Meager projections in orthocomplete homogeneous effect algebras
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 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 with a binary partial operation ⊕ and two nullary operations 0, 1, satisfying the following conditions for any :
- (E1)
if is defined, then is defined and ,
- (E2)
if and are defined, then and are defined and ,
- (E3)
for every there is a unique such that exists and ,
- (E4)
if is
Operation rules
Lemma 3.1 (Foulis and Bennett [3, Theorem 2.5]) Let E be an effect algebra and . implies , implies .
Lemma 3.2 (Jenča and Pulmannová [9, Lemma 3.1]) Let E be an effect algebra, with and . If exists, then . If exists, then .
Lemma 3.3 (Niederle and Paseka [16, Lemma 1.16]) Let E be a sharply dominating effect algebra and let . Then .
For convenience, we denote . It follows that and in
Meager projections
Definition 4.1 Let E be an orthocomplete homogeneous effect algebra and . Denote . We call the meager projection of x on y.
Let E be an orthocomplete homogeneous effect algebra and . It follows from Lemma 3.6 that exists. Moreover, we have exists from Lemma 3.9, and thus the meager projection is well defined. It is obvious that is a meager element and .
Let E be an orthocomplete atomic homogeneous effect algebra and . 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 is lattice-ordered, then E is a lattice effect algebra. It follows that E is lattice-ordered if and only if 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.
References (19)
Characterization of homogeneity in orthocomplete atomic effect algebras
Fuzzy Sets Syst.
(2014)- et al.
Open problems from the 12th International Conference on Fuzzy Set Theory and Its Applications
Fuzzy Sets Syst.
(2015) - et al.
Homogeneous orthocomplete effect algebras are covered by MV-algebras
Fuzzy Sets Syst.
(2013) Distributivity and associativity in effect algebras
Fuzzy Sets Syst.
(2016)- et al.
New Trends in Quantum Structures
(2000) - et al.
Filters and supports in orthoalgebras
Int. J. Theor. Phys.
(1992) - et al.
Effect algebras and unsharp quantum logics
Found. Phys.
(1994) - et al.
MV and Heyting effect algebras
Found. Phys.
(2000) The center of an effect algebra
Order
(1995)
Cited by (2)
Joins and meets in effect algebras
2024, Algebra UniversalisJoins and meets in effect algebras
2021, arXiv