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.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.

Applications of MathematicsJan 1, 19858 citations

Quantum Logics

Richard J. Greechie, Stanley Gudder

Springer eBooksJan 1, 19753 citations

Quantum Logics with Lattice State Spaces

Jiří Binder, Mirko Navara

Let $L$ be a quantum logic and let $S(L)$ denote the set of all states on $L$ . (By a state we mean a nonnegative bounded $\sigma$ -additive measure, not necessarily normalized.) We ask whether every logic whose state space is a lattice has to be Boolean. We prove that this is so for finite logics and "projection logics." On the other hand, we show that there exist even concrete non-Boolean logics with a lattice state space (in fact, we prove that every countable concrete logic can be enlarged to a logic with a lattice state space). In the appendix we shortly consider the lattice properties of the set of observables and correct the paper [10].Let $L$ be a quantum logic and let $S(L)$ denote the set of all states on $L$. (By a state we mean a nonnegative bounded $\sigma$-additive measure, not necessarily normalized.) We ask whether every logic whose state space is a lattice has to be Boolean. We prove that this is so for finite logics and "projection logics." On the other hand, we show that there exist even concrete non-Boolean logics with a lattice state space (in fact, we prove that every countable concrete logic can be enlarged to a logic with a lattice state space). In the appendix we shortly consider the lattice properties of the set of observables and correct the paper [10].

Proceedings of the American Mathematical SocietyJan 1, 19875 citations

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.

Let

L

be a quantum logic and let

S(L)

denote the set of all states on

L

. (By a state we mean a nonnegative bounded

\sigma

-additive measure, not necessarily normalized.) We ask whether every logic whose state space is a lattice has to be Boolean. We prove that this is so for finite logics and "projection logics." On the other hand, we show that there exist even concrete non-Boolean logics with a lattice state space (in fact, we prove that every countable concrete logic can be enlarged to a logic with a lattice state space). In the appendix we shortly consider the lattice properties of the set of observables and correct the paper [10].