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I created effect algebra.
I gave it an Idea-section and added some more references.
Thanks. I made the link spectrum pointing to spectrum of an operator.
When I looked for texts concerning effect algebras I often found formulations like “effect of a computation” (but this seems rather unrelated) and google put papers on functional programming on the first places; so I was a bit surprised that the idea section starts with “in quantum mechanics”. But of course quantum logic is also a motivation Bart Jacob has in “new directions in categorical logic”.
Thanks. We should also have more examples. The simplest example is the addition on real numbers in $[0,1]$.
When I looked for texts concerning effect algebras I often found formulations like “effect of a computation” (but this seems rather unrelated)
Yes, that’s unrelated. And you should add a warning to the entry that says so! These articles on “algebras with effects” are talking about things more like those at monad (in computer science).
so I was a bit surprised that the idea section starts with “in quantum mechanics”.
That’s where the notion of effect algebra originates. It’s a formalization of the behaviour of those self-adjoined operators which in quantum mechanics are called “effects”, a generalization of projectors.
(I am not a big fan of any of this terminology and notions, but in some circles it is standard.)
We should also have more examples.
There may already be some examples at coalgebra of an endofunctor; and an entry coalgebra of the real interval. I didn’t yet check which of those are effect algebras. But I created effect algebra of predicates since my interest was mainly in this example of the internal logic of extensive categories. In fact I would like to understand in which sense extensive categories can be seen as cohesive (so the effect algebras may be rather a side-effect).
I urge you to write out how the unit interval is naturally an effect algebra under addition. You don’t have to look at any other entries for this. Just look at the definitions that you wrote out at effect algebra.
Greetings from “some circles” ;-).
I do not think it is a good idea to denote the effect algebra operation by $\vee$. Besides being nonstandard, it gives a false impression that the $a\vee b$ is the supremum of the set $\{a,b\}$ with respect to the poset structure. This is not true in general. For example (in the real unit interval) $0.5 \vee 0.5=1$ looks silly, doesn’t it?
The standard symbol for the effect algebra operation is $\oplus$.
write out how the unit interval is naturally an effect algebra under addition
Ok, I will write this, then.
I do not think it is a good idea to denote the effect algebra operation by ∨
It is denoted by ’\ovee’ in “new directions” but this symbol is not available in itex. $\oplus$ is used only once in the reference denoting a direct sum of vector spaces. However ’\ovee’ has a complemetary symbol ’\owedge’, so $\oplus$ would then be complementary to $\ominus$ which is used for something different in the reference: In Lemma 6 p.25 are detailed some relations between the different operators.
Stephan,
once you write out the example of the unit interval you’ll see why $\oplus$ is a good idea. Also write out the example that gives the concept its name, that of effect operators. (Both are simple. Should just take a minute.)
I see that it is a partially defined addition of real numbers and I didn’t mean to insist on $\vee$. Parts of what I wrote above remain untouched, however.
I see that it is a partially defined addition of real numbers
Yup!
Could you also add the example of effect operators?
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