Coreflections in Algebraic Quantum Logic

Foundations of Physics, May 2012

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.

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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 - 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)


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Bart Jacobs, Jorik Mandemaker. Coreflections in Algebraic Quantum Logic, Foundations of Physics, 2012, pp. 932-958, Volume 42, Issue 7, DOI: 10.1007/s10701-012-9654-8