- Email: [email protected]
The extension of measures on D-lattices
Available online at www.sciencedirect.com
Fuzzy Sets and Systems 244 (2014) 123–129 www.elsevier.com/locate/fss
The extension of measures on D-lattices Giuseppina Barbieri Dipartimento di Matematica e Informatica, Università di Udine, Via delle Scienze 206, 33100 Udine, Italy Received 15 July 2011; received in revised form 28 June 2013; accepted 1 July 2013 Available online 5 July 2013
Abstract A lattice ordered group valued measure is extended from a D-lattice into a σ -complete D-lattice. A D-lattice is a lattice with a greatest element 1 and a smallest element 0 endowed with an order-compatible operation, called a difference, which satisfies a list of axioms. The result generalizes the classical result known for measures on Boolean algebras. © 2013 Elsevier B.V. All rights reserved. Keywords: D-lattices; Measures; Lattice ordered groups
1. Introduction
The problem of extensions of measures defined on Boolean algebras to measures defined on σ -complete Boolean algebras has been solved in the real valued case by [1] and [4], in the lattice ordered valued case by [11]. We use and generalize some ideas from [10] in order to prove that a lattice ordered group valued measure can be extended from a D-lattice into a σ -complete D-lattice in a unique way as specified in Theorem 3.14. For this we use some results contained in Chapter 3 of the book by Rieˇcan and Neubrunn [12]. D-lattices are generalization of Boolean algebras. Examples of D-lattices are orthomodular lattices used in Quantum Mechanics and MV-algebras used in Mathematical Logic. Both orthomodular lattices and MV-algebras generalize Boolean algebras, but they are incompatible generalizations, in the sense that an orthomodular lattice need not be an MV-algebra, and conversely. D-lattices also have some applications in Mathematical Economics (see, for instance [7]). Our source of information for D-lattices is the book by Dvureˇcenskij and Pulmannová [5]. The paper is organized as follows: In Section 2 we give some preliminary definitions and stuff which we need in the sequel (namely -groups, D-lattices, measures on D-lattices). In Section 3 we give our results: We start with a sub-D-lattice A of a σ -continuous D-lattice H and a measure defined on A and we extend it to the σ -complete D-lattice σ (A) generated by A.
E-mail address: [email protected]. 0165-0114/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.fss.2013.07.002
124
G. Barbieri / Fuzzy Sets and Systems 244 (2014) 123–129
2. Preliminaries 2.1. -groups An Abelian group (G, +) is called a lattice ordered group G (briefly, -group) if it is a lattice and the following implication holds: a b implies a + c b + c. An -group is called σ -complete if every upper bounded countable subset of G has the supremum in G. An -group is called super Dedekind complete if, for any nonempty subset A bounded from above, there exists a countable subset A∗ such that sup A = sup A∗ . G is weakly σ -distributive if for every bounded double sequence (ai,j ) such that for ai,j ↓ 0 (j → ∞) we have
ai,φ(i) = 0.
φ∈NN i
The symbol xn ↓ x means xn xn+1 and x =
xn .
Theorem 2.1. Let G be a super Dedekind complete -group. Let an,i,j be a bounded sequence such that an,i,j ↓ 0 (j → ∞). Then element a ∈ G there exists a bounded sequence ai,j such that ai,j ↓ 0 (j → ∞) for every positive and a ∧ ( n i an,i,φ(i+n) ) i ai,φ(i) for every φ : N → N. From now on, if not otherwise specified, let G be a super Dedekind complete weakly σ -distributive -group. 2.2. D-lattices and measures Definition 2.2. Let (H, ) be a poset with a smallest element 0 and a greatest element 1 and let be a partial operation on H such that b a is defined if and only if a b and for all a, b, c ∈ H : If a b then b a b and b (b a) = a; If a b c then c b c a and (c a) (c b) = b a. Then (H, , ) is called a difference poset (D-poset for short), or a difference lattice (D-lattice for short) if H is a lattice. One defines in H a partial operation ⊕ as follows: a ⊕ b is defined and
a⊕b=c
if and only if c b is defined and c b = a.
The operation ⊕ is well-defined by the cancellation law [5, page 13] (a b, c and b a = c a imply b = c), and (H, ⊕, 0, 1) is an effect algebra (see [5, Theorem 1.3.4]), that is the following conditions are satisfied for all a, b, c ∈ H : If a ⊕ b is defined, then b ⊕ a is defined and a ⊕ b = b ⊕ a; If b ⊕ c is defined and a ⊕ (b ⊕ c) is defined, then a ⊕ b and (a ⊕ b) ⊕ c are defined, and a ⊕ (b ⊕ c) = (a ⊕ b) ⊕ c; There exists a unique a ⊥ ∈ E such that a ⊕ a ⊥ is defined and a ⊕ a ⊥ = 1; If a ⊕ 1 is defined, then a = 0. We say that two elements a and b of H are orthogonal ifa b⊥ . If a1 , . . . , an ∈ H , we inductively define nk=1 ak = ( n−1 k=1 ak ) ⊕ an provided that the right hand side exists. The definitionis independent of permutations of the elements. We say that a finite sequence {a1 , . . . , an } of H is n n orthogonal if a exists. A sequence (a ) is called orthogonal if k=1 k k=1 ak exists for every n ∈ N. If, moreover, n nn n∈N supn k=1 ak exists, we set an = supn k=1 ak . We give two examples of D-lattices: Definition 2.3. An orthomodular lattice is a lattice with a smallest element 0 and a greatest element 1, with a map ⊥ : H → H , called orthocomplementation with the following properties:
G. Barbieri / Fuzzy Sets and Systems 244 (2014) 123–129
(O1) (O2) (O3) (O4)
125
a ∨ a ⊥ = 1 and a ∧ a ⊥ = 0; a b implies a ⊥ b⊥ ; a ⊥⊥ = a; a b implies b = a ∨ (b ∧ a ⊥ ) for every a, b ∈ H .
A D-lattice is an orthomodular lattice if and only if orthogonal elements are disjoint. Definition 2.4. An MV-algebra (H, +,⊥ ; 0, 1) is a commutative semigroup (H, +) with 0, 1 and a unary operation ⊥ : H → H which satisfies the following axioms: (MV1) (MV2) (MV3) (MV4)
x + 1 = 1; x ⊥⊥ = x; 0⊥ = 1; (x ⊥ + y)⊥ + y = (x + y ⊥ )⊥ + x for every x, y ∈ H .
A D-lattice is an MV-algebra if and only if disjoint elements are orthogonal. Definition 2.5. A D-lattice H is said to be σ -complete if its underlying lattice is σ -complete, i.e., every nonempty countable subset of H has a supremum in H . A σ -complete lattice will be called σ -continuous if xn ↑ x implies xn ∧ y ↑ x ∧ y. From now on H is a σ -continuous D-lattice. We give the following lemma which will be useful in the sequel. Lemma 2.6. Let bn , f ∈ H . Then the following equalities hold: (i) (sup bn ) f = sup(bn f ); (ii) f (sup bn ) = inf(f bn ); (iii) f (inf bn ) = sup(f bn ); provided that the operations are defined. Proof. (i) Obviously, sup(bn f ) (sup bn ) f . Vice versa, let z bn f . Then z ⊕ f bn . Hence z ⊕ f sup bn , whence z (sup bn ) f . Therefore (sup bn ) f is the least upper bound of bn f . (ii) is proved in [3, Lemma 3.7]. (iii) Obviously, f (inf bn ) sup(f bn ). Now let w (f bn ). We get f bn = (f bn ) ∧ f w ∧ f f , which gives f (w ∧ f ) bn , therefore f (w ∧ f ) inf bn . Then we obtain f (inf bn ) w ∧ f w which implies that f (inf bn ) is the least upper bound of f bn . 2 Definition 2.7. A mapping μ : H → G is called a measure if (i) μ(0) = 0; (ii) If a, b ∈ H with a b, then μ(a) μ(b); (iii) If (an ) is an orthogonal sequence and an ∈ H , then μ( an ) = limn ni=1 μ(ai ). A measure is called modular if μ(a ∨ b) + μ(a ∧ b) = μ(a) + μ(b) for every a, b ∈ H .
126
G. Barbieri / Fuzzy Sets and Systems 244 (2014) 123–129
As observed in [9] every subadditive state on an orthomodular lattice (i.e., every function μ such that μ(a ∨ b) μ(a) + μ(b) for every a, b ∈ H and μ(a ∨ b) = μ(a) + μ(b) for every orthogonal element a, b ∈ H ) is a modular function. Every state on an MV-algebra is a modular function. Remark 2.8. The mapping μ is additive, i.e., μ(x ⊕ y) = μ(x) + μ(y) for x y ⊥ if and only if μ(y) = μ(y x) + μ(x) for x y ∈ H . 3. Extension We start with a sub-D-lattice A of a σ -continuous D-lattice H and a measure μ : A → G. We want to extend it to the σ -complete D-lattice σ (A) generated by A. As a byproduct we obtain two versions of the theorem valid for orthomodular lattices and MV-algebras. We begin with a lemma: Lemma 3.1. (See [12, Lemma 3.3.5].) If an , bn ∈ A and an ↑ a and bn ↑ b with a b, then lim μ(an ) lim μ(bn ). Proof. Evidently an ∧ bm ↑ an ∧ b = an . Hence μ(an ) = limm μ(an ∧ bm ) limm μ(bm ). Therefore limn μ(an ) limm μ(bm ). 2 Definition 3.2. (Cf. [12, Definition 3.3.6].) We put Aσ := {b ∈ H : ∃bn ∈ A, bn ↑ b}, Aδ := {c ∈ H : ∃cn ∈ A, cn ↓ c} and define mappings μσ : Aσ → G and μδ : Aδ → G by the formulas μσ (b) = limn μ(bn ), μδ (c) = limn μ(cn ). They are well-defined by Lemma 3.1. Lemma 3.3. The mapping μσ is additive. Proof. Let a, b ∈ Aσ with a ⊥ b. Then there exist an , bn ∈ A with an ↑ a and bn ↑ b. Then an ⊥ bn , since an a and bn b and a ⊥ b. Moreover, we have that an ⊕ bn ∈ A and an ⊕ bn ↑ a ⊕ b. Hence a ⊕ b ∈ Aσ and μσ (a ⊕ b) = lim μ(an ⊕ bn ) = lim μ(an ) + lim μ(bn ) = μσ (a) + μσ (b). 2 Notation 3.4. We put μ∗ (d) = inf{μσ (f ): f d, f ∈ Aσ } for d ∈ H . The mapping μ∗ is an extension of μσ and it is increasing. Lemma 3.5. Let a ∈ H . Then there exists an ∈ Aσ , such that an ↓, an a and μ∗ (a) = lim μ∗ (an ). σ Proof. Let a ∈ H . Then there existσ (bn ) ∈ A and a bounded double sequence (ai,j ) with ai,j ↓ 0 (j → ∞) such that ∗ bn a and μ (a) + ai,φ(i) > μ (bn ). Put an := kn bk . Then an ↓, an ∈ Aσ , an a and μσ (an ) μσ (bn ) μ∗ (a) + ai,φ(i) . Since G is weakly σ -distributive we get lim μ∗ (an ) μ∗ (a). Hence μ∗ (a) = lim μ∗ (an ). 2
Lemma 3.6. If a ∈ σ (A) and b ∈ Aσ , then μ∗ (a ⊕ b) μ∗ (a) + μ∗ (b). Proof. Set M := {b ∈ H : for every a ∈ σ (A) with a ⊥ b, μ∗ (a ⊕ b) μ∗ (a) + μ∗ (b)}. (1) We claim that A ⊆ M: Take a ∈ σ (A) and b ∈ A with a ⊥ b. Then by Lemma 3.5 there exists (an ) in Aσ such that an a and μ∗ (a) = lim μ∗ (an ). Set cn = an ∧ b⊥ . We get cn ∈ Aσ and cn a. Then μ∗ (a) μσ (cn ) μσ (an ), whence lim μσ (cn ) = μ∗ (a). Notice that cn ⊥ b and set bn = cn ⊕ b. By [5, 1.8.7] we have bn ∈ Aσ and bn a ⊕ b. Hence μσ (bn ) μ∗ (a ⊕ b). Then μ∗ (a) + μ∗ (b) = lim μσ (cn ) + μσ (b) lim μσ (bn ). Therefore μ∗ (a ⊕ b) μ∗ (a) + μ∗ (b). (2) We claim that Aσ ⊆ M: Let b ∈ Aσ . Then there exists a sequence bn ∈ A with bn ↑ b. Take an element a ∈ σ (M) orthogonal to b. Since bn b, then a ⊥ bn and by [5, 1.8.7] bn ⊕ a ↑ b ⊕ a. Therefore μ∗ (a ⊕ b) = lim μ∗ (a ⊕ bn ) lim μ(bn ) + μ∗ (a) = μσ (b) + μ∗ (a). 2
G. Barbieri / Fuzzy Sets and Systems 244 (2014) 123–129
127
Proposition 3.7. If b ∈ Aσ , c ∈ Aδ , c b, then μσ (b) μδ (c). Proof. Let cn ∈ A with cn ↓ c. Notice that μ∗ (b) = μσ (b) μ∗ (c) since μ∗ is increasing. It is sufficient to show that μ∗ (c) lim μ(cn ): As c1 cn ↑ c1 c we get μ∗ (c1 c) = μσ (c1 c) = lim μ(c1 cn ) = μ(c1 ) − lim μ(cn ). Now c1 = c ⊕ (c1 c). By 3.6 μ(c1 ) μ∗ (c) + μσ (c1 c). The opposite inequality lim μ(cn ) μ∗ (c) is clear. So μσ (b) μδ (c) = lim μ(cn ). 2 Remark 3.8. Notice that (H, ∧, ∨, +, −) assumes the role of the structure X in [12, 3.3.1 Assumptions] where ∧ and ∨ are the lattice operations, x + y := x ∨ y and x − y := x (x ∧ y) for x, y ∈ H . Thanks to Proposition 3.7 the measure μ : A → G satisfies the assumptions contained [12]. From now on we make use of the following notation: Notation 3.9. (See [12, Definition 3.3.9].) We say that x ∈ L, if there exist w ∈ G, bounded ai,j , bi,j ∈ G, ai,j > 0, φ φ φ φ φ bi,j > 0, ai,j ↓ 0, bi,j ↓ 0 (j → +∞) such that for every φ : N → N there are x1 , x2 ∈ H , x1 x x2 and μσ (x2 ) − δ φ i ai,φ(i) w μ (x1 ) + i bi,φ(i) . Proposition 3.10. (See [12, Proposition 3.3.10].) If x ∈ L, then inf{μσ (x2 ): x2 ∈ Aσ , x2 x} = sup{μδ (x1 ): x1 ∈ Aδ , x1 x}. Proposition 3.11. (See [12, Theorem 3.3.12].) If x, y ∈ L, x y, then y − x ∈ L and μ∗ (y − x) = μ∗ (y) − μ∗ (x). Proof. By the definition of L, there are positive elements ai,j , bi,j in G, and for every φ : N → N elements φ φ φ φ φ φ φ φ φ φ x1 , y1 ∈ Aδ and x2 , y2 ∈ Aσ such that x1 x x2 , y1 y y2 , and μσ (x2 ) − i ai,φ(i) μ∗ (x) μδ (x1 ) + φ φ φ φ φ φ φ σ φ ∗ δ φ δ i bi,φ(i) , μ (y2 ) − i ai,φ(i) μ (y) μ (y1 ) + i bi,φ(i) . Put z1 := y1 − x2 , z2 := y2 − x1 . Then z1 ∈ A , φ φ φ φ φ φ φ z ∈ Aσ , z1 = y1 − x2 y − x y2 − x1 = z2 . We can choose ci,j ↓ 0 (j → +∞) such that i ai,φ(i) + 2 σ φ σ φ δ φ i bi,φ(i) i ci,φ(i) for every φ : N → N. Then μ (z2 ) − i ci,φ(i) μ (y2 ) − μ (x1 ) − i ai,φ(i) − i bi,φ(i) φ φ φ μ∗ (y) − μ∗ (x) μδ (y1 ) − μσ (x2 ) + i ai,φ(i) + i bi,φ(i) μδ (z1 ) + i ci,φ(i) . Therefore y − x ∈ L and μ∗ (y − x) = μ∗ (y) − μ∗ (x). 2 Proposition 3.12. (See [12, Theorem 3.3.13].) Let (zn ) be a monotone sequence in L converging to an element z ∈ H . Then z ∈ L and μ∗ (z) = lim μ∗ (zn ). Proof. By assumptions there is a ∈ Aσ such that a z. By Proposition 3.11 z2 − z1 ∈ L, z3 − z2 ∈ L, . . . , zn − zn−1 ∈ L, . . . , hence there are an,i,j ∈ G, an,i,j↓ 0 (j → σ ∗ σ +∞) such that for every φ : N → N there is yn ∈ A , y1 z1 , yn zn − zn−1 , yn a, μ (z1 ) μ (y1 ) − i a1,i,φ(i) ∗ σ and μ (zn − zn−1 ) μ (yn ) − i a1,i,φ(i) . There exists +∞) a bounded sequence (ai,j ) in G, ai,j∗ ↓ 0 (j → ∗ (z )) ∧ ( ∗ (z ) + a ) a for every φ : N → N. Therefore μ (z ) = μ such that (μ(a) − μ 1 n 1 nn i σ n,i,φ(i+n−1) n i i,φ(i) n n ∗ (z ) − μ∗ (z σ i i−1 )) i=2 i=1 μ (yi ) − k=1 i ak,i,φ(i+k−1) μ (un ) − k=1 i ak,i,φ(i+k−1) where un = (μ σ ∗ ∗ ), hence μσ (un ) − μ∗ (zn ) i ai,φ(i) . a ∧ ni=1 y i . Then μ (un ) − μ (zn ) μ(a) − μ (z1 σ σ Since zn ∈ L there are bn,i,j ↓ 0 (j → Put u = un . Then u ∈ A , u z and μ (u) − i ai,φ(i) limn μ∗ (zn ). +∞) such that for every φ : N → N there are vn ∈ Aδ , vn zn with μδ (vn ) + i bn,i,φ(n+i−1) μ∗ (zn ). Put b0,i,j := limn μ∗ (zn ) − μ∗ (zj ). Now fix j φ(1) (hence b0,i,j b0,i,φ(1) ). − μδ (v)) ∧ ( n i b n,i,φ(i+n−1) ) Put v := vj . By Theorem 2.1 there are bi,j∗ ↓ 0 (j →∗ +∞) such that (μ(a) δ (v) + b for every φ : N → N. Since lim μ (z ) = μ (z ) + b μ b + n j 0,i,j i,φ(i) k,i,φ(i+j −1) i i i b0,i,φ(i) and lim μ∗ (zn ) − μδ (v) μ(a) − μδ (v) we obtain lim μ∗ (zn ) − μδ (v) i bi,φ(i) . We have shown that there is an element w = limn μ∗ (zn ) and there are sequences ai,j , bi,j with ai,j↓ 0, bi,j ↓ 0 (j → +∞) such that for every φ : N → N there are u ∈ Aσ and v ∈ Aδ with v z u and μσ (u) − i ai,φ(i) δ w μ (v) + i bi,φ(i) . Therefore z ∈ L and μ∗ (z) = w = limn μ∗ (zn ). 2
128
G. Barbieri / Fuzzy Sets and Systems 244 (2014) 123–129
An alternative proof of the following proposition is contained in [2]. Proposition 3.13. Let σ (A) be the σ -complete D-lattice generated by A, M(A) the least set over A closed under monotone sequences. Then σ (A) = M(A). Proof. It can be proved in a standard way like in [8, Theorem B]. Since a D-lattice σ -complete is closed under monotone sequences, we have σ (A) ⊆ M(A). The proof will be completed by showing that M(A) is a σ -complete D-lattice. For f ∈ H put K(f ) := {e ∈ H : e ∨ f, e − f, f − e ∈ M(A)}. If K(f ) is not empty, then we claim that it is closed under monotone sequences: If (en ) is an increasing sequence in K(f ) with en ↑ e, then en − f ↑ e − f ; so e − f ∈ M(A) and f − en ↓ f − e so e − f ∈ M(A). If e, f ∈ A, then e ∈ K(f ); since this is true for every e ∈ A, then A ⊆ K(f ). Since M(A) is the smallest set closed under monotone sequences containing A, we have M(A) ⊆ K(f ). If e ∈ M(A) and f ∈ A, then e ∈ K(f ) and therefore f ∈ K(e). Since this is true for every f ∈ A, it follows that M(A) ⊆ K(e). Since this is true for every e ∈ M(A) it follows that M(A) is a D-lattice. A D-lattice which is closed under monotone sequences is a σ -complete D-lattice. 2 Theorem 3.14. Let H be a σ -continuous D-lattice, let A be a sub-D-lattice. Let σ (A) be the σ -complete sub-Dlattice of H generated by A. Then there exists only one measure μ¯ : σ (A) → G that is an extension of the measure μ : A → G. Proof. Existence: Evidently σ (A) = M(A) ⊆ L. Put μ¯ = μ∗ |σ (A) . By Proposition 3.12 and Proposition 3.11 μ¯ is a measure. Uniqueness: Let ν : σ (A) → G be a measure ν|A = μ. ¯ = ν(x)}. Evidently σ (A) ⊇ A, K is closed under limits of monotone sequences. TherePut K = {x ∈ σ (A): μ(x) fore K ⊇ M(A) = σ (A). 2 Corollary 3.15. Let H be a σ -complete MV-algebra, let A be a sub-MV-algebra. Let σ (A) be the σ -complete subMV-algebra of H generated by A. Then there exists only one measure μ¯ : σ (A) → G that is an extension of the measure μ : A → G. Corollary 3.16. Let H be a σ -continuous orthomodular lattice, let A be an orthomodular sublattice. Let σ (A) be the σ -complete orthocomplemented sublattice of H generated by A. Then there exists only one measure μ¯ : σ (A) → G that is an extension of the measure μ : A → G. 4. Conclusion In this paper we have generalized the well-known Caratheodory’s extension theorem to the framework of measures on D-lattices. A question worth asking: Is the same true for a measure defined on a pseudo-effect algebras which is a noncommutative version of a D-lattice [6]? Acknowledgement I would like to express my gratitude to the anonymous referees for their help. References [1] V.N. Aleksjuk, F.D. Beznosikov, Extension of a continuous outer measure on a Boolean algebra, Izv. Vysš. Uˇcebn. Zaved., Mat. 4 (119) (1972) 3–9 (in Russian). [2] A. Avallone, A. de Simone, P. Vitolo, Effect algebras and extensions of measures, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 9 (2) (2006) 423–444. [3] A. Avallone, P. Vitolo, Lyapunov decomposition of measures on effect algebras, Sci. Math. Jpn. (2008) 633–641.
G. Barbieri / Fuzzy Sets and Systems 244 (2014) 123–129
129
[4] L. Drewnowski, Topological rings of sets, continuous set functions, integration. I, Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 20 (1972) 269–276; L. Drewnowski, Topological rings of sets, continuous set functions, integration. II, Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 20 (1972) 277–286. [5] A. Dvureˇcenskij, S. Pulmannová, New Trends in Quantum Structures, Kluwer Academic Publishers, Bratislava, 2000. [6] A. Dvureˇcenskij, T. Vetterlein, Pseudoeffect algebras. I. Basic properties, Int. J. Theor. Phys. 40 (3) (2001) 685–701. [7] L. Epstein, J. Zhang, Subjective probabilities on subjectively unambiguous events, Econometrica 69 (2001) 65–306. [8] P.R. Halmos, Measure Theory, D. Van Nostrand Company, Inc., New York, NY, 1950. [9] P. Pták, S. Pulmannová, A measure-theoretic characterization of Boolean algebras among orthomodular lattices, Comment. Math. Univ. Carol. 35 (1) (1994) 205–208. [10] B. Rieˇcan, The measure extension theorem for subadditive probability measures in orthomodular σ -continuous lattices, Comment. Math. Univ. Carol. 20 (2) (1979) 309–316. [11] B. Rieˇcan, Carathéodory measurability revisited, Tatra Mt. Math. Publ. 34 (II) (2006) 321–332. [12] B. Rieˇcan, T. Neubrunn, Integral, Measure, and Ordering, Mathematics and Its Applications, vol. 411, Kluwer Academic Publishers/Ister Science, Dordrecht/Bratislava, 1997.
Fuzzy Sets and Systems 244 (2014) 123–129 www.elsevier.com/locate/fss
The extension of measures on D-lattices Giuseppina Barbieri Dipartimento di Matematica e Informatica, Università di Udine, Via delle Scienze 206, 33100 Udine, Italy Received 15 July 2011; received in revised form 28 June 2013; accepted 1 July 2013 Available online 5 July 2013
Abstract A lattice ordered group valued measure is extended from a D-lattice into a σ -complete D-lattice. A D-lattice is a lattice with a greatest element 1 and a smallest element 0 endowed with an order-compatible operation, called a difference, which satisfies a list of axioms. The result generalizes the classical result known for measures on Boolean algebras. © 2013 Elsevier B.V. All rights reserved. Keywords: D-lattices; Measures; Lattice ordered groups
1. Introduction
The problem of extensions of measures defined on Boolean algebras to measures defined on σ -complete Boolean algebras has been solved in the real valued case by [1] and [4], in the lattice ordered valued case by [11]. We use and generalize some ideas from [10] in order to prove that a lattice ordered group valued measure can be extended from a D-lattice into a σ -complete D-lattice in a unique way as specified in Theorem 3.14. For this we use some results contained in Chapter 3 of the book by Rieˇcan and Neubrunn [12]. D-lattices are generalization of Boolean algebras. Examples of D-lattices are orthomodular lattices used in Quantum Mechanics and MV-algebras used in Mathematical Logic. Both orthomodular lattices and MV-algebras generalize Boolean algebras, but they are incompatible generalizations, in the sense that an orthomodular lattice need not be an MV-algebra, and conversely. D-lattices also have some applications in Mathematical Economics (see, for instance [7]). Our source of information for D-lattices is the book by Dvureˇcenskij and Pulmannová [5]. The paper is organized as follows: In Section 2 we give some preliminary definitions and stuff which we need in the sequel (namely -groups, D-lattices, measures on D-lattices). In Section 3 we give our results: We start with a sub-D-lattice A of a σ -continuous D-lattice H and a measure defined on A and we extend it to the σ -complete D-lattice σ (A) generated by A.
E-mail address: [email protected]. 0165-0114/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.fss.2013.07.002
124
G. Barbieri / Fuzzy Sets and Systems 244 (2014) 123–129
2. Preliminaries 2.1. -groups An Abelian group (G, +) is called a lattice ordered group G (briefly, -group) if it is a lattice and the following implication holds: a b implies a + c b + c. An -group is called σ -complete if every upper bounded countable subset of G has the supremum in G. An -group is called super Dedekind complete if, for any nonempty subset A bounded from above, there exists a countable subset A∗ such that sup A = sup A∗ . G is weakly σ -distributive if for every bounded double sequence (ai,j ) such that for ai,j ↓ 0 (j → ∞) we have
ai,φ(i) = 0.
φ∈NN i
The symbol xn ↓ x means xn xn+1 and x =
xn .
Theorem 2.1. Let G be a super Dedekind complete -group. Let an,i,j be a bounded sequence such that an,i,j ↓ 0 (j → ∞). Then element a ∈ G there exists a bounded sequence ai,j such that ai,j ↓ 0 (j → ∞) for every positive and a ∧ ( n i an,i,φ(i+n) ) i ai,φ(i) for every φ : N → N. From now on, if not otherwise specified, let G be a super Dedekind complete weakly σ -distributive -group. 2.2. D-lattices and measures Definition 2.2. Let (H, ) be a poset with a smallest element 0 and a greatest element 1 and let be a partial operation on H such that b a is defined if and only if a b and for all a, b, c ∈ H : If a b then b a b and b (b a) = a; If a b c then c b c a and (c a) (c b) = b a. Then (H, , ) is called a difference poset (D-poset for short), or a difference lattice (D-lattice for short) if H is a lattice. One defines in H a partial operation ⊕ as follows: a ⊕ b is defined and
a⊕b=c
if and only if c b is defined and c b = a.
The operation ⊕ is well-defined by the cancellation law [5, page 13] (a b, c and b a = c a imply b = c), and (H, ⊕, 0, 1) is an effect algebra (see [5, Theorem 1.3.4]), that is the following conditions are satisfied for all a, b, c ∈ H : If a ⊕ b is defined, then b ⊕ a is defined and a ⊕ b = b ⊕ a; If b ⊕ c is defined and a ⊕ (b ⊕ c) is defined, then a ⊕ b and (a ⊕ b) ⊕ c are defined, and a ⊕ (b ⊕ c) = (a ⊕ b) ⊕ c; There exists a unique a ⊥ ∈ E such that a ⊕ a ⊥ is defined and a ⊕ a ⊥ = 1; If a ⊕ 1 is defined, then a = 0. We say that two elements a and b of H are orthogonal ifa b⊥ . If a1 , . . . , an ∈ H , we inductively define nk=1 ak = ( n−1 k=1 ak ) ⊕ an provided that the right hand side exists. The definitionis independent of permutations of the elements. We say that a finite sequence {a1 , . . . , an } of H is n n orthogonal if a exists. A sequence (a ) is called orthogonal if k=1 k k=1 ak exists for every n ∈ N. If, moreover, n nn n∈N supn k=1 ak exists, we set an = supn k=1 ak . We give two examples of D-lattices: Definition 2.3. An orthomodular lattice is a lattice with a smallest element 0 and a greatest element 1, with a map ⊥ : H → H , called orthocomplementation with the following properties:
G. Barbieri / Fuzzy Sets and Systems 244 (2014) 123–129
(O1) (O2) (O3) (O4)
125
a ∨ a ⊥ = 1 and a ∧ a ⊥ = 0; a b implies a ⊥ b⊥ ; a ⊥⊥ = a; a b implies b = a ∨ (b ∧ a ⊥ ) for every a, b ∈ H .
A D-lattice is an orthomodular lattice if and only if orthogonal elements are disjoint. Definition 2.4. An MV-algebra (H, +,⊥ ; 0, 1) is a commutative semigroup (H, +) with 0, 1 and a unary operation ⊥ : H → H which satisfies the following axioms: (MV1) (MV2) (MV3) (MV4)
x + 1 = 1; x ⊥⊥ = x; 0⊥ = 1; (x ⊥ + y)⊥ + y = (x + y ⊥ )⊥ + x for every x, y ∈ H .
A D-lattice is an MV-algebra if and only if disjoint elements are orthogonal. Definition 2.5. A D-lattice H is said to be σ -complete if its underlying lattice is σ -complete, i.e., every nonempty countable subset of H has a supremum in H . A σ -complete lattice will be called σ -continuous if xn ↑ x implies xn ∧ y ↑ x ∧ y. From now on H is a σ -continuous D-lattice. We give the following lemma which will be useful in the sequel. Lemma 2.6. Let bn , f ∈ H . Then the following equalities hold: (i) (sup bn ) f = sup(bn f ); (ii) f (sup bn ) = inf(f bn ); (iii) f (inf bn ) = sup(f bn ); provided that the operations are defined. Proof. (i) Obviously, sup(bn f ) (sup bn ) f . Vice versa, let z bn f . Then z ⊕ f bn . Hence z ⊕ f sup bn , whence z (sup bn ) f . Therefore (sup bn ) f is the least upper bound of bn f . (ii) is proved in [3, Lemma 3.7]. (iii) Obviously, f (inf bn ) sup(f bn ). Now let w (f bn ). We get f bn = (f bn ) ∧ f w ∧ f f , which gives f (w ∧ f ) bn , therefore f (w ∧ f ) inf bn . Then we obtain f (inf bn ) w ∧ f w which implies that f (inf bn ) is the least upper bound of f bn . 2 Definition 2.7. A mapping μ : H → G is called a measure if (i) μ(0) = 0; (ii) If a, b ∈ H with a b, then μ(a) μ(b); (iii) If (an ) is an orthogonal sequence and an ∈ H , then μ( an ) = limn ni=1 μ(ai ). A measure is called modular if μ(a ∨ b) + μ(a ∧ b) = μ(a) + μ(b) for every a, b ∈ H .
126
G. Barbieri / Fuzzy Sets and Systems 244 (2014) 123–129
As observed in [9] every subadditive state on an orthomodular lattice (i.e., every function μ such that μ(a ∨ b) μ(a) + μ(b) for every a, b ∈ H and μ(a ∨ b) = μ(a) + μ(b) for every orthogonal element a, b ∈ H ) is a modular function. Every state on an MV-algebra is a modular function. Remark 2.8. The mapping μ is additive, i.e., μ(x ⊕ y) = μ(x) + μ(y) for x y ⊥ if and only if μ(y) = μ(y x) + μ(x) for x y ∈ H . 3. Extension We start with a sub-D-lattice A of a σ -continuous D-lattice H and a measure μ : A → G. We want to extend it to the σ -complete D-lattice σ (A) generated by A. As a byproduct we obtain two versions of the theorem valid for orthomodular lattices and MV-algebras. We begin with a lemma: Lemma 3.1. (See [12, Lemma 3.3.5].) If an , bn ∈ A and an ↑ a and bn ↑ b with a b, then lim μ(an ) lim μ(bn ). Proof. Evidently an ∧ bm ↑ an ∧ b = an . Hence μ(an ) = limm μ(an ∧ bm ) limm μ(bm ). Therefore limn μ(an ) limm μ(bm ). 2 Definition 3.2. (Cf. [12, Definition 3.3.6].) We put Aσ := {b ∈ H : ∃bn ∈ A, bn ↑ b}, Aδ := {c ∈ H : ∃cn ∈ A, cn ↓ c} and define mappings μσ : Aσ → G and μδ : Aδ → G by the formulas μσ (b) = limn μ(bn ), μδ (c) = limn μ(cn ). They are well-defined by Lemma 3.1. Lemma 3.3. The mapping μσ is additive. Proof. Let a, b ∈ Aσ with a ⊥ b. Then there exist an , bn ∈ A with an ↑ a and bn ↑ b. Then an ⊥ bn , since an a and bn b and a ⊥ b. Moreover, we have that an ⊕ bn ∈ A and an ⊕ bn ↑ a ⊕ b. Hence a ⊕ b ∈ Aσ and μσ (a ⊕ b) = lim μ(an ⊕ bn ) = lim μ(an ) + lim μ(bn ) = μσ (a) + μσ (b). 2 Notation 3.4. We put μ∗ (d) = inf{μσ (f ): f d, f ∈ Aσ } for d ∈ H . The mapping μ∗ is an extension of μσ and it is increasing. Lemma 3.5. Let a ∈ H . Then there exists an ∈ Aσ , such that an ↓, an a and μ∗ (a) = lim μ∗ (an ). σ Proof. Let a ∈ H . Then there existσ (bn ) ∈ A and a bounded double sequence (ai,j ) with ai,j ↓ 0 (j → ∞) such that ∗ bn a and μ (a) + ai,φ(i) > μ (bn ). Put an := kn bk . Then an ↓, an ∈ Aσ , an a and μσ (an ) μσ (bn ) μ∗ (a) + ai,φ(i) . Since G is weakly σ -distributive we get lim μ∗ (an ) μ∗ (a). Hence μ∗ (a) = lim μ∗ (an ). 2
Lemma 3.6. If a ∈ σ (A) and b ∈ Aσ , then μ∗ (a ⊕ b) μ∗ (a) + μ∗ (b). Proof. Set M := {b ∈ H : for every a ∈ σ (A) with a ⊥ b, μ∗ (a ⊕ b) μ∗ (a) + μ∗ (b)}. (1) We claim that A ⊆ M: Take a ∈ σ (A) and b ∈ A with a ⊥ b. Then by Lemma 3.5 there exists (an ) in Aσ such that an a and μ∗ (a) = lim μ∗ (an ). Set cn = an ∧ b⊥ . We get cn ∈ Aσ and cn a. Then μ∗ (a) μσ (cn ) μσ (an ), whence lim μσ (cn ) = μ∗ (a). Notice that cn ⊥ b and set bn = cn ⊕ b. By [5, 1.8.7] we have bn ∈ Aσ and bn a ⊕ b. Hence μσ (bn ) μ∗ (a ⊕ b). Then μ∗ (a) + μ∗ (b) = lim μσ (cn ) + μσ (b) lim μσ (bn ). Therefore μ∗ (a ⊕ b) μ∗ (a) + μ∗ (b). (2) We claim that Aσ ⊆ M: Let b ∈ Aσ . Then there exists a sequence bn ∈ A with bn ↑ b. Take an element a ∈ σ (M) orthogonal to b. Since bn b, then a ⊥ bn and by [5, 1.8.7] bn ⊕ a ↑ b ⊕ a. Therefore μ∗ (a ⊕ b) = lim μ∗ (a ⊕ bn ) lim μ(bn ) + μ∗ (a) = μσ (b) + μ∗ (a). 2
G. Barbieri / Fuzzy Sets and Systems 244 (2014) 123–129
127
Proposition 3.7. If b ∈ Aσ , c ∈ Aδ , c b, then μσ (b) μδ (c). Proof. Let cn ∈ A with cn ↓ c. Notice that μ∗ (b) = μσ (b) μ∗ (c) since μ∗ is increasing. It is sufficient to show that μ∗ (c) lim μ(cn ): As c1 cn ↑ c1 c we get μ∗ (c1 c) = μσ (c1 c) = lim μ(c1 cn ) = μ(c1 ) − lim μ(cn ). Now c1 = c ⊕ (c1 c). By 3.6 μ(c1 ) μ∗ (c) + μσ (c1 c). The opposite inequality lim μ(cn ) μ∗ (c) is clear. So μσ (b) μδ (c) = lim μ(cn ). 2 Remark 3.8. Notice that (H, ∧, ∨, +, −) assumes the role of the structure X in [12, 3.3.1 Assumptions] where ∧ and ∨ are the lattice operations, x + y := x ∨ y and x − y := x (x ∧ y) for x, y ∈ H . Thanks to Proposition 3.7 the measure μ : A → G satisfies the assumptions contained [12]. From now on we make use of the following notation: Notation 3.9. (See [12, Definition 3.3.9].) We say that x ∈ L, if there exist w ∈ G, bounded ai,j , bi,j ∈ G, ai,j > 0, φ φ φ φ φ bi,j > 0, ai,j ↓ 0, bi,j ↓ 0 (j → +∞) such that for every φ : N → N there are x1 , x2 ∈ H , x1 x x2 and μσ (x2 ) − δ φ i ai,φ(i) w μ (x1 ) + i bi,φ(i) . Proposition 3.10. (See [12, Proposition 3.3.10].) If x ∈ L, then inf{μσ (x2 ): x2 ∈ Aσ , x2 x} = sup{μδ (x1 ): x1 ∈ Aδ , x1 x}. Proposition 3.11. (See [12, Theorem 3.3.12].) If x, y ∈ L, x y, then y − x ∈ L and μ∗ (y − x) = μ∗ (y) − μ∗ (x). Proof. By the definition of L, there are positive elements ai,j , bi,j in G, and for every φ : N → N elements φ φ φ φ φ φ φ φ φ φ x1 , y1 ∈ Aδ and x2 , y2 ∈ Aσ such that x1 x x2 , y1 y y2 , and μσ (x2 ) − i ai,φ(i) μ∗ (x) μδ (x1 ) + φ φ φ φ φ φ φ σ φ ∗ δ φ δ i bi,φ(i) , μ (y2 ) − i ai,φ(i) μ (y) μ (y1 ) + i bi,φ(i) . Put z1 := y1 − x2 , z2 := y2 − x1 . Then z1 ∈ A , φ φ φ φ φ φ φ z ∈ Aσ , z1 = y1 − x2 y − x y2 − x1 = z2 . We can choose ci,j ↓ 0 (j → +∞) such that i ai,φ(i) + 2 σ φ σ φ δ φ i bi,φ(i) i ci,φ(i) for every φ : N → N. Then μ (z2 ) − i ci,φ(i) μ (y2 ) − μ (x1 ) − i ai,φ(i) − i bi,φ(i) φ φ φ μ∗ (y) − μ∗ (x) μδ (y1 ) − μσ (x2 ) + i ai,φ(i) + i bi,φ(i) μδ (z1 ) + i ci,φ(i) . Therefore y − x ∈ L and μ∗ (y − x) = μ∗ (y) − μ∗ (x). 2 Proposition 3.12. (See [12, Theorem 3.3.13].) Let (zn ) be a monotone sequence in L converging to an element z ∈ H . Then z ∈ L and μ∗ (z) = lim μ∗ (zn ). Proof. By assumptions there is a ∈ Aσ such that a z. By Proposition 3.11 z2 − z1 ∈ L, z3 − z2 ∈ L, . . . , zn − zn−1 ∈ L, . . . , hence there are an,i,j ∈ G, an,i,j↓ 0 (j → σ ∗ σ +∞) such that for every φ : N → N there is yn ∈ A , y1 z1 , yn zn − zn−1 , yn a, μ (z1 ) μ (y1 ) − i a1,i,φ(i) ∗ σ and μ (zn − zn−1 ) μ (yn ) − i a1,i,φ(i) . There exists +∞) a bounded sequence (ai,j ) in G, ai,j∗ ↓ 0 (j → ∗ (z )) ∧ ( ∗ (z ) + a ) a for every φ : N → N. Therefore μ (z ) = μ such that (μ(a) − μ 1 n 1 nn i σ n,i,φ(i+n−1) n i i,φ(i) n n ∗ (z ) − μ∗ (z σ i i−1 )) i=2 i=1 μ (yi ) − k=1 i ak,i,φ(i+k−1) μ (un ) − k=1 i ak,i,φ(i+k−1) where un = (μ σ ∗ ∗ ), hence μσ (un ) − μ∗ (zn ) i ai,φ(i) . a ∧ ni=1 y i . Then μ (un ) − μ (zn ) μ(a) − μ (z1 σ σ Since zn ∈ L there are bn,i,j ↓ 0 (j → Put u = un . Then u ∈ A , u z and μ (u) − i ai,φ(i) limn μ∗ (zn ). +∞) such that for every φ : N → N there are vn ∈ Aδ , vn zn with μδ (vn ) + i bn,i,φ(n+i−1) μ∗ (zn ). Put b0,i,j := limn μ∗ (zn ) − μ∗ (zj ). Now fix j φ(1) (hence b0,i,j b0,i,φ(1) ). − μδ (v)) ∧ ( n i b n,i,φ(i+n−1) ) Put v := vj . By Theorem 2.1 there are bi,j∗ ↓ 0 (j →∗ +∞) such that (μ(a) δ (v) + b for every φ : N → N. Since lim μ (z ) = μ (z ) + b μ b + n j 0,i,j i,φ(i) k,i,φ(i+j −1) i i i b0,i,φ(i) and lim μ∗ (zn ) − μδ (v) μ(a) − μδ (v) we obtain lim μ∗ (zn ) − μδ (v) i bi,φ(i) . We have shown that there is an element w = limn μ∗ (zn ) and there are sequences ai,j , bi,j with ai,j↓ 0, bi,j ↓ 0 (j → +∞) such that for every φ : N → N there are u ∈ Aσ and v ∈ Aδ with v z u and μσ (u) − i ai,φ(i) δ w μ (v) + i bi,φ(i) . Therefore z ∈ L and μ∗ (z) = w = limn μ∗ (zn ). 2
128
G. Barbieri / Fuzzy Sets and Systems 244 (2014) 123–129
An alternative proof of the following proposition is contained in [2]. Proposition 3.13. Let σ (A) be the σ -complete D-lattice generated by A, M(A) the least set over A closed under monotone sequences. Then σ (A) = M(A). Proof. It can be proved in a standard way like in [8, Theorem B]. Since a D-lattice σ -complete is closed under monotone sequences, we have σ (A) ⊆ M(A). The proof will be completed by showing that M(A) is a σ -complete D-lattice. For f ∈ H put K(f ) := {e ∈ H : e ∨ f, e − f, f − e ∈ M(A)}. If K(f ) is not empty, then we claim that it is closed under monotone sequences: If (en ) is an increasing sequence in K(f ) with en ↑ e, then en − f ↑ e − f ; so e − f ∈ M(A) and f − en ↓ f − e so e − f ∈ M(A). If e, f ∈ A, then e ∈ K(f ); since this is true for every e ∈ A, then A ⊆ K(f ). Since M(A) is the smallest set closed under monotone sequences containing A, we have M(A) ⊆ K(f ). If e ∈ M(A) and f ∈ A, then e ∈ K(f ) and therefore f ∈ K(e). Since this is true for every f ∈ A, it follows that M(A) ⊆ K(e). Since this is true for every e ∈ M(A) it follows that M(A) is a D-lattice. A D-lattice which is closed under monotone sequences is a σ -complete D-lattice. 2 Theorem 3.14. Let H be a σ -continuous D-lattice, let A be a sub-D-lattice. Let σ (A) be the σ -complete sub-Dlattice of H generated by A. Then there exists only one measure μ¯ : σ (A) → G that is an extension of the measure μ : A → G. Proof. Existence: Evidently σ (A) = M(A) ⊆ L. Put μ¯ = μ∗ |σ (A) . By Proposition 3.12 and Proposition 3.11 μ¯ is a measure. Uniqueness: Let ν : σ (A) → G be a measure ν|A = μ. ¯ = ν(x)}. Evidently σ (A) ⊇ A, K is closed under limits of monotone sequences. TherePut K = {x ∈ σ (A): μ(x) fore K ⊇ M(A) = σ (A). 2 Corollary 3.15. Let H be a σ -complete MV-algebra, let A be a sub-MV-algebra. Let σ (A) be the σ -complete subMV-algebra of H generated by A. Then there exists only one measure μ¯ : σ (A) → G that is an extension of the measure μ : A → G. Corollary 3.16. Let H be a σ -continuous orthomodular lattice, let A be an orthomodular sublattice. Let σ (A) be the σ -complete orthocomplemented sublattice of H generated by A. Then there exists only one measure μ¯ : σ (A) → G that is an extension of the measure μ : A → G. 4. Conclusion In this paper we have generalized the well-known Caratheodory’s extension theorem to the framework of measures on D-lattices. A question worth asking: Is the same true for a measure defined on a pseudo-effect algebras which is a noncommutative version of a D-lattice [6]? Acknowledgement I would like to express my gratitude to the anonymous referees for their help. References [1] V.N. Aleksjuk, F.D. Beznosikov, Extension of a continuous outer measure on a Boolean algebra, Izv. Vysš. Uˇcebn. Zaved., Mat. 4 (119) (1972) 3–9 (in Russian). [2] A. Avallone, A. de Simone, P. Vitolo, Effect algebras and extensions of measures, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 9 (2) (2006) 423–444. [3] A. Avallone, P. Vitolo, Lyapunov decomposition of measures on effect algebras, Sci. Math. Jpn. (2008) 633–641.
G. Barbieri / Fuzzy Sets and Systems 244 (2014) 123–129
129
[4] L. Drewnowski, Topological rings of sets, continuous set functions, integration. I, Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 20 (1972) 269–276; L. Drewnowski, Topological rings of sets, continuous set functions, integration. II, Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 20 (1972) 277–286. [5] A. Dvureˇcenskij, S. Pulmannová, New Trends in Quantum Structures, Kluwer Academic Publishers, Bratislava, 2000. [6] A. Dvureˇcenskij, T. Vetterlein, Pseudoeffect algebras. I. Basic properties, Int. J. Theor. Phys. 40 (3) (2001) 685–701. [7] L. Epstein, J. Zhang, Subjective probabilities on subjectively unambiguous events, Econometrica 69 (2001) 65–306. [8] P.R. Halmos, Measure Theory, D. Van Nostrand Company, Inc., New York, NY, 1950. [9] P. Pták, S. Pulmannová, A measure-theoretic characterization of Boolean algebras among orthomodular lattices, Comment. Math. Univ. Carol. 35 (1) (1994) 205–208. [10] B. Rieˇcan, The measure extension theorem for subadditive probability measures in orthomodular σ -continuous lattices, Comment. Math. Univ. Carol. 20 (2) (1979) 309–316. [11] B. Rieˇcan, Carathéodory measurability revisited, Tatra Mt. Math. Publ. 34 (II) (2006) 321–332. [12] B. Rieˇcan, T. Neubrunn, Integral, Measure, and Ordering, Mathematics and Its Applications, vol. 411, Kluwer Academic Publishers/Ister Science, Dordrecht/Bratislava, 1997.