• Corpus ID: 26619089

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}
}
  • Z. Riecanová
  • Published 2011
  • Computer Science, Mathematics
  • Kybernetika
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… 
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Background Citations
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2 Citations

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

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

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

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

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

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.

Pseudocomplemented lattice effect algebras and existence of states

S-Dominating Effect Algebras

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

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

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

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

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

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…