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
σ-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
E be the event space of an experiment that can be indefinitely repeated. A natural question arises: given a countable cardinal
κ, which is the event space of the
κ-times repeated experiment? In the case of classical experiments, where
E is a (complete) Boolean algebra on some set
S, i.e. a classical or distributive logic, the answer is more or less known: the (complete) Boolean algebra on
Sκ generated by
Eκ. But, what if
E is not a Boolean algebra? In this paper we give a constructive answer to this question for any
κ and in the context of general orthocomplemented complete lattices, i.e. general logics. Concretely, given a general logic
E defining the event space of a given experiment, we construct a logic
Uκ(E) representing the event space of the
κ-times repeated experiment, in such a way that
Uκ(E) and
E are isomorphic if
κ=1, and such that
Uκ(E) is distributive if and only if so is
E. We also extend our construction to the case in which the event space changes from one repetition to another and the cardinal
κ is arbitrary. This gives rise to tensor products
⨂α∈κEα of families
{Eα}α∈κ of orthocomplemented complete lattices, in terms of which
Uκ(E)=⨂α∈κE.
Let S be a set of states of a physical system and p(s) the probability of the occurrence of an event when the system is in state s in S. Such a function p from S to [0,1] is known as a numerical event or more accurately an S-probability. A set P of numerical events including the constant functions 0 and 1 and 1-p with every p in P becomes a poset when ordered by the order of real functions and can serve as a general setting for quantum logics. We call such a poset P a general set of events (GSE). The thoroughly investigated algebras of S-probabilities (including Hilbert logics), concrete logics and Boolean algebras can all be represented within this setting. In this paper we study various classes of GSEs, in particular those that are orthoposets and their interrelations and connections to known logics. Moreover, we characterize GSEs as posets by means of states and discuss the situation for GSEs to be lattices.