CJM Chinese Journal of Mathematics 2314-8071 Hindawi Publishing Corporation 570496 10.1155/2013/570496 570496 Research Article A Note of Filters in Effect Algebras http://orcid.org/0000-0002-8447-6051 Meng Biao Long 1 Xin Xiao Long 2 Li Z.-Y. Liu C.-s. 1 Department of Mathematics Xi’an University of Science and Technology Xi’an 710054 China xaut.edu.cn 2 Department of Mathematics Northwest University Xi’an 710127 China nwu.edu.cn 2013 10 11 2013 2013 29 07 2013 23 09 2013 2013 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.

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) E 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 E is an effect algebra filter if and only if E is an orthomodular lattice. Every lattice filter in an MV-algebra E is an MV-filter if and only if E is a Boolean algebra.

1. Introduction

The notion of effect algebras has been introduced by Foulis and Bennett  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 . The two notions are categorically equivalent. Many results with respect to effect algebras and D-posets have been obtained (see ). A comprehensive introduction about effect algebras can been found in the monograph . 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 E (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 E is an effect algebra filter if and only if E is an orthomodular lattice. A lattice filter in an MV-algebra E is an MV-filter if and only if E is a Boolean algebra.

Definition 1 (see [<xref ref-type="bibr" rid="B7">1</xref>]).

An effect algebra is an algebraic structure (E;,0,1), where E is a nonempty set, 0 and 1 are distinct elements of E, and is a partial binary operation on E that satisfies the following conditions.

Commutative Law. If ab is defined, then ba is defined and ab=ba.

Associative Law. If ab and (ab)c are defined, then bc and a(bc) are defined and (ab)c=a(bc).

Orthosupplementation Law. For any aE, there is a unique aE, such that aa=1.

Zero-One Law. If a1 is defined, then a=0.

When the hypotheses of (E2) are satisfied, we write abc for element (ab)c=a(bc) in E.

For simplicity, we use the notation E for an effect algebra. If ab is defined, we write ab and whenever we write ab we are implicitly assuming that ab. A partial ordering on an effect algebra is defined by ab if and only if there is a cE, such that ac=b. Such an element c is unique (if it exists) and is denoted by ba. Then (E;,0,1) is a difference poset (D-poset, for short) . In this case we note a:=1a. It is known that for any aE, a′′:=(a)=a and ab implies ba.

Definition 2 (see [<xref ref-type="bibr" rid="B6">4</xref>]).

An orthoalgebra is an algebraic structure (E;,0,1) satisfies (E1)–(E3) and the following condition.

Consistency Law. If aa is defined, then a=0.

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 E be an orthoalgebra. If a,bE and there are l.u.d{a,b} and g.l.d{a,b}, then we denote ab:=l.u.d{a,b} and ab:=g.l.d{a,b}. Then the map :aa is an orthocomplementation on the bounded poset (E;), that is, the following conditions hold: for all a,bE,

a′′:=(a)=a,

abba,

aa=0,

aa=1.

Definition 3 (see [<xref ref-type="bibr" rid="B8">5</xref>]).

An orthocomplementation lattice (E;,,,0,1) is called an orthomodular lattice if it satisfies the orthomodular Law: (1)for  anya,bE,abimpliesb=a(ab)or  equivalently  b=a(ba).

Lemma 4.

In an effect algebra E, ab if and only if ab.

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 [<xref ref-type="bibr" rid="B3">3</xref>]).

Let (E;,0,1) be an effect algebra. In the terms of the effect algebra operation , a partial operation can be defined as follows: for any a,bE(2)ab  exists  if  and  only  if  ab,ab=def(ab).

The following assertions are obvious and the proofs are omitted.

Proposition 7.

Let E be an effect algebra. Then for any a,b,cE:

If ab exists, then ba exists and ab=ba.

If bc and a(bc) exist, then ab and (ab)c exist and a(bc)=(ab)c.

For any a there is a b, such that ab=0 and b=a.

If 0a exists, then a=1.

For any aE, aa exists and aa=0.

ab exists if and only if ab.

If E is a lattice ordered effect algebra, then for all a,bE with ab, we have abab.

Definition 8 (see [<xref ref-type="bibr" rid="B11">9</xref>]).

Let (E;,0,1) be an effect algebra and F be a nonempty subset of E. Then F is called an effect algebra filter on E if for all a,bE with ab,  a,bF if and only if abF. An effect algebra filter F of E is called to be proper if FE.

Obviously, an effect algebra filter F of E is proper if and only if 0F.

Definition 9 (see [<xref ref-type="bibr" rid="B6">4</xref>]).

Let (L;,,0,1) be a lattice. A nonempty subset F of L is called a lattice filter of L if for all a,bL,a,bF if and only if abF.

Proposition 10.

Let (E;,0,1) be an effect algebra. A nonempty subset F of E is an effect algebra filter of E if and only if F satisfies:

if aF and ab, then bF.

For a,bE with ba, then (ab)F and aF imply bF.

Proof.

Let F be an effect algebra filter of E. If aF and ab, then (ab)b=((ab)b)=aF. So bF. (EF1) holds. Suppose that ba and (ab)F, aF. Then (ba)F, and b=((ba)a)=(ab)aF. (EF2) holds.

Conversely suppose F satisfies (EF1) and (EF2). Let a,bF with ab. Since (b(ab))=((ab)b)=((ab)b)=aF and bF, by (EF2) we obtain abI. On the other hand, if a,bE with ab and abF, then by aba,b and (EF1) we have a,bF. Hence, F be an effect algebra filter of E.

Proposition 11.

Let (L;,,0,1) be a lattice. A nonempty subset F of L is a lattice filter of L if and only if F satisfies the following.

If aF and ab, then bF.

If a,bF, then abF.

Proof.

Let F be a lattice filter of L. If aF and ab, then ab=aF. So bI, (LF1) holds. (LF2) holds obviously.

Conversely suppose F satisfies (LF1) and (LF2). Let a,bF. By (LF2) we obtain abF. On the other hand, if a,bE with abF, then by aba,b and (LF1) we have a,bF. Hence, F is a lattice filter of L.

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 E={0,a,b,1} with ab, a=a, b=b, 1=aa=bb=ab, and ab, x0=x  (xE). The order relations are as the following picture (3)

Then (E;,0,1) is a lattice ordered effect algebra. F={a,1} is a lattice filter of E, but F is not an effect algebra filter of E because aF, aa=(aa)=1=0F.

By the way we point out that in the example, the orthosupplement of a in the effect algebra is a and the orthocomplement of a in the lattice is b; they are different.

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

Theorem 13.

Let (E;,0,1) be a lattice ordered effect algebra. Then the following conditions are equivalent.

(a) Every lattice filter is an effect algebra filter.

(b) For all a,bE with ab, ab=ab.

(c) E is a lattice ordered orthoalgebra.

Proof.

(a) (b). Assume (a) and a,bE with ab. Since [ab,1] is a lattice filter of E and a,b[ab,1], by the assumption [ab,1] is also an effect algebra filter of E, it follows that ab[ab,1], and so abab. Thus, ab=ab. (b) holds.

(b) (c). Assume (b) and a is any element of E. By (b) and Proposition 7(v), we have aa=aa=0. By Part (iii) of Proposition  1.5.3 in , E is an orthoalgebra. (c) holds.

(c) (a) Suppose E is an orthoalgebra and F is any lattice filter of E. If a,bF with ab, then abF. Because (ab)=abab=(ab), it follows from Part (ii) of Proposition  1.5.3 in  that (ab)=(ab), and so ab=ab. Therefore, abF. Conversely if abF, by aba,b and (LF1), we have a,bF. This prove that F is an effect algebra filter of E. (a) holds.

By Lemma 5 and Theorem 13 we have the following.

Corollary 14.

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

3. Filters in MV-Algebras

In 1959, Chang  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 (E;  +,,0,1) is called an MV-algebra if E is a nonempty set, 0 and 1 are distinct elements of E, + is a total binary operation on E, and is a unary operation on E satisfying

(a+b)+c=a(b+c),

a+0=a,

a+b=b+a,

a+1=1,

a′′:=(a)=a,

0=1,

a+a=1,

a+(a+b)=b+(b+a).

In an MV-algebra, we can define total operations ·, −, , and as follows: for any a,bE, a·b:=(a+b); ab:=(a+b)·b; ab:=(a·b)+b; b-a:=(a+b); ab if and only if a=ab. Thus, (E;,-,0,1) is a Boolean D-poset ([7, Theorem 1.8.16]). 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 . It is easy to prove the following.

In an MV-algebra E, a-b=a-(ab) for any a,bE.

Definition 15.

Let (E;+,0,1) be an MV-algebra and F be a nonempty subset of E. Then F is called an MV-filter on E if for all a,bE, a,bF if and only if a·bF.

Proposition 16.

Let (E;+,0,1) be an MV-algebra. A nonempty subset F of E is an MV-filter of E if and only if F satisfies

if aF and ab, then bF.

For a,bE, then (a-b)F and aF imply bF.

Proof.

It is analogous to Proposition 10 and omitted.

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

Proposition 17.

Let (E;+,0,1) be an MV-algebra. If F is an MV-filter of E, then F is also a lattice filter of E.

Proof.

Suppose F is an MV-filter of E. It follows from Proposition 16 that F satisfies (LF1). For any a,bE we have (4)a=(1-a)(b-a)=(b(ab)). Let a,bF. Then (b-(ab))F by (MF1). By using (MF2) and bF, we obtain abF. (LF2) holds. From Proposition 11 it follows that F is a lattice filter of E.

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

Example 18.

Let E={0,a,b,1}. The Hasse diagram, the tables of operations − and + are as follows: (5)

It is easy to check that E is an MV-algebra and F={b,1} is a lattice filter of E, but F is not an effect algebra filter of E because (b-a)=a=bF and bF, while aF. That is, F does not satisfy (MF2).

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

Proposition 19.

Let (E;+,0,1) be an MV-algebra. Then the following assertions are equivalent.

Every lattice filter of E is an MV-filter of E.

For any a, aa=aa.

E is a Boolean algebra.

Proof.

(i) (ii). Assume (i) and aE. Since F:=[aa,1] is a lattice filter of E and aaa,a, by (MF2) we have aaF. Hence, aaaa. (ii) holds.

(ii) (iii). Assume (ii) and aE. Then aa=(aa)=(aa)=a+a=1 by De Morgan Law and (MV8). E is a complemented distributive lattice, that is, a Boolean algebra. (iii) holds.

(iii) (i). Suppose that E is a Boolean algebra and F is any lattice filter of E. It is obvious that F satisfies (MF1). Now let (a-b)F and aF, by (LF2) (a-b)aF. Because (6)(a-b)a=((a-b)a)((a-b)(a-b))=((aa)-b)=(1-b)=b, by (MF1) we have bF. F satisfies (MF2). This shows that F is an MV-filter filter of E. (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 E is an effect algebra filter if and only if E is an orthomodular lattice. A lattice filter in an MV-algebra E is an MV-filter if and only if E 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.

Foulis D. J. Bennett M. K. Effect algebras and unsharp quantum logics Foundations of Physics 1994 24 10 1331 1352 10.1007/BF02283036 MR1304942 ZBL1213.06004 Chovanec F. Kôpka F. D -lattices International Journal of Theoretical Physics 1995 34 8 1297 1302 10.1007/BF00676241 MR1353674 Chovanec F. Rybáriková E. Ideals and filters in D-posets International Journal of Theoretical Physics 1998 37 1 17 22 10.1023/A:1026648803063 MR1637144 Foulis D. J. Greechie R. J. Rüttimann G. T. Filters and supports in orthoalgebras International Journal of Theoretical Physics 1992 31 5 789 807 10.1007/BF00678545 MR1162623 ZBL0764.03026 Jenča G. Pulmannová S. Ideals and quotients in lattice ordered effect algebras Soft Computing 2001 5 5 376 380 10.1007/s005000100139 Ma Z. Note on ideals of effect algebras Information Sciences 2009 179 5 505 507 10.1016/j.ins.2008.07.018 MR2490189 ZBL1166.03037 Dvurečenskij A. Pulmannová S. New Trends in Quantum Structures 2000 516 Dordrecht, The Netherlands Kluwer Academic Publishers xvi+541 Mathematics and Its Applications MR1861369 Liu D. L. Wang G. J. Fuzzy filters in effect algebras Fuzzy Systems and Mathematics 2009 23 3 6 16 MR2547361 ZBL1264.06016 Wang L. Zhou X. N. Prime filters, congruences and quotients in effect algebras Mathematical Theory and Applications 2007 27 1 78 81 MR2343182 Chang C. C. A new proof of the completeness of the Łukasiewicz axioms Transactions of the American Mathematical Society 1959 93 74 80 MR0122718 ZBL0093.01104 Cignoli R. L. O. D'Ottaviano I. M. L. Mundici D. Algebraic Foundations of Many-Valued Reasoning 2000 7 Dordrecht, The Netherlands Kluwer Academic Publishers x+231 Trends in Logic—Studia Logica Library MR1786097