Spahn effect algebra (Rev #3, changes)

Showing changes from revision #2 to #3: Added | Removed | Changed

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\vee y^\perp)^\perp

and

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

and

yx=z:y=xzy\ominus 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 Hence these functors are afrobenius 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 16:07:50 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.