A generalization of set-difference

DOI: 10.2478/s12175-011-0051-0 Math. Slovaca 61 (2011), No. 6, 835–850 A GENERALIZATION OF SET-DIFFERENCE Paolo Vitolo (Communicated by Anatolij Dvurečenskij ) ABSTRACT. A structure called d0 -algebra is defined and studied, which leads to a new description of D-lattices and generalized D-lattices by means of only one totally defined operation. c
2011 Mathematical Institute Slovak Academy of Sciences Introduction Since the first years of 20th century, algebras of sets have been generalized to Boolean algebras, mainly for applications in Measure Theory and Probabilty. Later on, due also to the developments of Quantum Mechanics, orthomodular lattices were introduced. Independently, MV-algebras were defined in order to investigate suitable extensions of the classical structures of 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. More recenty (see [2, 3] or the book [4]), D-lattices (i.e. lattice-ordered effect algebras) have been introduced, which are a common generalization of orthomodular lattices and MV-algebras, and hence of Boolean algebras, too. These structures have application also in Mathematical Economics (see, for instance [5]). The structure of D-lattice consists of an order relation and a partially defined difference related to each other; the equivalent structure of a lattice-ordered effect algebra has a partially defined sum and an order relation induced by it, but one has to require that the structure be a lattice in the induced order: thus both approaches are somewhat unsatisfactory from an algebraic point of view. 2010 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n: Primary 03G05; Secondary 03G25, 81P10. K e y w o r d s: set-difference, Boolean algebra, D-lattice, D-poset, generalized D-lattice, congruence, ideal. PAOLO VITOLO The idea of this article originates from the aim of describing D-lattices by making use of just one (totally defined) operation—called difference, and denoted by “\”—and a list of axioms. This, in fact, leads quite naturally to a further generalization of the structure of a D-lattice, which is called d0 -algebra. The reader should note that a description of D-lattices with a totally defined operation has been already done in [4, Theor. 1.8.11]. In the present article, we choose a different operation, which has the advantage of allowing description of structures without a greatest element, such as generalized D-lattices [4, §1.8]. We also stress the fact that the order relation (and hence the semilattice or lattice structure) can be derived from the operation. The paper is organized as follows: In the first section we introduce d0 -algebras, and establish basic properties. In the second section we investigate the correspondence between congruences and ideals, thus extending some results of [1]. Next, in §3, we study the ordering relation. Section 4 deals with another difference, denoted by “−”, and the relationship between the two operations. In the final section the definition of D-lattice and generalized D-lattice are recalled, and these structures are characterized as d0 -algebras satisying some additional assumptions. 1. Definition of d0 -algebra First of all, we define the structure we deal with throughout the paper.
 A d0 -algebra is a nonempty set A, with a distiguished element 1.1. 0 and a binary operation \, called difference, such that the following properties hold, for all a, b, c ∈ A: (d1) a \ 0 = a;
 (d2) a \ (b \ a) \ a = 0;

 (d3) a \ (a \ b) \ (b \ a) = b \ (b \ a) \ (a \ b) ; (d4) (a \ b) \ (c \ b) = (a \ c) \ (b \ c);



 (d5) a \ (a \ b) = b\ (b\ a) \ (b\ a) \ b and (a \ b) \ a = (b\ a) \ b \ b\ (b\ a) . It is not difficult to see that this is in fact a generalization of set-difference, i.e. if one takes as A any nonempty collection of sets closed under set-difference, then, by interpreting the symbol “\” just as set-difference and 0 as the empty set, all the above-listed axioms are satisfied. Actually we will prove a more general statement later on (see Theorem 5.6 below). 836 A GENERALIZATION OF SET-DIFFERENCE 1.2.     In a d0 -algebra A the following hold, for all a, b, c ∈ A: (i) 0 \ a = 0 and a \ a = 0; (ii) a \ b = b \ a =⇒ a = b; (iii) a \ b = 0 =⇒ (a \ c) \ (b \ c) = 0; (iv) a \ b = 0 =⇒ (b \ a) \ b = 0 and b \ (b \ a) = a;
 (v) (a \ b) \ b \ (a \ b) = 0. P r o o f.
 (i) By (d1) and (d2), we have 0 \ a = 0 \ (a \ 0) \ 0 = 0. This implies in particular that (a
 \ 0) \
(0 \ a) = a. Itfollows, applying (d3), that a \ a = a \ (a \ 0) \ (0 \ a) = 0 \ (0 \ a) \ (a \ 0) = 0. (ii) Indeed, if a \ b = b \ a then (a \ b) \ (b \ a) = (b \ a) \ (a \ b) = 0. Hence, from (d3), we get a = b. (iii) By (d4), we have (a \ c) \ (b \ c) = (a \ b) \ (c \ b) = 0 \ (c \ b) = 0. (iv) Applying the second equality in (d5) (with a and b interchanged) and the first equality in (i), we get



 (b \ a) \ b = (a \ b) \ a \ a \ (a \ b) = (0 \ a) \ a \ (a \ b) = 0 \ a \ (a \ b) = 0. Next, from the first equality in (d5), it follows that

 b \ (b \ a) = a \ (a \ b) \ (a \ b) \ a) = (a \ 0) \ (0 \ a) = a. (v) Applying first the second equality in (d5) (with a \ b in place of a), and then (d2) (with a and b interchanged), we get


  

 (a \ b) \ b \ (a \ b) = b \ (a \ b) \ b \ b \ b \ (a \ b)

 = 0 \ b \ b \ (a \ b) = 0.
1.3.     In a d0 -algebra A the following holds, for all a, b, c ∈ A: a \ b = b \ c = 0 =⇒ a \ c = 0. P r o o f. It follows immediately from Proposition 1.2(iii) and (d1).
Besides the rather easy properties established above, the following consequences of the axioms will be useful in the sequel. 1.4.     In a d0 -algebra A the following holds, for all a, b ∈ A:
 (a \ b) \ (b \ a) \ a = 0. 837 PAOLO VITOLO P r o o f. Applying (d2), (d4), the first equality in (d5), then Proposition 1.2(v) and (d4) twice and, finally, (d2) again, we have:
 (a \ b) \ (b \ a) \ a

 

  = (a \ b) \ (b \ a) \ a \ a \ (b \ a) \ a




 = (a \ b) \ (b \ a) \ a \ (b \ a) \ a \ a \ (b \ a)


 = (a \ b) \ a \ (b \ a) \ a 



 \ (b \ a) \ (b \ a) \ a \ (b \ a) \ a \ (b \ a)




 = (a \ b) \ a \ (b \ a) \ a \ (b \ a) \ (b \ a) \ a



   = (a \ b) \ a \ (b \ a) \ (b \ a) \ a \ (b \ a)

  = (a \ b) \ a \ (b \ a)

  = (a \ b) \ b \ (a \ b) = 0.
In a d0 -algebra A the following hold, for all a, b ∈ A:

  a \ (a \ b) \ (b \ a) \ a = 0 and a \ a \ (a \ b) \ (b \ a) = (a \ b) \ (b \ a). 1.5.
   
P r o o f. By Proposition 1.4, we can apply Proposition 1.2(iv) (with a in place of b and (a \ b) \ (b \ a) in place of a).
1.6. In a d0 -algebra A the following holds, for all a, b ∈ A: 
 (a \ b) \ (b \ a) \ (b \ a) \ (a \ b) = a \ b.    

 P r o o f. Let c = a\ (a\b)\(b\a) . Applying the second equality in Corollary
 1.5, we get (a \ b) \ (b \ a) = a \ c. Since by (d3) we have b \ (b \ a) \ (a \ b) = c, arguing similarly as above we also get (b \ a) \ (a \ b) = b \ c. Hence, by (d4),

 (a \ b) \ (b \ a) \ (b \ a) \ (a \ b) = (a \ c) \ (b \ c) = (a \ b) \ (c \ b) = a \ b where the last equality follows because c \ b = 0 (apply the first equality in Corollary 1.5 with a and b interchanged).
1.7.     In a d0 -algebra A the following holds, for all a, b, c ∈ A: a \ c = b \ c =⇒ (c \ b) \ (c \ a) = (a \ b) \ (b \ a). P r o o f. By (d4) we have (b \ a) \ (c \ a) = (b \ c) \ (a \ c) = 0 and, similarly, (a \ b) \ (c \ b) = 0. Hence, from Proposition 1.2(iv), we obtain

 (c \ a) \ (b \ a) \ (c \ a) = 0, (c \ b) \ (a \ b) \ (c \ b) = 0 (1.1) 838 A GENERALIZATION OF SET-DIFFERENCE and
 (c \ a) \ (c \ a) \ (b \ a) = b \ a,
 (c \ b) \ (c \ b) \ (a \ b) = a \ b. Now observe that (c \ b) \ (a \ b) = (c \ a) \ (b \ a), by (d4). Thus, applying (d4) again and taking (1.1) into account, it follows that



 (a \ b) \ (b \ a) = (c \ b) \ (c \ b) \ (a \ b) \ (c \ a) \ (c \ a) \ (b \ a)



 = (c \ b) \ (c \ a) \ (b \ a) \ (c \ a) \ (c \ a) \ (b \ a) 



= (c \ b) \ (c \ a) \ (c \ a) \ (b \ a) \ c \ a))
 = (c \ b) \ (c \ a) \ 0 = (c \ b) \ (c \ a).
2. Congruences and ideals We define ideals of d0 -algebras, in order to establish a canonical one-to-one correspondence between congruences and ideals. Of course, congruences are defined in the usual way.
 2.1. A congruence of a d0 -algebra A is an equivalence relation ∼ on A such that, for every a, b, c, d ∈ A
 (a ∼ b & c ∼ d) =⇒ a \ c ∼ b \ d. 2.2. An ideal of a d0 -algebra A is a nonempty subset I ⊆ A such that, for every a, b ∈ A: (I1) If a ∈ I then a \ b ∈ I; (I2) If a ∈ I and b \ a ∈ I then b ∈ I. One easily sees that every ideal must contain 0 (just put b = a into (I1) and recall Proposition 1.2(i)).     2.3. Let ∼ be a congruence of a d0 -algebra A, and let I be the ∼-equivalence class of 0. Then I is an ideal of A. P r o o f. Let a ∈ I and b ∈ A; then a\b ∼ 0\b = 0, which gives (I1). Suppose now that b\ a ∈ I;
from we have just proved it follows in particular that (b\ a) \ (a \ b) ∈ I and a \ (a \ b) \ (b \ a) ∈ I: hence, by (d1) and (d3),

 b = b \ 0 ∼ b \ (b \ a) \ (a \ b) = a \ (a \ b) \ (b \ a) ∈ I.
Now we construct the relation associated to an ideal.
 2.4. Let I be an ideal of a d0 -algebra A. The relation associated to I on A, denoted by ∼I , is defined as follows: ∀a, b ∈ A : a ∼I b ⇐⇒ (a \ b ∈ I & b \ a ∈ I). 839 PAOLO VITOLO In order to prove the main theorem of this section, we need a preliminary result. 2.5. 
 Let a, b be elements of a d0 -algebra A, with a \ b = 0. Then, for every c ∈ A, we have



 (c \ b) \ (c \ a) = (b \ a) \ (c \ a) \ (b \ a) \ (b \ a) \ (b \ a) \ (c \ a) and
 


(c \ a) \ (c \ b) = (b \ a) \ (b \ a) \ (c \ a) \ (b \ a) \ (c \ a) \ (b \ a) . P r o o f. Applying (d1), then (d4) (with a and c interchanged), and finally the second equality in (d5) (with c \ a in place of a and b \ a in place of b), we have
 (c \ b) \ (c \ a) = (c \ b) \ (a \ b) \ (c \ a)
 = (c \ a) \ (b \ a) \ (c \ a)



 = (b \ a) \ (c \ a) \ (b \ a) \ (b \ a) \ (b \ a) \ (c \ a) . Similarly, applying (d1), then (d4) (with a and c interchanged), and finally the first equality in (d5) (with c \ a in place of a and b \ a in place of b), we have
 (c \ a) \ (c \ b) = (c \ a) \ (c \ b) \ (a \ b)
 = (c \ a) \ (c \ a) \ (b \ a)



 = (b \ a) \ (b \ a) \ (c \ a) \ (b \ a) \ (c \ a) \ (b \ a) .
2.6. 

 Let A be a d0 -algebra. For every ideal I of A, the relation ∼I (see Definition 2.4) is a congruence of A, whose equivalence class of 0 coincides with I. Conversely, given any congruence ∼ of A, its equivalence class of 0 is an ideal, whose associated relation is ∼. P r o o f. Given an ideal I, we first prove that ∼I is an equivalence relation. By Proposition 1.2(i), for every a ∈ A we have a \ a = 0 ∈ I, i.e. a ∼I a. Symmetry of ∼I follows immediately from its definition. To see that ∼I is transitive, consider a, b, c ∈ I with a ∼I b and b ∼I c: by (d4) and (I1) we have (a \ c) \ (b \ c) = (a \ b) \ (c \ b) ∈ I, hence a \ c ∈ I by (I2); similarly one gets c \ a ∈ I, so that a ∼I c. To complete the proof that ∼I is a congruence, is suffices to show that a ∼I b implies a\c ∼I b\c and c\a ∼I c\b for every c ∈ A. To get the former implication note that we have (a \ c) \ (b \ c) = (a \ b) \ (c \ b) ∈ I by (d4) and (I1), and similarly (b \ c) \ (a \ c) ∈ I. To get the latter implication, consider first the case in which a \ b = 0: applying Lemma 2.5 and taking (I1) into account, we have both (c \ b) \ (c \ a) ∈ I and (c \ a) \ (c \ b) ∈ I, whence c \ a ∼I c \ b. Now we obtain the general case if we can find d ∈ A such that d \ a = d \ b = 0 and 840 A GENERALIZATION OF SET-DIFFERENCE
 a ∼I d ∼I b. This is accomplished by setting d = a \ (a \ b) \ (b \ a) . Indeed, from Corollary 1.5, we get d
\ a = 0 ∈ I and a \ d = (a \ b) \ (b \ a) ∈ I. Since by (d3) d is also equal to b \ (b \ a) \ (a \ b) , in the same way as above we get d \ b = 0 ∈ I and b \ d = (b \ a) \ (a \ b) ∈ I. Finally, from (d1) and Definition 2.4 it follows that a ∼I 0 if and only if a ∈ I, i.e. the ∼I -equivalence class of 0 coincides with I. Conversely, let ∼ be a congruence, and let I be the ∼-equivalence class of 0: by Proposition 2.3, I is an ideal; hence ∼I is a congruence by the arguments above. It remains to prove that ∼ and ∼I coincide. Suppose first that a ∼ b: then a\b ∈ I and b\a ∈ I so that a ∼I b. Conversely, if a ∼I b, then both a \ b and b \ a belong to I: hence (a \ b) \ (b \ a) ∼ 0 and (b \ a) \ (a \ b) ∼ 0, too; therefore, applying (d1) and (d3), we have

 a = a \ 0 ∼ a \ (a \ b) \ (b \ a) = b \ (b \ a) \ (a \ b) ∼ b \ 0 = b.
3. The ordering relation In this section we are going to see that d0 -algebras have a canonical ordering relation. 3.1. 

 In a d0 -algebra A, letting for every a, b ∈ A a ≤ b ⇐⇒ a \ b = 0 defines a partial order, and 0 is the least element of A with respect to this order. P r o o f. Reflexivity follows from the second equality in Proposition 1.2(i), antisymmetry from Proposition 1.2(ii), and transitivity from Corollary 1.3. The fact that 0 is the least element is an immediate consequence of the first equality in Proposition 1.2(i).
The notation a ≤ b, with a and b elements of a d0 -algebra A, will be used— from now on—always to denote the partial order introduced above (and we could also write, equivalently, b ≥ a). With this notation most of the previous formulas could be rewritten in a slightly simpler form. For instance, Proposition 1.2(iii) says that a ≤ b implies a \ c ≤ b \ c for every a, b, c ∈ A. In the sequel such a simpler form will be tacitly used most of the times. 3.2.     If I is an ideal of a d0 -algebra A, then for every a ∈ I and every b ∈ A with b ≤ a one has b ∈ I. P r o o f. Let a ∈ I, b ∈ A, and suppose that b ≤ a. Applying Proposition 1.2(iv), we get a \ (a \ b) = b. Hence b ∈ I by (I1).
841 PAOLO VITOLO
 3.3. A unit of a d0 -algebra A is an element u ∈ A such that a \ u = 0 for every a ∈ A. A unit of a d0 -algebra A, if it exists, is always unique (as follows from Proposition 1.2(ii)) and is the greatest element of A. In the sequel it will be always denoted by 1.
 3.4. We say that a d0 -algebra A is directed if for every a, b ∈ A there exists c ∈ A such that a \ c = b \ c = 0 In other words a d0 -algebra A is directed if and only if (A, ≥) is a directed set in the usual sense. A d0 -algebra with unit is trivially directed. 3.5. Example. The d0 -algebra V1 consisting of the set {0, a, b}, with a \ b = b and b \ a = a (see Table 1) is not directed. Table 1. The Cayley table of the d0 -algebra V1 introduced in Example 3.5 \ 0 a b 0 0 a b a 0 0 a b 0 b 0 4. The second-order difference In this section we are going to introduce an operation on a d0 -algebra, which is to be considered as another difference.
 4.1. In a d0 -algebra A, the second-order difference is the operation −, defined as follows: 4.2.     ∀a, b ∈ A : a − b = (a \ b) \ (b \ a) (4.1) In a d0 -algebra A, the following hold, for all a, b ∈ A: (i) a ≥ b =⇒ a − b = a \ b; (ii) a ≤ b ⇐⇒ a − b = 0. P r o o f. (1) It follows immediately from the definition of −. (2) If a \ b = 0 then a − b = (a \ b) \ (b \ a) = 0 \ (b \ a) = 0. Conversely, if a − b = 0, since by Corollary 1.5 we have b \ (b − a) ≤ b, applying (d3) we obtain a = a \ 0 = a \ (a − b) = b \ (b − a) ≤ b. 842
A GENERALIZATION OF SET-DIFFERENCE In the sequel we will tacitly substitute an expression like (a \ b) \ (b \ a) with its simpler form a − b in all formulas where we find it convenient. For example the statement of Proposition 1.6 would be rewritten as (a \ b) − (b \ a) = a \ b or, equivalently, (a − b) \ (b − a) = a \ b. 4.3.     In a d0 -algebra A, the following holds, for all a, b ∈ A: (a − b) − (b − a) = a − b. P r o o f. Apply Proposition 1.6 with a \ b in place of a and b \ a in place of b.

4.4. We say that a d0 -algebra has the Riesz property if the operations \ and − coincide. 4.5. Example. The d0 -algebra V2 consisting of the set {0, a, b}, with a \ b = a and b \ a = b (see Table 2) has the Riesz property. On the other hand, the d0 -algebra V1 of Example 3.5 does not have the Riesz property. Note that the difference of V2 coincides with the second-order difference of V1 . Table 2. The Cayley table of the d0 -algebra V2 introduced in Example 4.5 \ 0 a b
 4.6. a, b, c ∈ A, 0 0 a b a 0 0 b b 0 a 0 We say that a d0 -algebra A is monotone if, for every b ≤ c =⇒ a − c ≤ a − b. In other words, A is monotone if and only if, for every a ∈ A, the mapping x → a − x is decreasing. 4.7. 

 Every directed d0-algebra is monotone. P r o o f. Let A be a directed d0 -algebra, and let a, b, c ∈ A with b ≤ c. Take m ∈ A such that m ≥ a and m ≥ c (i.e. a \ m = b \ m = c \ m = 0). Applying Proposition 1.7 with m in place of c, we get (m \ b) \ (m \ a) = a − b; similarly (m \ c) \ (m \ a) = a − c and (m \ c) \ (m \ b) = b − c = 0. 843 PAOLO VITOLO Hence, applying (d4),

 (a − c) \ (a − b) = (m \ c) \ (m \ a) \ (m \ b) \ (m \ a)

 = (m \ c) \ (m \ b) \ (m \ a) \ (m \ b)
 = 0 \ (m \ a) \ (m \ b) = 0.
4.8. 
 In a d0 -algebra A the following hold, for all a, b ∈ A:
 a \ (a − b) ≤ a, a \ a \ (a − b) = a − b and a \ (a − b) ≤ b. P r o o f. Let i = a \ (a − b): by Corollary 1.5 we have i ≤ a and a \ i = a − b; similarly, since i = b \ (b − a) by (d3), we also have i ≤ b.
4.9. 

 Every monotone d0 -algebra A is a semilattice, where ∀a, b ∈ A : a ∧ b = a \ (a − b). (4.2) P r o o f. Let i = a \ (a − b): by the previous lemma we have i ≤ a, a \ i = a − b and i ≤ b. Now let c ∈ A and suppose that c ≤ a and c ≤ b: we show that c ≤ i, too, which completes the proof. Note that c \ a = i \ a = 0; applying first Proposition 1.7 (with a in place of c, i in place of b and c in place of a), then Proposition 4.2(i) and monotonicity, we have c − i = (a \ i) \ (a \ c) = (a − b) \ (a − c) = 0 which means that c ≤ i, by virtue of Proposition 4.2(ii).
4.10.     Every directed d0 -algebra is a semilattice. P r o o f. It follows immediately from Theorem 4.7 and Theorem 4.9.
Note that, by Proposition 1.4 and Proposition 4.2(i), the equality in (4.2) could be rewritten as a ∧ b = a − (a − b). 4.11.     In a monotone d0 -algebra A, we have the following ∀a, b ∈ A : a − b = a \ (a ∧ b) = a − (a ∧ b). P r o o f. Since by Proposition 1.4, a − b ≤ a, we may apply Proposition 1.2(iv) (with
a − b in place of a and a in place of b) and we obtain a \ (a ∧ b) = a \ a \ (a − b) = a − b. The other equality follows form Proposition 4.2(i).
Clearly, in a d0 -algebra A, an equation of the form a∧b=0 (4.3) makes sense, even if A is not a semilattice. As usual, if a, b ∈ A satisfy (4.3), we say that the are disjoint. 844 A GENERALIZATION OF SET-DIFFERENCE 4.12.     In a d0 -algebra A, for every a, b ∈ A we have the following implication: a ∧ b = 0 =⇒ a − b = a. Moreover, if A is monotone, the converse implication also holds. P r o o f. Suppose that a ∧ b = 0. By Lemma 4.8, we have a \ (a − b) = 0, i.e. a ≤ a − b; by Proposition 1.4 we have a − b ≤ a, too, hence equality follows. Conversely, if a − b = a, applying Proposition 1.2(iv) (with a ∧ b in place of a and a in place of b) and Corollary 4.11, we obtain
 a ∧ b = a \ a \ (a ∧ b) = a \ (a − b) = 0.
4.13. Example. Let Y1 and Y2 be the d0 -algebras whose underlying set is {0, a, b, c} and whose operations are given by the tables in Table 3. Both Y1 and Y2 do not verify (4.2) (though they are semilattices) and hence they are not monotone. In particular Y2 shows that even a d0 -algebra with the Riesz property need not be monotone. Table 3. Two non-monotone d0 -algebras on the same set Y1 : \ 0 a b c 0 0 a b c a 0 0 a 0 b 0 b 0 0 c 0 c c 0 Y2 : \ 0 a b c 0 0 a b c a 0 0 b 0 b 0 a 0 0 c 0 c c 0 4.14. 
 In a monotone d0 -algebra A the following implication holds, for all a, b ∈ A: a \ b ≤ a =⇒ a \ b ≥ a − b. P r o o f. Suppose that a \ b ≤ a, and put c = a \ (a \ b). By Proposition 1.2(iv) (with a \ b in place of a and a in place of b) we have
 c = a \ (a \ b) ≤ a and a \ c = a \ a \ (a \ b) = a \ b. (4.4) Now, applying the first equality in (d5), we get c = a \ (a \ b) = b − (b \ a). Hence, by Proposition 1.4, one obtains that c ≤ b, too, and therefore c ≤ a ∧ b (here Theorem 4.9 is taken into account). Applying the second equality in (4.4), Proposition 4.2(i), then monotonicity and Corollary 4.11, we have a \ b = a \ c = a − c ≥ a − (a ∧ b) = a − b.
845 PAOLO VITOLO     4.15. In a monotone d0 -algebra A the following equivalence holds, for all a, b ∈ A: a \ b ≤ a ⇐⇒ a \ b = a − b. P r o o f. The implication ⇐= follows from Proposition 1.4. So, suppose that a \ b ≤ a: by the previous lemma we have (a \ b) \ (b \ a) = a − b ≤ a \ b. Now apply the lemma again, with a \ b in place of a and b \ a in place of b, and recalling Proposition 1.6, we have a − b = (a \ b) \ (b \ a) ≥ (a \ b) − (b \ a) = a \ b. Hence equality follows.
In view of the above result, a monotone d0 -algebra A has the Riesz property if and only if a \ b ≤ a for every a, b ∈ A. In fact the monotonicity assumption is unnecessary, as we are going to see. 4.16.     A d0 -algebra A has the Riesz property if and only if ∀a, b ∈ A : a \ b ≤ a. (4.5) P r o o f. To show the only nontrivial implication, suppose that (4.5) holds, and take any a, b ∈ A. Applying (4.5) (with a \ b in place of a and b \ a in place of b), we get a − b = (a \ b) \ (b \ a) ≤ a \ b. (4.6) On the other hand, by Proposition 1.6 and (4.5) (with a − b in place of a and b − a in place of b), we also have a \ b = (a − b) \ (b − a) ≤ a − b, whence the equality.
5. D-posets and D-lattices
 5.1. A poset with difference is a structure of the form (P, ≤, ⊖), where P is nonempty set, ≤ is a partial orded on P and ⊖ a partially defined operation on P such that the following conditions are satisfied for all a, b, c ∈ P : (DP1) a ⊖ b is defined if and only if b ≤ a; (DP2) If a ≤ b then b ⊖ a ≤ b and b ⊖ (b ⊖ a) = a; (DP3) If a ≤ b ≤ c then c ⊖ b ≤ c ⊖ a and (c ⊖ a) ⊖ (c ⊖ b) = b ⊖ a. Note that if a poset with difference (P, ≤, ⊖) has the smallest element, denoted by 0 as usual, then a ⊖ a = 0 for each a ∈ P . 846 A GENERALIZATION OF SET-DIFFERENCE     5.2. all a, b, c ∈ P : In a poset with difference (P, ≤, ⊖), the following hold, for (i) If P has the smallest element 0, then a ⊖ 0 = a; (ii) If P has the smallest element 0, and b ≤ a, then a ⊖ b = 0 ⇐⇒ a = b; (iii) If c ≤ b ≤ a then b ⊖ c ≤ a ⊖ c and (a ⊖ c) ⊖ (b ⊖ c) = a ⊖ b. P r o o f. (i) Recall that 0 = a ⊖ a and apply (DP2). (ii) One implication is immediate. Conversely, assume b ≤ a and a ⊖ b = 0; then a = a ⊖ 0 = a ⊖ (a ⊖ b) = b by (i) and (DP2). (iii) See [4, Prop. 1.1.2(i)].


 Every d0-algebra A can be viewed as a poset with difference by 5.3. letting, ∀a, b ∈ A such that b ≤ a, define a ⊖ b = a \ b. (5.1) P r o o f. Indeed, (DP1) holds by construction and (DP2) by Proposition 1.2(iv). To complete the proof, take a, b, c ∈ A with a ≤ b ≤ c, and apply Proposition 1.7: we have (c ⊖ b) \ (c ⊖ a) = (c \ b) \ (c \ a) = a − b = 0, i.e. c ⊖ b ≤ c ⊖ a; now interchange the roles of a and b and apply Proposition 1.7 again: we have (taking Proposition 4.2(i) into account) (c ⊖ a) ⊖ (c ⊖ b) = (c \ a) \ (c \ b) = b − a = b \ a = b ⊖ a.

5.4. A poset with difference is called D-poset if it has (the smallest and) the greatest element. A poset with difference which is also a lattice is called a generalized D-lattice. A D-poset which is also a lattice is called a D-lattice. Observe that a generalized D-lattice must have the smallest element.     5.5. all a, b, c ∈ P : In a generalized D-lattice (P, ≤, ⊖), the following hold, for (i) If a ≥ b ∨ c then a ⊖ (b ∧ c) = (a ⊖ b) ∨ (a ⊖ c); (ii) If a ≥ b ∨ c then a ⊖ (b ∨ c) = (a ⊖ b) ∧ (a ⊖ c); (iii) If a ≤ b ∧ c then (b ∧ c) ⊖ a = (b ⊖ a) ∧ (c ⊖ a); (iv) If a ≤ b ∧ c then (b ∨ c) ⊖ a = (b ⊖ a) ∨ (c ⊖ a). P r o o f. See [4, Prop. 1.8.2 and Prop. 1.8.4].
In view of Theorem 5.3 and Definition 5.4, it is apparent that a d0 -algebra A which is a lattice in its canonical ordering can be viewed as a generalized D-lattice, defining ⊖ as in (5.1). The following crucial result shows that in fact that each generalized D-lattice can be canonically identified with a directed d0 -algebra. 847 PAOLO VITOLO 5.6. 

 Every directed d0 -algebra A can be viewed as a generalized D-lattice, by defining ⊖ as in (5.1). Conversely if (A, ≤, ⊖) is a generalized D-lattice, then letting ∀a, b ∈ A : a \ b = (a ∨ b) ⊖ b (5.2) defines a (directed) d0 -algebra whose canonical order coincides with ≤. Hence this d0 -algebra, when viewed as a poset with difference, gives back (A, ≤, ⊖). P r o o f. Suppose that A is a directed d0 -algebra; we already know (see Corollary 4.10 above) that A is a semilattice, hence it suffices to prove that a ∨ b exists for every a, b ∈ A. Actually, given a, b ∈ A, we show that
 a ∨ b = m \ (m \ a) ∧ (m \ b) , (5.3) where m ∈ A is such that m ≥ a and m ≥ b.
 Fix m ∈ A with m ≥ a and m ≥ b and let s = m \ (m \ a) ∧ (m \ b) . Since (m \ a) ∧ (m \ b) ≤ m \ a, applying Proposition 1.2(iv) and Proposition 1.7, we have a − s = (m \ s) \ (m \ a) 

 = m \ m \ (m \ a) ∧ (m \ b) \ (m \ a)
 = (m \ a) ∧ (m \ b) \ (m \ a) = 0 so that a ≤ s; similarly one sees that b ≤ s. Now let c ∈ a with a ≤ c and b ≤ c: we prove that s ≤ c. Set d = c ∧ m; clearly it suffices to show that s ≤ d. Since a ≤ d and d ≤ m, by Proposition 1.7 it follows that (m \ d) \ (m \ a) = a − d = 0, i.e. m \ d ≤ m \ a; analogously one sees that m \ d ≤ m \ b, and therefore m \ d ≤ (m \ a) ∧ (m \ b). Finally, applying Proposition 1.7 again, we obtain s − d = (m \ d) \ (m \ s) 

 = (m \ d) \ m \ m \ (m \ a) ∧ (m \ b)
 = (m \ d) \ (m \ a) ∧ (m \ b) = 0, thus (by virtue of Proposition 4.2(ii)) we have s ≤ d, as claimed. Conversely let (A, ≤, ⊖) be a generalized D-lattice. Let us begin by observing that ∀a, b ∈ A : (a ∨ b) ⊖ b = 0 ⇐⇒ a ≤ b. (5.4) Define the operation \ as in (5.2): we show that all the conditions listed in Definition 1.1 are satisfied. 848 A GENERALIZATION OF SET-DIFFERENCE Condition (d1) follows readily from Proposition 5.2(i). To prove (d2), that in the light of (5.4) can be rewritten as a \ (b \ a) ≤ a, we proceed as follows:


 a \ (b \ a) = a ∨ (b ∨ a) ⊖ a ⊖ (b ∨ a) ⊖ a


 ≤ (b ∨ a) ∨ (b ∨ a) ⊖ a ⊖ (b ∨ a) ⊖ a
 = (b ∨ a) ⊖ (b ∨ a) ⊖ a = a, where the inequality is a consequence of Proposition 5.2(iii). In order to simplify the other verifications, we note that ∀a, b ∈ A : b ≤ a =⇒ a \ b = a ⊖ b. (5.5) Define the operation − as in (4.1): we claim that ∀a, b ∈ A : a − b = a ⊖ (a ∧ b). (5.6) Indeed, applying Proposition 5.5(i) and Proposition 5.2(iii), we obtain



 (a \ b) \ (b \ a) = (a ∨ b) ⊖ b ∨ (b ∨ a) ⊖ a ⊖ (b ∨ a) ⊖ a

 = (a ∨ b) ⊖ (a ∧ b) ⊖ (a ∨ b) ⊖ a = a ⊖ (a ∧ b).

 Since
by (DP3)we have a ⊖ a ⊖ (a ∧ b) = a ∧ b, it follows that a ⊖ a ⊖ (a ∧ b) = b ⊖ b ⊖ (b ∧ a) and this, in view of (5.5) and (5.6), is equivalent to (d3). Now observe that, by Proposition 5.5(iv) and Proposition 5.2(iii), we have



 (a \ b) \ (c \ b) = (a ∨ b) ⊖ b ∨ (c ∨ b) ⊖ b ⊖ (c ∨ b) ⊖ b


  = (a ∨ b) ∨ (c ∨ b) ⊖ b ⊖ (c ∨ b) ⊖ b
 = (a ∨ b) ∨ (c ∨ b ) ⊖ (c ∨ b) = (a ∨ b ∨ c) ⊖ (b ∨ c) and therefore (a \ c) \ (b \ c) = (a ∨ b ∨ c) ⊖ (b ∨ c), too, so that we get (d4). Finally, let us prove (d5). Recalling the definition of − and (5.6), we obtain the first equality as follows:

 b \ (b \ a) \ (b \ a) \ b = b − (b \ a)

  = b ⊖ (a ∨ b) ⊖ a ∧ b 


 = b ⊖ (a ∨ b) ⊖ a ∧ (a ∨ b) ⊖ (a ∨ b) ⊖ b 

 = b ⊖ (a ∨ b) ⊖ a ∨ (a ∨ b) ⊖ b



   = (a ∨ b) ⊖ (a ∨ b) ⊖ b ⊖ (a ∨ b) ⊖ a ∨ (a ∨ b) ⊖ b


 = a ∨ (a ∨ b) ⊖ b ⊖ (a ∨ b) ⊖ b = a \ (a \ b), 849 PAOLO VITOLO where we have applied (DP2), Proposition 5.5(ii), then (DP2) again and (DP3). Similarly, we obtain the second equality in (d5) as follows:

 (b \ a) \ b \ b \ (b \ a) = (b \ a) − b


 = (a ∨ b) ⊖ a ⊖ b ∧ (a ∨ b) ⊖ a


 
= (a ∨ b) ⊖ a ⊖ (a ∨ b) ⊖ (a ∨ b) ⊖ b ∧ (a ∨ b) ⊖ a 


  = (a ∨ b) ⊖ a ⊖ (a ∨ b) ⊖ (a ∨ b) ⊖ b ∨ a

  = (a ∨ b) ⊖ b ∨ a ⊖ a = (a \ b) \ a. This completes the proof that A is a d0 -algebra. The canonical order of this d0 -algebra, by virtue of (5.4), coincides with the order ≤ of the given generalized D-lattice (thus, in particular, A is directed).
5.7.     Every d0 -algebra with unit can be viewed as a D-lattice. Conversely, every D-lattice, with the difference defined as in (5.2), is a d0 -algebra (with unit). P r o o f. Straightforward consequence of the previous theorem.
REFERENCES [1] AVALLONE, A.—VITOLO, P.: Congruences and ideals of effect algebras, Order 20 (2003), 67–77. [2] BENNET, M. K.—FOULIS, D. J.: Effect algebras and unsharp quantum logics, Found. Phys. 24 (1994), 1331–1352. [3] CHOVANEC, F.—KÔPKA, F.: D-posets, Math. Slovaca 44 (1994), 21–34. [4] DVUREČENSKIJ, A.—PULMANNOVÁ, S.: New Trends in Quantum Structures, Kluwer Academic Publishers, Dordrecht, 2000. [5] EPSTEIN, L. G.—ZHANG, J.: Subjective probabilities on subjectively unambiguous events, Econometrica 69 (2001), 265–306. Received 6. 4. 2009 Accepted 3. 10. 2009 850 Dipartimento di Matematica e Informatica Università della Basilicata viale Ateneo Lucano 10 Contrada Macchia Romana I–85100 Potenza ITALY E-mail : [email protected]