Add Semifield and CommutativeSemifield to algebra

[armanbilge]

Introduces a Semifield and CommutativeSemifield to algebra. I'm going by the second definition on Wikipedia:

• a commutative monoid for addition (+)
• a group for multiplication (*)

Thanks in advance!

[armanbilge]

@johnynek would you mind reviewing this? Thanks!

[johnynek]

Looks good to me.

Are there examples you have in mind that are not other structures?

Are there any instances we can create that we couldn't fully express before?

[armanbilge]

@johnynek Thanks for the review! I'm using semifields for non-negative reals in my probabilistic programming library, where probabilities are added, multiplied, and divided (but never negative). For example, I have a LogDouble type which will have this: https://github.com/armanbilge/schrodinger/blob/main/log-double/src/main/scala/schrodinger/LogDouble.scala#L67

Not sure if this is anything we can add to Algebra itself.

[johnynek]

I can't restart the microsite build. Can you push again? I want to make sure it was a flake.

[armanbilge]

Absolutely. But this was a known issue, see #3974 (comment)

[armanbilge]

@johnynek good to go.

[benhutchison]

Thank you @armanbilge & @johnynek for adding Semifield & friends 🙏

I'm guessing the algebra type-classes must sometimes feel obscure and even that no one notices/appreciates the work. So I thought I'd share my story..

Having defined a non-negative sub-species of BigDecimal, I went looking for the right algebra to implement +, * and /, without negation. And lo and behold, I discovered Semifield was exactly what I needed. At first, I assumed it was part of @non's original vision, but then noticed it is missing from the algebra docs and got curious how/when it came to be.

On docs, I do think an extra two rows are needed in the Ring table. But then it further qualifies the statement that "every lower structure is also an instance of the structures above it"; I actually wonder if with another two exceptions, that over-generalization has gone past time?

[armanbilge]

Thanks!

I do think an extra two rows are needed in the Ring table.

Indeed!

I actually wonder if with another two exceptions, that over-generalization has gone past time?

Yes, agree, probably an inheritance diagram would be more useful at this point :) (in addition to the existing table.)