• Relative Entropy (Part 1): how various structures important in probability theory arise naturally when you do linear algebra using only the nonnegative real numbers.

• Relative Entropy (Part 2): a category related to statistical inference, and how relative entropy defines a functor on this category.

But then Tobias Fritz noticed a big problem.

]]>Numbers in are relative probabilities. In , 0 means ‘impossible’ and 1 means ‘possible’. So, the quotient map sends nonzero probabilities to ‘possible’, and zero probability to ‘impossible’.

]]>More generally, the -modules are precisely the join-semilattices, hence all these become -modules via the above homomorphism.

(Writing for the Boolean rig may be a bit misleading, since the addition is not the usual one!)

]]>Your second example is more interesting, though, because who ever thinks of as a -module?

]]>is confusing to me. Is that quotient a quotient of sets, or of -modules? Since you’re trying to create a -module I’ll assume the latter.

If so, you’re taking viewed as a free -module with two generators, and you’re creating a quotient -module in which all elements of the form are set equal to zero.

But doing that forces

That is, all elements along diagonal lines get identified. So, every element becomes equivalent to one of the form or one of the form . And

so every element has a negative. Now I’m thinking Tobias was right and your gadget is isomorphic to as a -module. Let me just check that addition works correctly: I want plus to equal if is bigger, and if is bigger. I’ll just do the first case:

Yup! So, you’ve got a topological 1-dimensional module that requires two generators, and it’s isomorphic to

Nice.

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