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

Title:

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

Creator:

Kalina, Martin

Identifier:

https://cdk.lib.cas.cz/client/handle/uuid:e9deb15d-afc7-48c9-b011-3822bc45cc3c
uuid:e9deb15d-afc7-48c9-b011-3822bc45cc3c

Subject:

lattice effect algebra, center, atom, and Mac Neille completion

Type:

model:article and TEXT

Format:

bez média and svazek

Description:

If element z of a lattice effect algebra (E,⊕,0,1) is central, then the interval [0,z] is a lattice effect algebra with the new top element z and with inherited partial binary operation ⊕. It is a known fact that if the set C(E) of central elements of E is an atomic Boolean algebra and the supremum of all atoms of C(E) in E equals to the top element of E, then E is isomorphic to a subdirect product of irreducible effect algebras (\cite{R2}). This means that if there exists a MacNeille completion E^ of E which is its extension (i.e. E is densely embeddable into E^) then it is possible to embed E into a direct product of irreducible effect algebras. Thus E inherits some of the properties of E^. For example, the existence of a state in E^ implies the existence of a state in E. In this context, a natural question arises if the MacNeille completion of the center of E (denoted as MC(C(E))) is necessarily the same as the center of E^, i.e., if MC(C(E))=C(E^) is necessarily true. We show that the equality is not necessarily fulfilled. We find a necessary condition under which the equality may hold. Moreover, we show also that even the completeness of C(E) and its bifullness in E is not sufficient to guarantee the mentioned equality.

Language:

English

Rights:

http://creativecommons.org/publicdomain/mark/1.0/
policy:public

Coverage:

[935]-947

Source:

Kybernetika | 2010 Volume:46 | Number:6

Harvested from:

CDK

Metadata only:

false