Elsevier

Fuzzy Sets and Systems

Volume 339, 15 May 2018, Pages 51-61
Fuzzy Sets and Systems

Meager projections in orthocomplete homogeneous effect algebras

https://doi.org/10.1016/j.fss.2017.12.005Get rights and content

Abstract

We define meager projections on homogeneous effect algebras and discuss their properties. As an application, we prove that, if E is an orthocomplete homogeneous effect algebra such that S(E) is lattice-ordered, then E is a lattice effect algebra, which gives an affirmative answer to an open problem stated by Mesiar and Stupňanová.

Introduction

In the nineties of the twentieth century, two equivalent quantum structures, D-posets [13] and effect algebras [3] were extensively studied, which were considered as “unsharp” generalizations of the structures which arise in quantum mechanics (orthomodular lattices, orthomodular posets, orthoalgebras) incorporating some fuzzy logics (MV-algebras).

Effect algebras are partially ordered by a natural way, and if they are lattices then they are called lattice effect algebras. The set of all sharp elements of a lattice effect algebra E is an orthomodular lattice, being a sub-effect algebra and a sublattice of E [7]. In [8], a new class of effect algebras, called homogeneous effect algebras, was introduced. The class of homogeneous effect algebras includes orthoalgebras, effect algebras satisfying the Riesz decomposition property and lattice effect algebras. The set of all sharp elements of a homogeneous effect algebra is its sub-effect algebra [8].

Below we present Problem 10.1 from [14], which was posed by Jenča during The Twelfth International Conference on Fuzzy Set Theory and Applications 2014.

Problem 1.1

([14]) Prove or disprove: If E is an orthocomplete homogeneous effect algebra such that S(E) is lattice-ordered, then E is a lattice effect algebra.

In this paper, we give an affirmative answer to this problem using the properties related to meager projections on orthocomplete homogeneous effect algebras.

Section snippets

Preliminaries and basic facts

Effect algebras have been introduced by Foulis and Bennett to study the foundations of quantum mechanics [3]. An effect algebra is a partial algebra (E;,0,1) with a binary partial operation ⊕ and two nullary operations 0, 1, satisfying the following conditions for any x,y,zE:

  • (E1)

    if xy is defined, then yx is defined and xy=yx,

  • (E2)

    if xy and (xy)z are defined, then yz and x(yz) are defined and (xy)z=x(yz),

  • (E3)

    for every xE there is a unique xE such that xx exists and xx=1,

  • (E4)

    if x1 is

Operation rules

Lemma 3.1

(Foulis and Bennett [3, Theorem 2.5]) Let E be an effect algebra and x,y,zE.

  • (i)

    xy=xz implies y=z,

  • (ii)

    xyxz implies yz.

Lemma 3.2

(Jenča and Pulmannová [9, Lemma 3.1]) Let E be an effect algebra, with x,yE and c,dC(E).

  • (i)

    If xy exists, then c(xy)=(cx)(cy).

  • (ii)

    If cd exists, then x(cd)=(xc)(xd).

Lemma 3.3

(Niederle and Paseka [16, Lemma 1.16]) Let E be a sharply dominating effect algebra and let xE. Then xx=xx=xx.

For convenience, we denote xN=xx. It follows that xNM(E) and xN=xM=xx in

Meager projections

Definition 4.1

Let E be an orthocomplete homogeneous effect algebra and x,yE. Denote π(x,y)=xMxMyM. We call π(x,y) the meager projection of x on y.

Let E be an orthocomplete homogeneous effect algebra and x,yE. It follows from Lemma 3.6 that xMyM exists. Moreover, we have xMxMyM exists from Lemma 3.9, and thus the meager projection is well defined. It is obvious that π(x,y) is a meager element and π(x,y)xM.

Let E be an orthocomplete atomic homogeneous effect algebra and xE. The atomic decomposition

Concluding remarks

We have introduced and discussed meager projections on homogeneous effect algebras. By the properties of meager projections, we prove that, if E is an orthocomplete homogeneous effect algebra such that S(E) is lattice-ordered, then E is a lattice effect algebra. It follows that E is lattice-ordered if and only if S(E) is lattice-ordered. We hope this study will provide a new method to study effect algebras and related structures.

Acknowledgements

The research was supported by National Natural Science Foundation of China (Grant No. 11401128) and Doctoral Starting up Foundation of Guilin University of Technology. The author is highly grateful to the editors and the anonymous referees for their valuable comments and suggestions.

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