Chinese Journal of MathematicsVolume 2013 (2013), Article ID 570496, 4 pageshttp://dx.doi.org/10.1155/2013/570496

Research Article

A Note of Filters in Effect Algebras

Biao Long Meng¹ and Xiao Long Xin²

¹Department of Mathematics, Xi’an University of Science and Technology, Xi’an 710054, China²Department of Mathematics, Northwest University, Xi’an 710127, China

Received 29 July 2013; Accepted 23 September 2013

Academic Editors: Z.-Y. Li and C.-s. Liu

Copyright © 2013 Biao Long Meng and Xiao Long Xin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We investigate relations of the two classes of filters in effect algebras (resp., MV-algebras). We prove that a lattice filter in a lattice ordered effect algebra (resp., MV-algebra) does not need to be an effect algebra filter (resp., MV-filter). In general, in MV-algebras, every MV-filter is also a lattice filter. Every lattice filter in a lattice ordered effect algebra is an effect algebra filter if and only if is an orthomodular lattice. Every lattice filter in an MV-algebra is an MV-filter if and only if is a Boolean algebra.

1. Introduction

The notion of effect algebras has been introduced by Foulis and Bennett [1] as an algebraic structure providing an instrument for studying quantum effects that may be unsharp. One of D-posets has been introduced by Chovanec and Kôpka [2]. The two notions are categorically equivalent. Many results with respect to effect algebras and D-posets have been obtained (see [1–6]). A comprehensive introduction about effect algebras can been found in the monograph [7]. The filter theory of effect algebras is an important objects of investigation (see [3, 4, 8, 9]). It is well known that a lattice ordered effect algebra contains both a lattice structure and an effect algebra structure; hence, the notions of lattice filters and effect algebra filters are investigated, respectively. One would ask: what relations are there between lattice filters and effect algebra filters? In this paper we discuss this problem in a lattice ordered effect algebra (resp., an MV-algebra). A lattice filter in a lattice ordered effect algebra does not need to be an effect algebra filter (resp., MV-filter). In general, lattice filter in a lattice ordered effect algebra is an effect algebra filter if and only if is an orthomodular lattice. A lattice filter in an MV-algebra is an MV-filter if and only if is a Boolean algebra.

Definition 1 (see [1]). An effect algebra is an algebraic structure , where is a nonempty set, and are distinct elements of , and is a partial binary operation on that satisfies the following conditions.(E1) Commutative Law. If is defined, then is defined and .(E2) Associative Law. If and are defined, then and are defined and .(E3) Orthosupplementation Law. For any , there is a unique , such that .(E4) Zero-One Law. If is defined, then .

When the hypotheses of (E2) are satisfied, we write for element in .

For simplicity, we use the notation for an effect algebra. If is defined, we write and whenever we write we are implicitly assuming that . A partial ordering on an effect algebra is defined by if and only if there is a , such that . Such an element is unique (if it exists) and is denoted by . Then is a difference poset (D-poset, for short) [2]. In this case we note . It is known that for any , and implies .

Definition 2 (see [4]). An orthoalgebra is an algebraic structure satisfies (E1)–(E3) and the following condition. Consistency Law. If is defined, then .

An orthoalgebra is always an effect algebra. An effect algebra (orthoalgebra, resp.) with lattice order is called a lattice ordered effect algebra (lattice ordered orthoalgebra, resp.).

Let be an orthoalgebra. If and there are and , then we denote and . Then the map is an orthocomplementation on the bounded poset (), that is, the following conditions hold: for all ,(1),(2),(3),(4).

Definition 3 (see [5]). An orthocomplementation lattice is called an orthomodular lattice if it satisfies the orthomodular Law:

Lemma 4. In an effect algebra , if and only if .

Lemma 5. A lattice ordered orthoalgebra is just an orthomodular lattice.

Hence, in an orthomudular lattice, the complementation operation in a lattice is the same as the orthosupplement operation in an effect algebra.

2. Filters in Lattice Effect Algebras

Definition 6 (see [3]). Let be an effect algebra. In the terms of the effect algebra operation , a partial operation can be defined as follows: for any

The following assertions are obvious and the proofs are omitted.

Proposition 7. Let be an effect algebra. Then for any : (i) If exists, then exists and .(ii) If and exist, then and exist and .(iii) For any there is a , such that and .(iv) If exists, then .(v) For any , exists and .(vi) exists if and only if .(vii) If is a lattice ordered effect algebra, then for all with , we have .

Definition 8 (see [9]). Let be an effect algebra and be a nonempty subset of . Then is called an effect algebra filter on if for all with ,   if and only if . An effect algebra filter of is called to be proper if .

Obviously, an effect algebra filter of is proper if and only if .

Definition 9 (see [4]). Let be a lattice. A nonempty subset of is called a lattice filter of if for all if and only if .

Proposition 10. Let be an effect algebra. A nonempty subset of is an effect algebra filter of if and only if satisfies: (EF ₁) if and , then .(EF ₂) For with , then and imply .

Proof. Let be an effect algebra filter of . If and , then . So . (EF₁) holds. Suppose that and , . Then , and . (EF₂) holds.Conversely suppose satisfies (EF₁) and (EF₂). Let with . Since and , by (EF₂) we obtain . On the other hand, if with and , then by and (EF₁) we have . Hence, be an effect algebra filter of .

Proposition 11. Let be a lattice. A nonempty subset of is a lattice filter of if and only if satisfies the following. (LF ₁) If and , then .(LF ₂) If , then .

Proof. Let be a lattice filter of . If and , then . So , (LF₁) holds. (LF₂) holds obviously.Conversely suppose satisfies (LF₁) and (LF₂). Let . By (LF₂) we obtain . On the other hand, if with , then by and (LF₁) we have . Hence, is a lattice filter of .

It is worth noting that in a lattice ordered effect algebra, a lattice filter does not need to be an effect algebra filter.

Example 12. Let with , , , , and , . The order relations are as the following picture (3)Then is a lattice ordered effect algebra. is a lattice filter of , but is not an effect algebra filter of because , By the way we point out that in the example, the orthosupplement of in the effect algebra is and the orthocomplement of in the lattice is ; they are different.

In order to make a lattice filter of also being an effect algebra filter of , we must add stronger conditions on .

Theorem 13. Let be a lattice ordered effect algebra. Then the following conditions are equivalent.  (a) Every lattice filter is an effect algebra filter. (b) For all with , . (c) is a lattice ordered orthoalgebra.

Proof. (a) ⇒ (b). Assume (a) and with . Since is a lattice filter of and , by the assumption is also an effect algebra filter of , it follows that , and so . Thus, . (b) holds.(b) ⇒ (c). Assume (b) and is any element of . By (b) and Proposition 7(v), we have . By Part (iii) of Proposition   in [7], is an orthoalgebra. (c) holds.(c) ⇒ (a) Suppose is an orthoalgebra and is any lattice filter of . If with , then . Because , it follows from Part (ii) of Proposition   in [7] that , and so . Therefore, . Conversely if , by and (LF₁), we have . This prove that is an effect algebra filter of . (a) holds.

By Lemma 5 and Theorem 13 we have the following.

Corollary 14. Let be a lattice ordered effect algebra. Then every lattice filtar is an effect algebra filter if and only if is an orthomodular lattice.

3. Filters in MV-Algebras

In 1959, Chang [10] introduced MV-algebras, which play an important role in many valued logic. An MV-algebra is a very important special example of an effect algebra (equivalently, D-poset). An algebraic structure is called an MV-algebra if is a nonempty set, and are distinct elements of , + is a total binary operation on , and is a unary operation on satisfying(MV1), (MV2), (MV3),(MV4),(MV5),(MV6),(MV7),(MV8).

In an MV-algebra, we can define total operations , −, , and as follows: for any , ; ; ; ; if and only if . Thus, is a Boolean D-poset ([7, Theorem ]). An MV-algebra is a bounded distributive lattice with respect to and . Hence, Definition 8 still applies to MV-algebras. The details on MV-algebras can be found in [11]. It is easy to prove the following.

In an MV-algebra , for any .

Definition 15. Let be an MV-algebra and be a nonempty subset of . Then is called an MV-filter on if for all , if and only if .

Proposition 16. Let be an MV-algebra. A nonempty subset of is an MV-filter of if and only if satisfies (MF ₁) if and , then .(MF ₂) For , then and imply .

Proof. It is analogous to Proposition 10 and omitted.

Observe that in Definition 15 (resp., in Proposition 16), we do not require .

Proposition 17. Let be an MV-algebra. If is an MV-filter of , then is also a lattice filter of .

Proof. Suppose is an MV-filter of . It follows from Proposition 16 that satisfies (LF₁). For any we have Let . Then by (MF₁). By using (MF₂) and , we obtain . (LF₂) holds. From Proposition 11 it follows that is a lattice filter of .

Even in an MV-algebra, a lattice filter does not need to be an MV-filter.

Example 18. Let . The Hasse diagram, the tables of operations − and + are as follows: (5)It is easy to check that is an MV-algebra and is a lattice filter of , but is not an effect algebra filter of because and , while . That is, does not satisfy (MF₂).

In what follows we give an interesting result, which is another main conclusion in this paper.

Proposition 19. Let be an MV-algebra. Then the following assertions are equivalent. (i) Every lattice filter of is an MV-filter of .(ii) For any , .(iii) is a Boolean algebra.

Proof. (i) ⇒ (ii). Assume (i) and . Since is a lattice filter of and , by (MF₂) we have . Hence, . (ii) holds.(ii) ⇒ (iii). Assume (ii) and . Then by De Morgan Law and (MV8). is a complemented distributive lattice, that is, a Boolean algebra. (iii) holds.(iii) ⇒ (i). Suppose that is a Boolean algebra and is any lattice filter of . It is obvious that satisfies (MF₁). Now let and , by (LF₂) . Because by (MF₁) we have . satisfies (MF₂). This shows that is an MV-filter filter of . (i) holds.

4. Conclusion

Ideals and filters play prominent roles in the study of effect algebras. There are various notions of ideals and filters in great literature on effect algebras. The relations among these notions are extensively investigated. It is known that in an effect algebra, lattice filters and effect algebra filters are two important notions. In this paper we show that a lattice filter in a lattice ordered effect algebra is not an effect algebra filter (resp., MV-filter). In general, a lattice filter in a lattice ordered effect algebra is an effect algebra filter if and only if is an orthomodular lattice. A lattice filter in an MV-algebra is an MV-filter if and only if is a Boolean algebra. We will deeply work in this aspect.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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