Pseudo-D-Lattices and Topologies Generated by Measures

italian journal of pure and applied mathematics – n. 29−2012 (25−42) 25 PSEUDO-D-LATTICES AND TOPOLOGIES GENERATED BY MEASURES Anna Avallone Paolo Vitolo Dipartimento di Matematica, Informatica ed Economia Universitá della Basilicata Viale dell’Ateneo Lucano 85100 Potenza Italy e-mail: [email protected] [email protected] Abstract. We prove that every modular measure on a pseudo-D-lattice L generates on L a lattice uniformity which makes uniformly continuous the pseudo-D-lattice operations. As an application, we obtain a uniqueness theorem for modular measures on pseudo D-lattices. Keywords: modular measures, pseudo-effect algebras, uniqueness theorem. AMS Classification: 28B05, 06C15. 1. Introduction Effect algebras (alias D-posets) have been independently introduced in 1994 by D.J. Foulis and M.K. Bennett in [9] and by F. Chovanek and F. Kopka in [11] for modelling unsharp measurement in a quantum mechanical system. They are a generalization of many structures which arise in Quantum Physics (see [14]) and in Mathematical Economics (see [18], [22], [10]), in particular of orthomodular posets and MV-algebras. After 1994, a great number of papers concerning effect algebras have been published. In 2001, G. Georgescu and A. Jogulescu in [20] introduced the concept of a pseudo-MV-algebra, which is a non-commutative generalization of an MV-algebra, and A. Dvurecenskij and T. Vetterlein in [15] introduced the more general structure of a pseudo-effect algebra, which is a non-commutative generalization of effect algebra. The study of these structures is motivated by the non-commutative nature of certain psychological processes and quantum mechanical experiments (see [13]) and there even exists a programming language based on a non-commutative 26 anna avallone, paolo vitolo logic (see [7]). For a study, see for example [15], [16], [12], [13], [23], [26] and many others. In the study of modular measures on lattice ordered effect algebras, essential tools are topological methods based on the theory of uniform lattices introduced by H. Weber in [24] (see for example [6], [4], [2], [5], [3]). In particular, a starting point for these topological methods was the result that the lattice uniformity generated by a modular measure on a lattice-ordered effect algebra E makes uniformly continuous the effect algebra operations of E (see [4]). The aim of this paper is to set up the basis for topological methods in the more general study of modular measures on pseudo-D-lattices (i.e. lattice-ordered pseudo-effect algebras), for future development of a measure theory in pseudo-Dlattices. Thus, in the first part of this paper, we prove that every modular measure on a pseudo-D-lattice L generates on L a D-uniformity, i.e. a lattice uniformity which makes uniformly continuous the pseudo-effect algebra operations, and we study D-uniformities on L. In the second part, we first prove a uniqueness theorem for measures on pseudo-effect algebras which extends previous results of [21] for a particular case of effect algebra and of [4] in arbitrary effect algebras; then we give a first example of application of the results of the first part, proving that, for modular measures on pseudo-D-lattices, the above uniqueness theorem holds without completeness assumptions on L. 1. Preliminaries A partial algebra (E, +, 0, 1), where + is a partial binary operation and 0, 1 are constants, is called a pseudo-effect algebra if, for all a, b, c ∈ E, the following properties hold: (1) a + b and (a + b) + c exist if and only if b + c and a + (b + c) exist and in this case (a + b) + c = a + (b + c). (2) For any a ∈ E, there exist exactly one d ∈ E and exactly one e ∈ E such that a + d = e + a = 1. (3) If a + b exists, there are d, e ∈ E such that a + b = d + a = b + e. (4) If 1 + a or a + 1 exists, then a = 0. We note that, if + is commutative, then E becomes an effect algebra. If we define a ≤ b if and only if there exists c ∈ E such that a + c = b, then ≤ is a partial ordering on E such that 0 ≤ a ≤ 1 for any a ∈ E. If E is a lattice with respect to this order, then we say that E is a lattice pseudo-effect algebra or a pseudo-D-lattice. If E is a pseudo-effect algebra, we can define two partial binary operations on E such that, for a, b ∈ E, a/b is defined if and only if b\a is defined if and only pseudo-D-lattices and topologies generated by measures 27 if a ≤ b and in this case we have (b\a) + a = a + (a/b) = b. In particular, we set ⊥ a = 1\a and a⊥ = a/1. In the sequel E is a pseudo-effect algebra, L is a pseudo-D-lattice and (G, +) is a topological Abelian group. For basic properties of pseudo-effect algebras we refer to [15], [13] and [26]. In particular, we need the following (see 1.4 and 1.6 of [15], 2.7, 2.9, 2.10 and 2.11 of [26] and [13], pp. 32 and 33). Proposition 1.1. Let a, b, c ∈ E. Then: (1) a + 0 = 0 + a = a. (2) ⊥ (a⊥ ) = (⊥ a)⊥ = a. (3) a + b = c if and only if a = ⊥ (b + c⊥ ) if and only if b = (⊥ c + a)⊥ . (4) a + b = a + c implies b = c; b + a = c + a implies b = c. (5) a + b exists if and only if a ≤ ⊥ b if and only if b ≤ a⊥ . (6) a ≤ b if and only if ⊥ b ≤ ⊥ a if and only if b⊥ ≤ a⊥ . (7) If b + c exists, then a ≤ b if and only if (a + c exists and) a + c ≤ b + c. (8) If c + b exists, then a ≤ b if and only if (c + a exists and) c + a ≤ c + b. (9) If a ≤ b ≤ c, then c\b ≤ c\a and b/c ≤ a/c. (10) If a ≤ b ≤ c, then b\a ≤ c\a and a/b ≤ a/c. (11) If a ≤ b ≤ c, then (c\b)/(c\a) = b\a and (a/c)\(b/c) = a/b. (12) If a ≤ b ≤ c, then (c\a)\(b\a) = c\b and (a/b)/(a/c) = b/c. Proposition 1.2. Let a, b, c ∈ L. Then: (1) If a ≤ c and b ≤ c, then c\(a∧b) = (c\a)∨(c\b) and (a∧b)/c = (a/c)∨(b/c). (2) If a ≤ c and b ≤ c, then c\(a∨b) = (c\a)∧(c\b) and (a∨b)/c = (a/c)∧(b/c). (3) If c ≤ a and c ≤ b, then (a∧b)\c = (a\c)∧(b\c) and c/(a∧b) = (c/a)∧(b/c). (4) If c ≤ a and c ≤ b, then (a∨b)\c = (a\c)∨(b\c) and c/(a∨b) = (c/a)∨(c/b). (5) ((a ∨ b)\a) ∧ ((a ∨ b)\b) = 0 and (a/(a ∨ b)) ∧ (b/(a ∨ b)) = 0. In the sequel, we set ∆ = {(a, b) ∈ E × E : a = b}. If a ≤ b, we set [a, b] = {c ∈ E : a ≤ c ≤ b}. If (an )n∈N is a sequence in E and a ∈ E, we write an ↑ a (respectively, an ↓ a) if (an )n∈N is increasing and a = supn an (respectively, (an )n∈N is decreasing and a = inf n an ). If a1 , . . . , an ∈ E, we inductively define a1 + · · · + an = (a1 + · · · + an−1 ) + an , provided that the right hand side exists. We say that the finite sequence (a1 , ..., an ) of E is orthogonal if a1 + · · · + an exists. Given an infinite sequence (an )n∈N , we say that it is orthogonal if, for every positive P integer n, a1 + · · · + an exists. If, moreover, supn∈N (a1 + · · · + an ) exists, we set ∞ n=1 an = supn∈N (a1 + · · · + an ). We say that E is Archimedean if, for every a ∈ E with a 6= 0, there exists an integer k > 0 such that ka exists and (k +1)a does not exist, where ka = a+· · ·+a k times. 28 anna avallone, paolo vitolo P We say that E is σ-complete if, for every orthogonal sequence (an ), ∞ n=1 an exists. If E is a pseudo-D-lattice, we set a ∗△ b = (a ∨ b)\(a ∧ b) and a △∗ b = (a ∧ b)/(a ∨ b). A function µ : E → G is said to be a measure if, for every a, b ∈ E with a ≤ b, µ(b) − µ(a) = µ(b\a) = µ(a/b). It is easy to see that µ is a measure if and only if, for every a, b ∈ E such that a + b exists, µ(a + b) = µ(a) + µ(b). We say µ Pthat ∞ is σ-additive if,Pfor every orthogonal sequence (an ) in E such that a = n=1 an exists, µ(a) = ∞ n=1 µ(an ). If µ : L → G, we say that µ is modular if, for every a, b ∈ L, µ(a ∨ b) + µ(a ∧ b) = µ(a) + µ(b). A uniformity U on L is said to be a lattice uniformity if the lattice operations ∨ and ∧ are uniformly continuous with respect to U. For a study, see [24]. As proved in [19], if L1 is a lattice, every modular function µ : L1 → G generates on L1 a lattice uniformity U(µ), called µ-uniformity, which is the weakest lattice uniformity which makes µ uniformly continuous and a basis of U(µ) is the family consisting of the sets UW = {(a, b) ∈ L1 × L1 : µ(c) − µ(d) ∈ W ∀ c, d ∈ [a ∧ b, a ∨ b], d ≤ c}, where W is a neighbourhood of 0 in G. A lattice uniformity U on L1 is said to be exhaustive if every monotone sequence in L1 is a Cauchy sequence in U, σ-order-continuous (σ-o.c.) if an ↑ a or an ↓ a in L1 implies that {an } converges to a in U, and order-continuous (o.c.) if the same condition holds for nets. If µ : L1 → G is a modular function, µ is said to be exhaustive (respectively, σ-o.c. or o.c.) if U(µ) is exhaustive (respectively, σ-o.c. or o.c.). By 3.5 and 3.6 of [25], we have that a modular function µ is exhaustive if and only if µ(an+1 ) − µ(an ) converges to 0 for every monotone sequence (an )n∈N in L1 , µ is σ-o.c. if and only if an ↑ a or an ↓ a imply that (µ(an )) converges to µ(a), and µ is o.c. if and only if, for every monotone net (aα )α∈J order convergent to a, (µ(aα )) converges to µ(a). 2. D-uniformities and modular measures In this section we introduce the concept of D-uniformity on L, which arises in a natural way from the study of modular measures since, as we will see in Theorem 2.9, every G-valued modular measure µ on L generates on L a D-uniformity. First we need some preliminaries. Lemma 2.1. Let a, b, c ∈ E. (1) If a+b exists and a+b ≤ c, then c\(a+b) = (c\b)\a and (a+b)/c = b/(a/c). (2) If a + b exists, then a + b = (⊥ b\a)⊥ = ⊥ (b/a⊥ ). (3) If a ≤ b, then b\a = ⊥ (a + b⊥ ) and a/b = (⊥ b + a)⊥ . pseudo-D-lattices and topologies generated by measures 29 Proof. (1) Set d = c\(a+b). Then c = d+(a+b) = (d+a)+b, whence d+a = c\b. Therefore d = (c\b)\a. In a similar way, setting e = (a + b)/c, we have b + e = a/c and therefore e = b/(a/c). (2) Setting c = 1 in (1), we have ⊥ (a + b) = ⊥ b\a and (a + b)⊥ = b/a⊥ . Therefore, by Proposition 1.1-(2), we obtain a + b = (⊥ b\a)⊥ and a + b = ⊥ (b/a⊥ ). (3) By Proposition 1.1-(5), a ≤ b implies that a + b⊥ exists. Then (3) follows from (2) and Proposition 1.1-(2). Lemma 2.2. Let a, b ∈ E, with a ≤ b. Then a/b = a⊥ \b⊥ and b\a = ⊥ b/⊥ a. Proof. It is sufficient to set c = 1 in Proposition 1.1-(11). Lemma 2.3. If a, b ∈ L, then a △∗ b = a⊥ ∗△ b⊥ (and a ∗△ b = ⊥ a ∗△ ⊥ b). Proof. By Lemma 2.2, we have a △∗ b = (a ∧ b)/(a ∨ b) = (a ∧ b)⊥ \(a ∨ b)⊥ = (a⊥ ∨ b⊥ )\(a⊥ ∧ b⊥ ) = a⊥ ∗△ b⊥ . The other equality can be proved in a similar way. Lemma 2.4. Let c, d ∈ E be such that c ≤ d. Set Ic,d = {a ∈ E : ∃ r, s ∈ [c, d] : r ≤ s, a = s\r} and Jc,d = {a ∈ E : ∃ r, s ∈ [c, d] : r ≤ s, a = r/s}. Then Ic,d = [0, d\c] and Jc,d = [0, c/d]. Proof. We prove the first equality. The other equality can be proved in a similar way. Let a ∈ Ic,d and choose r, s ∈ E such that c ≤ r ≤ s ≤ d and a = s\r. By Proposition 1.1-(9) and (10), we obtain a ≤ d\r ≤ d\c. Conversely, let a ∈ [0, d\c]. Then, by Proposition 1.1-(8), a + c exists and a + c ≤ d. Set s = a + c and r = c. Then c = r ≤ s ≤ d and a = (a + c)\c = s\r. Proposition 2.5. Let U be a uniformity on E. Set E1 = {(a, b) ∈ E × E : b ≤ a} and E2 = {(a, b) ∈ E × E : a + b exists } (= {(a, b) ∈ E × E : b ≤ a⊥ }). Then the following conditions are equivalent: (1) The operations (a, b) ∈ E2 → a + b ∈ E, a ∈ E → ⊥ a ∈ E and a ∈ E → a⊥ ∈ E are uniformly continuous with respect to U. (2) The operations (a, b) ∈ E1 → a\b ∈ E and (a, b) ∈ E1 → b/a ∈ E are uniformly continuous with respect to U. 30 anna avallone, paolo vitolo (3) The operations (a, b) ∈ E1 → a\b ∈ E and a ∈ E → a⊥ ∈ E are uniformly continuous with respect to U. (4) The operations (a, b) ∈ E1 → b/a ∈ E and a ∈ E → ⊥ a ∈ E are uniformly continuous with respect to U. (5) The operation (a, b) ∈ E2 → a⊥ \b (= a/⊥ b) ∈ E is uniformly continuous with respect to U. Moreover, if E is a pseudo D-lattice and U is a lattice uniformity on E, each of the above conditions is equivalent to each of the following: (6) The operations (a, b) ∈ E × E → a ∗△ b ∈ E and a ∈ E → a⊥ ∈ E are uniformly continuous with respect to U. (7) The operations (a, b) ∈ E × E → a △∗ b ∈ E and a ∈ E → uniformly continuous with respect to U. ⊥ a ∈ E are (8) The operation (a, b) ∈ E × E → a⊥ ∗△ b (= a △∗ ⊥ b) ∈ E is uniformly continuous with respect to U. Proof. (1) ⇒ (2) The uniform continuity of \ and / follows from Lemma 2.1-(3). (2) ⇒ (3) it is trivial. (2) ⇒ (4) it is trivial. (3) ⇒ (1) It is clear that the operation a ∈ E → ⊥ a is uniformly continuous. The uniform continuity of + follows from Lemma 2.1-(2). (4) ⇒ (1) is similar to the proof of (3) ⇒ (1). (3) ⇒ (5) is trivial. (5) ⇒ (3) Set a ∗ b = a⊥ \b. Then, since a ∗ 0 = a⊥ and 0 ∗ a = ⊥ a, we obtain a\b = (⊥ a)⊥ \b = ⊥ a ∗ b = (0 ∗ a) ∗ b and therefore \ is uniformly continuous. (6) ⇒ (3), (7) ⇒ (4), (8) ⇒ (5) and (6) ⇒ (8) are trivial. (7) ⇒ (6) follows from Lemma 2.3 and the equality a⊥ = a △∗ 1. (4) ⇒ (7) follows from the definition of a △∗ b. Definition 2.6. We say that a lattice uniformity U on L is a D-uniformity if it satisfies one of the conditions in the above proposition (and hence all). Therefore, if L is a lattice-ordered effect algebra, a D-uniformity in the sense of Definition 2.6 is a D-uniformity according to [4]. If we set, for subsets U and V of L × L, U \V = {(a\c, b\d) : c ≤ a, d ≤ b, (a, b) ∈ U, (c, d) ∈ V }, U/V = {(c/a, d/b) : c ≤ a, d ≤ b, (a, b) ∈ U, (c, d) ∈ V }, U ⊥ = {(a⊥ , b⊥ ) ∈ L × L : (a, b) ∈ U }, ⊥ U = {(⊥ a, ⊥ b) ∈ L × L : (a, b) ∈ U } it is clear that a lattice uniformity U on L is a D-uniformity if and only if, for every U ∈ U, there exist V, W ∈ U such that V \V ⊆ U and W ⊥ ⊆ U if and only if, for every U ∈ U , there exist V, W ∈ U such that V /V ⊆ U and ⊥ W ⊆ U. pseudo-D-lattices and topologies generated by measures 31 Moreover the following result holds. Proposition 2.7. Let U be a lattice uniformity on L. Then U is a D-uniformity if and only if, for every U ∈ U, there exists V, W ∈ U such that V \∆ ⊆ U, ∆\V ⊆ U and W ⊥ ⊆ U. Proof. Suppose that the above conditions are satisfied and we prove that U is a lattice uniformity. Since it follows by assumption that ⊥ is uniformly continuous, we have only to prove that, for every U ∈ U, there exists V2 ∈ U such that V2 \V2 ⊆ U. Let U ∈ U and choose V, V1 , V2 ∈ U such that V ◦ V ◦ V ⊆ U, V1 \∆ ⊆ V, ∆/V1 ⊆ V and V2 ∧ V2 ⊆ V1 . We prove that V2 \V2 ⊆ U. Let (a, b), (c, d) be in V2 such that c ≤ a and d ≤ b. We prove that (a\c, b\d) ∈ U. Indeed, since (c, c ∧ d) ∈ ∆ ∧ V2 ⊆ V2 ∧ V2 ⊆ V1 , we have (∗) (a\c, a\(c ∧ d)) ∈ ∆\V1 ⊆ V. Moreover, from V2 ⊆ V1 , we get (∗∗) (a\(c ∧ d), b\(c ∧ d)) ∈ V1 \∆ ⊆ V. Finally, since (c ∧ d, d) ∈ V2 ∧ V2 ⊆ V1 , we have (∗ ∗ ∗) (b\(c ∧ d), b\d) ∈ ∆\V1 ⊆ V. From (∗), (∗∗) and (∗ ∗ ∗), we obtain (a\c, b\d) ∈ V ◦ V ◦ V ⊆ U. In the sequel, if µ : L → G is a function and W is a neighbourhood of 0 in G, we set UW = {(a, b) ∈ L × L : µ(s) − µ(r) ∈ W ∀ r, s ∈ [a ∧ b, a ∨ b], r ≤ s}, AW = {(a, b) ∈ L × L : µ(c) ∈ W ∀ c ≤ a ∗△ b}, BW = {(a, b) ∈ L × L : µ(c) ∈ W ∀ c ≤ a △∗ b}. Lemma 2.8. Let µ : L → G be a measure and W a neighbourhood of 0 in G. Then UW = AW = BW . Proof. We use the notations of Lemma 2.4. Applying Lemma 2.4, we have AW = {(a, b) ∈ L × L : ∀ c ∈ [0, a ∗△ b], µ(c) ∈ W } = {(a, b) ∈ L × L : ∀ c ∈ Ia∧b,a∨b , µ(c) ∈ W } = {(a, b) ∈ L × L : ∀ r, s ∈ [a ∧ b, a ∨ b], r ≤ s, µ(s\r) ∈ W } = UW = {(a, b) ∈ L × L : ∀ r, s ∈ [a ∧ b, a ∨ b], r ≤ s, µ(r/s) ∈ W } = {(a, b) ∈ L × L : ∀ c ∈ Ja∧b,a∨b , µ(c) ∈ W } = {(a, b) ∈ L × L : ∀ c ∈ [0, a △∗ b], µ(c) ∈ W } = BW . 32 anna avallone, paolo vitolo Theorem 2.9. Let µ : L → G be a modular measure. Then the µ-uniformity U(µ) is a D-uniformity on L and a base of U(µ) is the family consisting of the sets AW , where W is a neighbourhood of 0 in G. Proof. By Lemma 2.8, a base of U(µ) is the family consisting of the sets AW , where W is a neighbourhood of 0 in G. Then, by Proposition (2.7), it is sufficient to prove the following conditions: (1) A⊥ W = AW . (2) AW \∆ ⊆ AW . (3) ∆/AW ⊆ AW . Proof of (1). By Lemma 2.3 and 2.8, we have ⊥ ⊥ ⊥ ⊥ ⊥ ∗ A⊥ W = BW = {(a , b ) : (a, b) ∈ BW } = {(a , b ) : ∀ c ≤ a △ b, µ(c) ∈ W } = {(a⊥ , b⊥ ) : ∀ c ≤ a⊥ ∗△ b⊥ , µ(c) ∈ W } = AW . Proof of (2). It is sufficient to prove that, for every a, b, c in L with c ≤ a and c ≤ b, (a\c) ∗△ (b\c) = a ∗△ b. Set d = a\c and e = b\c. By Proposition 1.2-(3) and (4), we have d ∨ e = (a ∨ b)\c and d ∧ e = (a ∧ b)\c. Therefore, by Proposition 1.1-(12), we obtain d ∗△ e = ((a ∨ b)\c)\((a ∧ b)\c) = (a ∨ b)\(a ∧ b) = a ∗△ b. Proof of (3). Since AW = BW by Lemma 2.8, it is sufficient to prove that, for every a, b, c in L with a ≤ c and b ≤ c, (c\a) △∗ (c\b) = a ∗△ b. Set d = c\a and e = c\b. By Proposition 1.2-(1) and (2), we obtain d ∨ e = c\(a ∧ b) and d ∧ e = c\(a ∨ b). By Proposition 1.1-(11), we get d △∗ e = (d ∧ e)/(d ∨ e) = (c\(a ∨ b))/(c\(a ∧ b)) = (a ∨ b)\(a ∧ b) = a ∗△ b. Definition 2.10. A DV-congruence on L (after A. Dvurecenskij and T. Vetterlein) is an equivalence relation N which satisfies the following conditions: (a) For every a, b ∈ L, if (a, c) ∈ N, (b, d) ∈ N, a + b and c + d exist, then (a + b, c + d) ∈ N. (b) If a + b exists, then, for every c ∈ L such that (c, a) ∈ N, there exists d ∈ L such that (d, b) ∈ N and c + d exists; and, for every h ∈ L such that (h, b) ∈ N, there exists k ∈ L such that (k, a) ∈ N and k + h exists. Proposition 2.11. Let N be a DV-congruence on E. Define the operation + on the quotient E/N in the following way: For every â, b̂ ∈ E/N, â + b̂ = ĉ if and only if there exist a′ ∈ â, b′ ∈ b̂ and c′ ∈ ĉ such that a′ + b′ = c′ in E. Then: (1) + is well defined on E/N and (E/N, +, 0̂, 1̂) is a pseudo-effect algebra. c = ĉ\b̂ and b/c c = b̂/ĉ. (2) If c ≥ b, then c\b Proof. (1) has been proved in 3.3 of [16]. (2) follows from the definition of + in E/N, since, if we set a = c\b, we have â + b̂ = ĉ, whence â = ĉ\b̂. In a similar way we obtain the other equality. pseudo-D-lattices and topologies generated by measures 33 The aim of the next two theorems is to obtain also in pseudo D-lattices a technique based on the ”completion method” of H. Weber (see [24] and [25]) which allowed in many cases to reduce the study of exhaustive modular measures on D-lattices to the study of o.c. modular measures on complete D-lattices (see for example [1]–[5]). Theorem 2.12. Let U be a D-uniformity on L. Then the following properties hold: T (1) N (U) = {U : U ∈ U} is a DV-congruence and a lattice congruence. (2) The quotient L̂ = L/N (U) is a pseudo-D-lattice. b = {(â, b̂) ∈ L̂× L̂ : (a, b) ∈ U }, the quotient uniformity (3) Setting, for U ∈ U, U b : U ∈ U } is a Hausdorff D-uniformity on L̂. Ub = {U (4) If G is Hausdorff and µ : L → G is a modular measure which is uniformly continuous with respect to U, then the function µ̂ : L̂ → G defined as µ̂(â) = µ(a) for a ∈ â ∈ L̂ is a well defined modular measure on L̂ and the Db uniformity generated by µ̂ coincides with U. Proof. (1) N (U) is a lattice congruence by 1.2.2 of [24]. Moreover, it is clear that N (U) satisfies condition (a) of Definition 2.10. Indeed, it is sufficient to observe that, since U is a D-uniformity, then, for every U ∈ U, there exists V ∈ U such that V + V ⊆ U, where V + V = {(a + c, b + d) : (a, b) ∈ V, (c, d) ∈ V, a + c and b + d exist}. Now we prove condition (b) of Definition 2.10. Since U is a D-uniformity, it is clear that (a, b) ∈ N (U) implies (a⊥ , b⊥ ) ∈ N (U) and (⊥ a, ⊥ b) ∈ N (U). Now, suppose that a + b exists and let c ∈ L be such that (c, a) ∈ N (U). We prove that there exists d ∈ L such that (d, b) ∈ N (U) and c + d exists. Set d = c⊥ ∧ b. By Proposition 1.1-(5), c + d exists since d ≤ c⊥ . Let U ∈ U and choose V ∈ U such that V ∧∆ ⊆ U. Since a+b exists, by Proposition 1.1-(5) we have that b ≤ a⊥ . Therefore, we get (d, b) = (c⊥ ∧ b, a⊥ ∧ b) = (c⊥ , a⊥ ) ∧ (b, b) ∈ V ∧ ∆ ⊆ U. Hence, (d, b) ∈ N (U). In a similar way, let h ∈ L be such that (h, b) ∈ N (U). We prove that there exists k ∈ L such that (k, a) ∈ N (U) and k + h exists. Set k = ⊥ h ∧ a. Since k ≤ ⊥ h, by Proposition 1.1-(5) we have that h + k exists. Moreover, since a + b exists, we have a ≤ ⊥ b. Let U ∈ U and choose V ∈ U such that ∆ ∧ V ⊆ U. Then we obtain (k, a) = (a ∧ ⊥ h, a ∧ ⊥ b) = (a, a) ∧ (⊥ h, ⊥ b) ∈ ∆ ∧ V ⊆ U. (2) By Proposition 2.11, L̂ is a pseudo-effect-algebra. It remains to prove that L̂ is a pseudo-D-lattice. By (3) of [16] (page 7), we have that â ≤ b̂ if and only if there exists h ∈ L such that â + ĥ = b̂ if and only if there exist c, d ∈ L such that (c, a) ∈ N (U), (d, h) ∈ N (U), c + d exists and (c + d, b) ∈ N (U). Moreover, by 1.2.3 of [24], L̂ is a lattice with respect to the following order: ′ â ≤ b̂ if and only if there exist c, k ∈ L such that (c, a) ∈ N (U), (k, b) ∈ N (U) and c ≤ k. Therefore, it is sufficient to observe that ≤ and ≤′ coincide. 34 anna avallone, paolo vitolo (3) It is known by Proposition 1.2.4 of [24] that Ub is a Hausdorff lattice uniformity. To prove that Ub is a D-uniformity, we apply Proposition 2.7. Let b ∈ Ub and choose W ∈ U such that W ⊥ ⊆ U. Applying Proposition 2.11-(3), we U d⊥ and therefore W c⊥ = W c⊥ ⊆ U b. obtain that W b ⊆U b . Let Now, choose V ∈ U closed such that V \V ⊆ U. We prove that Vb \∆ b. a, b, c ∈ L be such that (â, b̂) ∈ Vb , ĉ ≤ â and ĉ ≤ b̂. We prove that (â\ĉ, b̂\ĉ) ∈ U ˆ Since ĉ ≤ â and ĉ ≤ b̂, by 1.2.3 of [24] we can find d, e, r, s ∈ L such that d = ĉ, ê = â, r̂ = ĉ, ŝ = b̂, d ≤ e and r ≤ s. Since (ê, ŝ) = (â, b̂) ∈ Vb , by Proposition 1.2.4 of [24] we obtain that (e, s) ∈ V. Moreover, since dˆ = r̂, we have that (d, r) ∈ N (U). Therefore, we get (e\d, s\r) ∈ V \V ⊆ U. Set h = e\d and k = s\r. b . Now it is sufficient to observe that (ĥ, k̂) = (â\ĉ, b̂\ĉ) Hence we have (ĥ, k̂) ∈ U by Proposition 2.11-(2). The other condition of Proposition 2.7 can be proved in a similar way. (4) is known by the theory of uniform lattices (see [25]). Indeed, since U(µ) is the weakest lattice uniformity which makes µ uniformly continuous, we have U(µ) ≤ U, from which N (U) ⊆ N (U(µ)). By Propositions 2.5 and 3.1 of [25], (a, b) ∈ N (U(µ)) if and only if µ is constant on the interval [a ∧ b, a ∨ b]. Then, if â = b̂, we have µ(a) = µ(b) and therefore µ̂ is well defined on L̂. It is also known b Here we have only to observe that µ̂ is a modular function, too, and U(µ̂) = U. that, because of the definition of + in L̂ (see Proposition 2.11), µ̂ is a measure, too. e be the Theorem 2.13. Let U be a Hausdorff D-uniformity on L and let (L̃, U) uniform completion of (L, U). Then the following properties hold: (1) The lattice operations ∨ and ∧ and the pseudo-D-lattice operations \ and / can be extended in a unique way such that L̃ becomes a pseudo-D-lattice. (2) Ue is a D-uniformity on L̃. (3) If U is exhaustive, then L̃ is complete as lattice and Ue is o.c. (4) If G is complete and Hausdorff and µ : L → G is a modular measure which is uniformly continuous with respect to U, then µ can be extended in a unique way to a modular measure µ̃ : L̃ → G which is uniformly continuous with respect to Ue and o.c. and µ(L) is dense in µ̃(L̃). Proof. (1) By Proposition 1.3.1 of [24], it is known that the lattice operations ∨ and ∧ can be extended in a unique way such that L̃ becomes a lattice and Ue is a lattice uniformity. Then the set L̃′ = {(a, b) ∈ L̃ × L̃ : b ≤ a} coincides with the closure in (L̃, Ũ) of the set {(a, b) ∈ L × L : b ≤ a}. Denote again by / and \ the uniformly continuous extensions, respectively, of / and \ to L̃′ . To prove that L̃ is a pseudo-D-lattice, it is sufficient, by Theorem 2.7 of [26], to prove that \ and / have the following properties: pseudo-D-lattices and topologies generated by measures 35 (a) If a ≤ b ≤ c, then b/c ≤ a/c, c\b ≤ c\a, (a/c)\(b/c) = a/b, (c\b)/(c\a) = b\a. (b) For every a ∈ L, a\0 = 0/a = a. (a) Let a, b, c in L̃ such that a ≤ b ≤ c. Choose nets (aα ), (bα ) and (cα ) in L e Without loss of generality, we convergent, respectively, to a, b and c in (L̃, U). may assume that they are indexed in the same way. Moreover, we may suppose that aα ≤ bα ≤ cα for each α, since (aα ) can be replaced by (aα ∧ bα ) and (bα ) by (bα ∧ cα ). Therefore, by the definition of \ and /, we obtain b/c = limα (bα /cα ) ≤ limα (aα /cα ) = a/c and c\b = limα (cα \bα ) ≤ limα (cα \aα ) = c\a. Moreover, since bα /cα ≤ aα /cα , we have (a/c)\(b/c) = limα ((aα /cα )\(bα /cα )) = limα (aα /bα ) = a/b. In a similar way, since cα \bα ≤ cα \aα , we obtain that (c\b)/(c\a) = b\a. In the same way we obtain (b). (2) It is known that a base of Ue consists of the sets {U : U ∈ U}, where U e Then, to prove that Ue is a D-uniformity, it is sufficient is the closure of U in U. to prove that, for every U ∈ U , there exist V, W ∈ U such that V \V ⊆ U and W /W ⊆ U. Let U ∈ U and choose V ∈ U such that V \V ⊆ U. Let (a, b) ∈ V and (c, d) ∈ V be such that c ≤ a and d ≤ b. Choose nets ((aα , bα )) and ((cα , dα )) in e We may suppose that, V convergent, respectively, to (a, b) and (c, d) in Ue × U. for each α, cα ≤ aα and dα ≤ bα . Then ((aα \cα , bα \dα )) ∈ V \V ⊆ U and, by the definition of \, converges to (a\c, b\d). Therefore we get (a\c, b\d) ∈ U . In a similar way we prove that there exists W ∈ U such that W /W ⊆ U . (3) and (4) have been proved in Proposition 3.7 of [25]. We have only to observe that the continuity of µ̃ and the definition of \ and / on L̃ imply that µ̃ is a measure. Remark. In Theorem 4.6 of [17], it is proved that every Archimedean (and, therefore, every σ-complete) pseudo-MV-algebra is commutative. This is not true if L is an Archimedean pseudo-effect algebra, as the next examples prove. Let E = {0, 1, a, b, c}, where a, b, and c are not comparable. Define a + b = b + c = c + a = 1, while b + a, c + b and a + c are undefined. Then, E is a complete modular pseudo-D-lattice which is not commutative. Moreover let µ : E → [0, 1] be defined as µ(a) = µ(b) = µ(c) = 1/2, µ(0) = 0 and µ(1) = 1. Then µ is a modular measure on E with N (U(µ)) = ∆ and therefore Ẽ = E/N (U(µ)) = E. Now we obtain an infinite example considering the set F of all sequences with values in E, in which we define (an ) + (bn ) if and only if, for each n ∈ N, an + bn exists and in this case (an ) + (bn ) = (an + bn ). It is easy to see that, since E is finite, FP is a complete pseudo-D-lattice. n Moreover, if we define λ : F → [0, 1] as λ(a) = ∞ n=1 µ(an )/2 for a = (an ) ∈ F, we obtain a modular measure on F with λ(a) > 0 for every a ∈ F with a 6= 0. Then F̃ = F/N (U(λ)) = F. 36 anna avallone, paolo vitolo 3. Uniqueness theorems In this section we prove a uniqueness theorem for measures on pseudo-effectalgebras and we apply the results of the previous section to prove that, for modular measures on pseudo D-lattices, the uniqueness theorem holds without completeness assumptions. We say that E has the interpolation property if, for every sequences (an )n∈N and (bn )n∈N in E, with an ≤ an+1 ≤ bn+1 ≤ bn for each n, there exists a ∈ E such that, for each n, an ≤ a ≤ bn . It is clear that, if E is σ-complete, then E has the interpolation property. If µ : E → G is a measure, we say that: • E is µ-chained if, for every neighbourhood W of 0 in G and every a ∈ E, there exist a0 , a1 , . . . , ar in E such that 0 = a0 ≤ a1 ≤ . . . ≤ ar = a and µ(c) − µ(d) ∈ W whenever c, d ∈ [ai−1 , ai ] for some i ∈ {1, . . . , r}. • µ is strongly continuous if, for every neighbourhood W of 0 in G and every a ∈ E, there exists an orthogonal finite family (b1 , . . . , br ) in E such that b1 + . . . + br = a and µ(b) ∈ W whenever b ≤ bi for some i ≤ r. • If G is a linear space, µ is convex-ranged if, for every a ∈ E, µ([0, a]) is convex. Lemma 3.1. Let a, b, c in E. (1) If c ≤ a and a + b exists, then (c/a) + b exists and c/(a + b) = (c/a) + b. (2) If c ≤ a and b + a exists, then b + (a\c) exists and (b + a)\c = b + (a\c). (3) If a ≤ b ≤ c, then (a/b) + (b/c) exists and (a/b) + (b/c) = a/c. (4) If a ≤ b ≤ c, then (c\b) + (b\a) exists and (c\b) + (b\a) = c\a. Proof. (1) Since a + b and c + (c/a) = a exist and + is associative, then d = (c/a) + b and c + d exist and we have c/(a + b) = c/((c + (c/a)) + b) = c/(c + d) = d = (c/a) + b. (2) can be proved as (1). (3) Since b + (b/c) = c exists, by (1) we obtain that (a/b) + (b/c) exists and (a/b) + (b/c) = a/(b + (b/c)) = a/c. (4) In a similar way as (3), we obtain (4) applying (2). Lemma 3.2. Let h, k, r, s and a, b, c, d be in E. Then: (1) If h+k and r+s exist and h+k ≤ r+s, then k ≤ h/(r+s) and h ≤ (r+s)\k. (2) If b ≤ a and c ≤ a\b, then c + b exists, c + b ≤ a and b ≤ c/a. (3) If b ≤ a and c ≤ b/a, then b + c exists, b + c ≤ a and b ≤ a\c. pseudo-D-lattices and topologies generated by measures 37 Proof. (1) We first apply Proposition 1.1-(8) with a = k, b = h/(r + s) and c = h. Indeed, by assumption, r + s = h + (h/(r + s)) = c + b and h + k = c + a exist, and c + a ≤ c + b. Therefore, we have k = a ≤ b = h/(r + s). Now, we apply Proposition 1.1-(7) with a = h, c = k and b = (r + s)\k. By assumption, we have that r + s = ((r + s)\k) + k = b + c and h + k = a + c exist, and a + c ≤ b + c. Therefore, h = a ≤ b = (r + s)\k. (2) Since (a\b)+b = a exists and c ≤ a\b by assumption, then by Proposition 1.1-(7) we have that c + b exists and c + b ≤ a. By (1), we get b ≤ c/a. (3) Since b + (b/a) = a exists and c ≤ b/a by assumption, by Proposition 1.1-(8) we have that b + c exists and b + c ≤ a. By (1), we get b ≤ a\c. Lemma 3.3. Let a0 , a1 , ..., an be in E such that a0 ≤ a1 ≤ . . . ≤ an and, for every i ∈ {1, ..., n}, let bi = ai−1 /ai . Then (b1 , ..., bn ) is orthogonal and b1 + · · · + bn = a0 /an . Proof. By Lemma 3.1-(3), we get that b1 + b2 = (a0 /a1 ) + (a1 /a2 ) exists and it is equal to a0 /a2 . By induction, suppose that b1 + · · · bn−1 exists and it is equal to a0 /an−1 . Then, by Lemma 3.1-(3), we obtain that b1 + · · · bn−1 + bn = (a0 /an−1 ) + (an−1 /an ) exists and it is equal to a0 /an . Proposition 3.4. The following conditions are equivalent (1) E is σ-complete. (2) For every increasing sequence (an )n∈N in E, supn an exists. (3) For every decreasing sequence (an )n∈N in E, inf n an exists. Proof. The equivalence of (2) and (3) is trivial by Proposition 1.1-(6). (1) ⇒ (2) Let (an )n∈N be an increasing sequence in E and set bn = an−1 /an (where a0 = 0). By Lemma 3.3, P (bn ) is an orthogonal sequence and b1 + · · · + bn = an . Hence, by (1), a = supn i≤n bi = supn an exists. (2) ⇒ (1) Let (an )n∈N be an orthogonal sequence P in E. Set bn = a1 + . . . + an . Since (bn ) is an increasing sequence, we have that n∈N an = supn bn exists. Proposition 3.5. Let µ : E → G be a measure. Then the following conditions are equivalent: (1) µ is σ-additive. (2) For every sequence (an )n∈N in E, an ↑ a ⇒ µ(a) = limn µ(an ). (3) For every sequence (an )n∈N in E, an ↓ a ⇒ µ(a) = limn µ(an ). (4) For every sequence (an )n∈N in E, an ↓ 0 ⇒ limn µ(an ) = 0. Proof. (1) ⇒ (2) Let (an ) be such that an ↑ a. For each n ∈ N, set bn = an−1 /an , where a0 = 0. By Lemma 3.3, (bn ) is orthogonal and, for each n ∈ N, b1 +· · ·+bn = P 0/an = P an . Therefore, a =Psupn an = n∈N bn . Since µ is σ-additive, we obtain n µ(a) = ∞ n=1 µ(bn ) = limn k=1 µ(bk ) = limn µ(b1 + · · · + bn ) = limµ (an ). (2) ⇒ (3) Let (an )n∈N be such that an ↓ a. By Proposition 1.1-(6), we get that ↑ a⊥ . By (2), we obtain µ(a⊥ ) = limn µ(a⊥ a⊥ n n ), from which µ(a) = limn µ(an ). 38 anna avallone, paolo vitolo (3) ⇒ (4) is trivial. P (4) ⇒ (1) Let (an ) be an orthogonal sequence in E such that a = n∈N an exists. Since a1 + · · · + an ≤ a, then, for each n ∈ N, bn = a\(a1 + · · · + an ) exists. By Proposition (1.1)-9, (bn ) is a decreasing sequence. Moreover we have that inf n bn = 0. Indeed, if c ≤ bn for each n, by Lemma 3.2-(2) we obtain that, for each n, a1 +. . .+an ≤ c/a, from which we get a = supn (a1 +. . .+an ) ≤ c/a. By Lemma 3.2-(3), we have c ≤ a\a = 0. Now, since bn ↓ 0, by (4) we limn µ(bn ) = 0. Phave ∞ Since lim Pn µ(bn ) = limn (µ(a) − µ(a1 + · · · + an )) = µ(a) − n=1 µ(an ), we obtain µ(a) = ∞ n=1 µ(an ). Proposition 3.6. Let µ : E → G be a measure. Then E is µ-chained if and only if µ is strongly continuous. Proof. Suppose that E is µ-chained. Let W be a neighbourhood of 0 in G and a ∈ E. Choose a0 , a1 , · · · , ar in E such that 0 = a0 ≤ a1 ≤ · · · ≤ ar = a and µ(h)− µ(k) ∈ W whenever h, k ∈ [ai−1 , ai ] for some i ∈ {1, ..., r}. Set bi = ai−1 /ai for each i ∈ {1, ..., r}. By Lemma 3.3, (b1 , ..., br ) is orthogonal and b1 + · · · + br = a. Let i ≤ r and choose b ≤ bi . Since ai−1 + bi exists, by Proposition 1.1-(8) ai−1 + b exists and ai−1 ≤ ai−1 + b ≤ ai−1 + bi = ai . Therefore, we obtain µ(b) = µ(ai−1 + b) − µ(ai−1 ) ∈ W. Now, suppose that µ is strongly continuous. Let W and V be neighbourhoods of 0 in G with V − V ⊆ W and a ∈ E. Choose an orthogonal family (b1 , . . . , br ) in E such that b1 + · · · + br = a and µ(b) ∈ V whenever b ≤ bi for some i ≤ r. Set a0 = 0 and ai = b1 + . . . + bi for every i ≤ r. Then, we have 0 = a0 ≤ a1 ≤ · · · ≤ ar = a. Let i ≤ r and choose h, k ∈ [ai−1 , ai ]. By Proposition 1.1(10), we have ai−1 /h ≤ ai−1 /ai = bi and ai−1 /k ≤ ai−1 /ai = bi . Therefore, µ(h) − µ(k) = µ(ai−1 /h) − µ(ai−1 /k) ∈ V − V ⊆ W. The following result can be derived by Theorems 4.2 and 4.4 of [8]. Theorem 3.7. Suppose that E has the interpolation property and let µ : E → Rn be a strongly continuous measure. Then, if µ has nonnegative components, µ(E) is star-shaped with respect to 0. Moreover, if E is a lattice and µ is modular, then µ is convex-ranged. Proof. By Theorem 4.2 of [8], µ(E ′ ) is star-shaped with respect to 0 if E ′ is a µ-chained poset with smallest element 0 and greatest element 1, with a binary relation ⊥ and a partially defined binary operation ⊕ satisfying the following properties: (a) a ⊥ b if and only if a ⊕ b exists. (b) a ⊕ 0 = 0 ⊕ a = a. (c) If a ≤ b, then there exists c in E ′ with a ⊥ c and a ⊕ c = b. (d) If a ⊥ b, c ≤ a and d ≤ b, then c ⊥ d and c ⊕ d ≤ a ⊕ b. Moreover, by Theorem 4.4 of [8], µ(E ′ ) is convex if E ′ satisfies the additional condition: pseudo-D-lattices and topologies generated by measures 39 (e) If a ≤ c ≤ a ⊕ b, then there exists d ≤ b such that a ⊕ d = c. Now, observe that a pseudo-effect algebra satisfies all the above conditions if we define a ⊕ b = a + b and a ⊥ b if and only if a + b exists. Indeed (a), (b) and (c) are trivial, (d) follows from Proposition 1.1-(7) and (8), and (e) follows from Proposition 1.1-(8). Moreover it is easy to see that, if d ∈ E, the interval [0, d] is a pseudo-effect-algebra if we define, for every a, b ∈ E, a + b = c if and only if a + b = c in E and c ≤ d. The assumptions on E imply that [0, d] has the interpolation property and it is µ-chained by Proposition 3.6. Therefore, we can apply to [0, d] Theorems 4.2 and 4.4 of [8]. Now, using the results of Section 3 instead of the corresponding results of [4], it is possible to prove the following Uniqueness theorem for measures on pseudoeffect algebras, proved in [21] for measures on particular effect algebras and in [4] for measures on arbitrary effect algebras. Theorem 3.8. Let µ and ν be [0, +∞[-valued measures on E which satisfy the following conditions: (a) µ is convex-ranged. (b) There exist α ∈]0, µ(1)[ and β ∈]0, ν(1)[ such that, for every a ∈ E, µ(a) = α implies ν(a) = β. Moreover suppose that one of the following conditions is satisfied: (1) E is σ-complete and ν is σ-additive. (2) E has the interpolation property and, for every a ∈ E, ν(a) = 0 implies µ(a) = 0. (3) E has the interpolation property and, for every a ∈ E, µ(a) = α if and only if ν(a) = β. Then µ = λν, where λ = µ(1) . ν(1) Proof. The proof is similar to the proof of Theorem 3.1 of [4]. Now, we apply the results of Section 2 to prove that, if µ is a modular measure on L, then the Uniqueness theorem holds without completeness assumptions on L. First, we need the following result. Proposition 3.9. Let µ : L → [0, +∞[ be a modular measure. Then the following conditions are equivalent: (1) µ is strongly continuous. (2) For every ε > 0, there exists an orthogonal family (a1 , ..., ar ) in L such that a1 + · · · + ar = 1 and µ(ai ) < ε for every i ≤ n. 40 anna avallone, paolo vitolo Proof. (1) ⇒ (2) is trivial. (2) ⇒ (1) Let a ∈ L and ε > 0. Choose an orthogonal family (a1 , ..., ar ) in L such that a1 + · · · + ar = 1 and µ(ai ) < ε for every i ≤ r. Set b0 = 0 and bi = a1 + · · · + ai for every i ≤ r. Then we have 0 = b0 ≤ b1 ≤ ... ≤ bn = 1. Since bi = bi−1 + ai , we have ai = bi−1 /bi , from which we obtain µ(bi ) − µ(bi−1 ) = µ(ai ) < ε for every i ≤ n. Setting ci = bi ∧ a, we can see as in (2.3) of [1], that 0 = c0 ≤ c1 ≤ · · · ≤ cr = a and µ(ci ) − µ(ci−1 ) < ε for each i ≤ r. Set d0 = 0 and di = ci−1 /ci for i ≤ r. Then µ(di ) < ε for each i ≤ r and, by Lemma 3.3, d1 + . . . + dr = a. Theorem 3.10. Let µ, ν : L → [0, +∞[ be modular measures with the following properties: (1) µ is strongly continuous. (2) There exist α ∈]0, µ(1)[ and β ∈]0, ν(1)[ such that, if (an ) is a sequence in L with limn µ(an ) = α, then limn ν(an ) = β. Then µ = λν, where λ = µ(1) . ν(1) Proof. Denote by U the supremum of the D-uniformities generated by µ and ν (see Theorem 2.9). It is clear that U is a D-uniformity. Moreover µ and ν are obviously exhaustive, since they are monotone real-valued. Then U is exhaustive, too. Set L̂ = L/N (U), µ̂(a) = µ(a) and ν̂(a) = ν(a) for a ∈ â ∈ L̂, and denote by µ̃ and ν̃ the uniformly continuous extensions, respectively, of µ̂ and ν̂ to the uniform completion (L̃, Ũ) of L̂ (see Theorems 2.12 and 2.13). By Theorem 2.13, L̃ is a complete D-lattice and µ̃, ν̃ are o.c. modular measures and therefore σ-additive by Proposition 3.5. By Proposition 3.9, it is clear that µ̃ is strongly continuous, too, since 1c L = 1L̃ . Therefore, by Theorem 3.7, µ̃ is convex-ranged. Now let a ∈ L̃ such that µ̃(a) = α. Choose (an ) in L̂ which converges to a in (L̃, Ũ). By the continuity of µ̃ and ν̃, we get limn µ̂(an ) = µ̃(a) = α and limn ν̂(an ) = ν̃(a). By (2), we get ν̃(a) = β. Then µ̃ and ν̃ verify the assumptions of Theorem 3.8. By Theorem 3.8, we get µ̃ = λν̃, from which µ = λν. References [1] Avallone, A., Barbieri, G., Range of finitely additive fuzzy measures, Fuzzy Sets and Systems, 89 (1997), 231-241. [2] Avallone, A., Barbieri, G., Vitolo, P., Hahn decomposition of modular measures and applications, Annales Soc. Math. Polon., Series I: Comment. Math., XLIII (2003), 149-168. [3] Avallone, A., Barbieri, G., Vitolo, P., Weber, H., Decomposition of effect algebras and the Hammer-Sobczyk theorem, Algebra Universalis, 60 (2009), 1-18. pseudo-D-lattices and topologies generated by measures 41 [4] Avallone, A., Basile, B., On a Marinacci uniqueness theorem for measures, J. Math. Anal. Appl., 286, no. 2 (2003), 378-390. [5] Avallone, A., De Simone, A., Vitolo, P., Effect algebras and extensions of measures, Bollettino U.M.I., 9-B (2006), 423-444. [6] Avallone, A., Vitolo, P., Decomposition and control theorems in effect algebras, Sci. Math. Japon., 58 (2003), 1-14. [7] Bandot, R., Non-commutative programming language, Symposium LICS, Santa Barbara, 1000, 3-9. [8] Barbieri, G., Liapunov’s theorem for measures on D-posets, Intern. J. Theoret. Phys., 43, 7/8 (2004), 1613–1623. [9] Bennett, M.K., Foulis, D.J., Effect algebras and unsharp quantum logics, Found. Phys. 24, no. 10 (1994), 1331-1352. [10] Butnariu, D., Klement, P., Triangular norm-based measures and games with fuzzy coalitions, Kluwer Acad. Publ., 1993. [11] Chovanek, F., Kopka, F., D-lattices, Intern. J. Theoret. Phys., 34 (1995), 1297-1302. [12] Dvurecenskij, A., Central elements and Cantor-Bernstein’s theorem for pseudo-effect algebras, J. Austr. Math. Soc. 74 (2003), 121-143. [13] Dvurecenskij, A., New quantum structures, in Handbook of quantum logic and quantum structures. Edited by K. Engesser, D.M. Gabbay and D. Lehmann, Elsevier, 2007. [14] Dvurecenskij, A., S. Pulmannova, S., New trends in quantum structures, Kluwer Acad. Publ., 2000. [15] Dvurecenskij, A., Vetterlein, T., Pseudo-effect algebras I. Basic properties, Intern. J. Theoret. Phys., 40 (2001), 685-701. [16] Dvurecenskij, A., Vetterlein, T., Congruences and states on pseudoeffect algebras, Found Phys. Letters, 14 (2001), 425-446. [17] Dvurecenskij, A., Vetterlein, T., Archimedeanness and the MacNeille completion of pseudoeffect algebras and po-groups, Alg. Univ., 50, no. 2 (2003), 207-230. [18] Epstein, L.G., Zhang, J., Subjective probabilities on subjectively unambiguous events., Econometrica, 69, no. 2 (2001), 265–306. [19] Fleischer, I., Traynor, T., Group-valued modular functions, Alg. Univ., 14 (1982), 287-291. 42 anna avallone, paolo vitolo [20] Georgescu, G., Jogulescu, A., Pseudo MV-algebras, Multiple Val. Logic, 6 (2001), 95-135. [21] Marinacci, M., A uniqueness theorem for convex-ranged probabilities, Decis. Econom. Finance, 23, no. 2 (2000), 121–132. [22] Marinacci, M., Probabilistic sofistication and multiple priors, Econometrica, 70 (2002), 755-764. [23] Pulmannova, S., Generalized Sasaki projections and Riesz ideals in pseudoeffect algebras, Int. J. Theoret. Phys., 42, no. 7 (2003), 1413-1423. [24] Weber, H., Uniform Lattices I: A generalization of topological Riesz space and topological Boolean rings; Uniform lattices II, Ann. Mat. Pura Appl., 160 (1991), 347-370 and 165 (1993), 133-158. [25] Weber, H., On modular functions, Funct. et Approx., XXIV (1996), 35-52. [26] Shang Yun, Li Yongming, Chen Maoyin, Pseudo difference posets and Boolean D-posets, Intern. J. Theoret. Phys., 43, no. 12 (2004). Accepted: 09.09.2009