Great post! (I’ve used that notation for readability as well.)
This seems kinda like the full domain-theory version of Vanessa’s metathreat hierarchy without the restriction to operate on some fixed finite level, though I’m not sure how to get probabilities in there.
More later on how to get those domains into something you can actually work with on a computer, once I learn that.
I’m looking forward to these posts.
Questions:
One neat property of domains is that every continuous function from a domain to itself has a least fixed-point below all the other fixed points, and this least fixed-point also has a nice pretty description. The least fixed point of f∈[D→D], is given by the sup of the directed set A with elements: ⊥,f(⊥),f(f(⊥)),f(f(f(⊥)))...
Does this statement apply to the real numbers or a subset of them?
From now on, f;g;h will be a notation for “first apply f, then g, then h”. It’s the same as h∘g∘f, but without the need to take ten seconds to mentally reverse the order whenever it shows up.
Does the notation effect all the arrows in the proofs or just this part (below)?
Specifically, our candidate for the embedding is: F(e,e′)(f)=p;f;e′, and our candidate for the projection going the other way is to map g:D′→E′ to e;g;p′. Let’s call this function P(e,e′) for later use. It’s a bit of work to show that these are continuous functions, so I’m gonna skip that part, but I will show the part about these two things fulfilling the relevant conditions for embeddings and projections.
Comments:
Great post! (I’ve used that notation for readability as well.)
I’m looking forward to these posts.
Questions:
Does this statement apply to the real numbers or a subset of them?
Does the notation effect all the arrows in the proofs or just this part (below)?