On the concreteness of quantum logics

Pták, Pavel ; Wright, John D. Maitland

Applications of Mathematics, Tome 30 (1985) no. 4, pp. 274-285.

Résumé

It is shown that for any quantum logic $L$ one can find a concrete logic $K$ and a surjective homomorphism $f$ from $K$ onto $L$ such that $f$ maps the centre of $K$ onto the centre of $L$. Moreover, one can ensure that each finite set of compatible elements in $L$ is the image of a compatible subset of $K$. This result is "best possible" - let a logic $L$ be the homomorphic image of a concrete logic under a homomorphism such that, if $F$ is a finite subset of the pre-image of a compatible subset of $L$, then $F$ is compatible. Then $L$ must be concrete. In the second part one considers embeddings into concrete logics. It is shown that any concrete logic can be embedded into a concrete logic with preassigned centre and an abundance of two-valued measures. Finally, one proves that an arbitrary logic can be mapped into a concrete logic by a centrally additive mapping which preserves the ordering and complementation.

Détail

DOI : 10.21136/AM.1985.104150

Classification : 03G12, 06C15, 81B10
Mots-clés : orthomodular lattice; orthomodular poset; centres; orthocomplemented posets; concrete logics