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Cole Comfort (Dec 19 2020 at 11:39):Hi all. @Chad Nester and I are interested in effect algebras and we’re wondering what the motivating examples are. Where do they show up?
Perhaps @John van de Wetering can weigh in, because I know you are very familiar with these things.
John van de Wetering (Dec 19 2020 at 12:07):So there is the original motivation and there is a post priori categorical motivation.
John van de Wetering (Dec 19 2020 at 12:07):The original motivation was to abstract the unit interval of a C*-algebra and hence model the effects of a quantum system. This came forth out of a desire for ever more general 'quantum logics' which started with orthomodular lattices, which became orthomodular posets, which became orthoalgebras, which then became 'fuzzy orthoalgebras' which are now known as effect algebras
John van de Wetering (Dec 19 2020 at 12:08):A post priori categorical motivation is that there is a free-forgetful adjunction between the category of bounded posets (posets which have a minimum and maximum) and the category of orthomodular posets. The Eilenberg-Moore algebras of the resulting monad on bounded posets is isomorphic to the category of effect algebras
John van de Wetering (Dec 19 2020 at 12:10):Concrete examples are the real unit interval , the set of continuous functions from a compact Hausdorff space to , or the unit interval of effects in a C*-algebra, von Neumann algebra, Euclidean Jordan algebra or JB-algebra. All these examples are in fact 'convex effect algebras' that have the action of on them.
John van de Wetering (Dec 19 2020 at 12:11):Examples that aren't convex are for instance orthomodular posets (of which a special case is Boolean algebras). Also, given any ordered Abelian group, its unit interval is an effect algebra. Any MV-algebra is also an effect algebra. Finally, the predicate spaces in an effectus form an effect algebra, and any effect algebra can occur in such a way.
John van de Wetering (Dec 19 2020 at 12:12):Does that help?
Cole Comfort (Dec 19 2020 at 12:13):Yes thanks a lot! That gives quite a good summary.