# Lattice effect algebras densely embeddable into complete ones

@article{Riecanov2011LatticeEA, title={Lattice effect algebras densely embeddable into complete ones}, author={Zdenka Riecanov{\'a}}, journal={Kybernetika}, year={2011}, volume={47}, pages={100-109} }

An effect algebraic partial binary operation ⊕ defined on the underlying set E uniquely introduces partial order, but not conversely. We show that if on a MacNeille completion E of E there exists an effect algebraic partial binary operation ⊕ then ⊕ need not be an extension of ⊕. Moreover, for an Archimedean atomic lattice effect algebra E we give a necessary and sufficient condition for that ⊕ existing on E is an extension of ⊕ defined on E. Further we show that such ⊕ extending ⊕ exists at…

## 2 Citations

### Two Remarks to Bifullness of Centers of Archimedean Atomic Lattice Effect Algebras

- Mathematics, Computer Science
- 2011

It is shown that for atomic lattice effect algebras E (atomic orthomodular lattices) neither completeness (and atomicity) of C(E) nor σ -completeness of E are sufficient conditions for C( E) to be a bifull sublattice of E .

### Mac Neille completion of centers and centers of Mac Neille completions of lattice effect algebras

- MathematicsKybernetika
- 2010

It is shown that even the completeness of $C(E)$ and its bifullness in $E$ is not sufficient to guarantee the mentioned equality, and a necessary condition under which the equality may hold is found.

## 24 References

### Archimedean Atomic Lattice Effect Algebras with Complete Lattice of Sharp Elements

- Mathematics, Computer Science
- 2010

We study Archimedean atomic lattice effect algebras whose set of sharp elements is a complete lattice. We show properties of centers, compatibility centers and central atoms of such lattice effect…

### ORTHOGONAL SETS IN EFFECT ALGEBRAS

- Mathematics
- 2001

We show that for a lattice effect algebra two conceptions of completeness (cr-completeness) coincide. Moreover, a separable effect algebra is complete if and only if it is cr-complete. Further, in an…

### Smearings of States Defined on Sharp Elements Onto Effect Algebras

- Mathematics, Computer Science
- 2002

It is shown that if E is complete, atomic, and (o)-continuous, then a state on E exists iff there exists a state in S(E), and that such an effect algebra E is an algebraic lattice compactly generated by finite elements of E.

### S-Dominating Effect Algebras

- Mathematics, Computer Science
- 1998

It is shown that an S-dominating effect algebra P has a naturally defined Brouwer-complementation that gives P the structure of a Brou Wer–Zadeh poset, and it is proved that the sharp elements of P form anorthomodular lattice.

### Subdirect Decompositions of Lattice Effect Algebras

- Mathematics, Computer Science
- 2003

An application proves a theorem about subdirect decompositions of lattice effect algebras and proves a statement about the existence of subadditive state on some block-finite effect alGEbras.

### On central atoms of Archimedean atomic lattice effect algebras

- MathematicsKybernetika
- 2010

It is shown that there exists a lattice effect algebra (E, ⊕, 0, 1) with atomic C(E) which is not a bifull sublattice of E.

### Generalization of Blocks for D-Lattices and Lattice-Ordered Effect Algebras

- Mathematics, Computer Science
- 2000

We show that everyD-lattice (lattice-ordered effect algebra)P is a set-theoreticunion of maximal subsets of mutually compatible elements, called blocks.Moreover, blocks are sub-D-lattices and…

### Effect algebras and unsharp quantum logics

- Mathematics
- 1994

The effects in a quantum-mechanical system form a partial algebra and a partially ordered set which is the prototypical example of the effect algebras discussed in this paper. The relationships among…

### ARCHIMEDEAN AND BLOCK-FINITE LATTICE EFFECT ALGEBRAS

- Mathematics
- 2000

1. Basic definitions Effect algebras (introduced by Foulis D.J. and Bennett M.K. in [7], 1994) are important for modelling unsharp measurements in Hilbert space: The set of all effects is the set of…