and M. K. Bennett, Effect algebras and unsharp quantum logics, Found. Phys., 24 (1994), 1331–1352. [7] R. Giuntini and H. Greuling, Toward a formal language for unsharp properties, Found. Phys., 19 (1994), 769–780. [8] G. Gr¨ atzer, Universal Algebra, Springer-Verlag, second edition, 1979. [9] R. Greechie, D. Foulis, and S. Pulmannov´ a, The center of an effect algebra, Order, 12 (1995), 91–106.

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