• 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.

]]>So what’s really really going on is that is the rig of truth values under the operations OR and AND. This makes sense, since probability theory is generalized logic.

]]>You probably know this, but I should point out: if you think of as the rig that governs probability theory, then its quotient is the rig that governs possibility theory.

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’.

]]>Sorry, I should have explained that the homomorphism is the map which sends to and every positive number to .

]]>Actually what’s really going on here is that is the Boolean semiring (rig) with , which comes with a rig homomorphism . This is how the Boolean rig becomes a -module.

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!)

]]>Hey, you’re right! I didn’t think of that because I wasn’t sure if the real line would be finitely generated, but of course it is! It’s generated by 1 and -1. And that’s how I thought of my example: two generators such that the sum is zero.

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

]]>Hey—great to see you here, John! This example

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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