Spahn effect algebra (Rev #2, changes)

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Definition

Definition (p.22)

A partial commutative monoid (PCM) consists of a set MM with a zero element 0M0 \in M and a partial binary operation :M×MM\vee : M \times M \to M satisfying the three requirements below. They involve the notation xyx \perp y for: xyx \vee y is defined; in that case x,yx, y are called orthogonal.

  1. Commutativity: xyx\perp y implies yxy\perp x and xy=yxx\vee y=y\vee x.

  2. Associativity: yzy\perp z and x(yz)x\perp(y\vee z) implies xyx\perp y and (xy)z(x\vee y)\perp z and x(yz)=(xy)zx\vee (y\vee z)=(x\vee y)\vee z.

  3. Zero: 0x0\perp x and 0x=x0\vee x=x

Definition (p. 23)

An effect algebra is a PCM (E,0,)(E,0,\vee) with an orthocomplement. The latter is a unary operation () :EE(-)^\perp :E\to E satisfying:

  1. x Ex^\perp\in E is the unique element in EE with xx =1x\vee x^\perp=1, where 1=0 1=0^\perp.

  2. x1x=0x\perp 1\Rightarrow x=0.

For such an effect algebra one defines:

xy:=(x y ) x\wedge y=(x^\perp\wedge y:=(x^\perp\vee y^\perp)^\perp

and

xy: z.xz=y x\le y\Leftrightarrow y:\Leftrightarrow \exists_z.x\vee z=y

and

yx=z:y=xz y\ominus x=z\Leftrightarrow x=z:\Leftrightarrow y=x\vee z
Remark (p.25)

If we consider

()y:up(y)down(y )(-)\ominus y:up(y)\to down(y^\perp)

and

()y:down(y )up(-)\vee y:down(y^\perp)\to up

as functors between posets we have adjunctions

((y)(y)(y)((-\wedge y)\dashv (-\ominus y)\dashv (-\wedge y)

hence these functors are a frobenius pair.

Application

In new directions the approach to categorical logic where the substrate carrying the logical notions are Heyting algebras of subobjects in a topos is replaced by a new one where the substrate is effect algebras (of predicates) in extensive categories (extensive category).

Reference

  • Bart Jacobs, New Directions in Categorical Logic, for Classical, Probabilistic and Quantum Logic, 2012, arXiv:1205.3940

Revision on January 7, 2013 at 02:14:51 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.