Coreflections in Algebraic Quantum Logic
Bart Jacobs
Jorik Mandemaker
Various generalizations of Boolean algebras are being studied in algebraic quantum logic, including orthomodular lattices, orthomodular po-sets, orthoalgebras and effect algebras. This paper contains a systematic study of the structure in and between categories of such algebras. It does so via a combination of totalization (of partially defined operations) and transfer of structure via coreflections. The algebraic study of quantum logics focuses on structures like orthomodular lattices, orthomodular posets, orthoalgebras and effect algebras, see for instance [35, 11]. This paper takes a systematic categorical look at these algebraic structures, concentrating on (1) relations between these algebras in terms of adjunctions, and (2) categorical structure of the categories of these algebras. Typical of these algebraic structures is that they involve a partially defined sum operation that can be interpreted either as join of truth values (in orthomodular lattices/posets) or as sum of probabilities (in effect algebras). The leading example of such a partially defined sum is addition on the (real) unit interval [0, 1] of probabilities: for x, y [0, 1] the sum x y = x + y is defined only if x + y 1. Because this operation is so fundamental, the paper takes the
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notion of partial commutative monoid (PCM) as starting point. An effect algebra, for
instance, can then be understood as an orthosupplemented PCM, in which for each
element x there is a unique element x with x x = 1.
The paper studies algebraic quantum logics via a combination of:
totalization of the partially defined operation into a richer algebraic structure,
forming a coreflection with the original (partial) structures. Such a coreflection is
an adjunction where the left adjoint is a full and faithful functor;
transfer of structure along these coreflections. It is well-known (see [1, I,
Proposition 3.5.3]) that limits and colimits can be transferred from one category to another
if there is a coreflection between them. Here we extend this result to include also
transfer of adjunctions and of monoidal structure.
In Sect. 3 we show that effect algebras also form a reflection with test spaces
(following [8, 15]), so that we have a situation:
Both the reflection and the coreflection can be used to study the categorical structure
of the category of effect algebras. However, here we shall do so via the coreflection
only, because this coreflection involves total operations that are easy to work with.
In particular, we obtain tensors of effect algebras via this coreflection. They are not
new: they are constructed explicitly in [2, 6]. Here they simply arise from a transfer
result based on coreflections. The presence of these tensors is an important
advantage of effect algebras over orthomodular lattices [11]. They naturally lead to notions
like effect monoid (monoid in the category of effect algebras) and effect module
(action for such a monoid). For instance, the effects of a Hilbert spacepositive
operators below the identityform an example of such an effect module, with the effect
monoid [0, 1] as scalars. A systematic study of these structures will appear elsewhere.
2 Partial Commutative Monoids and Effect Algebras
Before introducing the main objects of study in this paper we first recall some basic
notions about commutative monoids and fix notation.
The free commutative monoid on a set A is written as M(A). It consists of
finite multisets n1a1 + + nk ak of elements ai A, with multiplicity ni N.
Such multisets may be seen as functions : A N with finite support, i.e. the set
sup() = {a A | (a) =0} is finite. The commutative monoid structure on M(A) is
then given pointwise by the structure in N, with addition ( + )(a) = (a) + (a)
and zero element 0(a) = 0. These operations can be understood as join of multisets,
with 0 as empty multiset.
The mapping A M(A) yields a left adjoint to the forgetful functor CMon
Sets from commutative monoids to sets. For a function f : A B we have
a homomorphism of monoids M(f ) : M(A) M(B) given by ( i ni ai )
( i ni f (ai )), or more formally, by M(f )()(b) = af 1(b) (a). The unit :
A M(A) of the adjunction may be written as (a) = 1a.
If M = (M, +, 0) is a commutative monoid we can interpret a multiset M(M)
over M as an element = xsup() (x) x, where n x is x + + x, n times.
In fact, this map is the counit of the adjunction mentioned before. Each monoid
M carries a preorder given by: x y iff y = x + z for some z M . In free
commutative monoids M(A) we get a poset order iff (a) (a) for all a A.
Homomorphisms of monoids are monotone functions wrt. this order . This applies
in particular to interpretations : M(M) M .
Definition 1 A partial commutative monoid, or PCM, is a triple (M, , 0) consisting
of a set M , an element 0 M and a partially defined binary operation such that
the three axioms below are satisfied. We let the expression x y mean x y is
defi (...truncated)