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• CommentRowNumber1.
• CommentAuthorKeithEPeterson
• CommentTimeDec 25th 2016

Noticed there was no page for the left adjoint to joins. Added subtraction.

• CommentRowNumber2.
• CommentAuthorTim_Porter
• CommentTimeDec 26th 2016
• (edited Dec 26th 2016)

Here is a question that would probably be easy to answer if some of the papers were not behind paywalls! Is there a link between this notion of subtraction and subtractive varieties/categories as in work by Ursini and also Z. Janelidze? I am asking because there may be a need for a page on some wider notion … subtractive algebra, subtractive object with the Heyting algebra subtraction as one example.

• CommentRowNumber3.
• CommentAuthorKeithEPeterson
• CommentTimeDec 26th 2016

I’m reading up on them in the work by Z. Janelidze. Subtractive Categories are neat!

• CommentRowNumber4.
• CommentAuthorTodd_Trimble
• CommentTimeDec 26th 2016
• (edited Dec 26th 2016)

We ought to consider what should go into an article titled “subtraction”. This is the sort of topic I can imagine Zoran or Toby editing, where one takes a concept of elementary arithmetic and goes on to explore its ramifications in higher mathematics. In the present case, it’s not just an operation on co-Heyting algebras or in something called “subtractive categories” (which most people never heard of): we have groups defined in terms of a difference operation, one can subtract virtual bundles, and torsors and even homs can be conceived in terms of generalized subtraction, as a small sample of the possibilities.

Edit: maybe “subtractive category” should be an article in its own right?

• CommentRowNumber5.
• CommentAuthorTim_Porter
• CommentTimeDec 26th 2016

In have started to add some stuff to subtraction. This has also meant creating subtractive variety.

• CommentRowNumber6.
• CommentAuthorKeithEPeterson
• CommentTimeDec 26th 2016

In any additive-category with quotient, subtraction, internal hom, and zero object, can’t I define a notion of simple derivative something like:

$\frac{([x+h, f]\otimes (x+h))\setminus [x,f]\otimes x}{(x+h)\setminus x}$ where $h := [-,[1,\varnothing] ]$

• CommentRowNumber7.
• CommentAuthorKeithEPeterson
• CommentTimeDec 26th 2016
• (edited Jan 7th 2017)

Wait a minute. If we identify subtraction with reverse-implication ⇐, we get out do-notation.

$do: [y \leftarrow x],[z \leftarrow w],\cdots \;\;\;:= ((y$$x)\oplus x);((z$$w)\oplus w);\cdots$

• CommentRowNumber8.
• CommentAuthorTim_Porter
• CommentTimeDec 27th 2016

I put a link to a paper that looks as if it is related to your last comment, Keith, at the page subtractive logic. That page might be a good place to put down some of your thoughts and to explain their significance.

• CommentRowNumber9.
• CommentAuthorKeithEPeterson
• CommentTimeDec 27th 2016

Yeah, I was reading the reference on that page, and then on to reading the reference that paper kept referring to, “Declarative Continuations” by Andrzej Filinski.

Usually that is how these things work. You read one paper, and then you end up reading about 10 more references, each with more references. Mathematics is dependency hell.

• CommentRowNumber10.
• CommentAuthorKeithEPeterson
• CommentTimeDec 29th 2016

Added the two canonical examples. Yes, I’m aware the definition of subtraction on the integers is a bit of a hack.

• CommentRowNumber11.
• CommentAuthorDavid_Corfield
• CommentTimeDec 29th 2016

pred(n) …is the inverse to the successor function

pred(0)?

Then

Subtraction in the abelian group of the integers $\mathbb{Z}$, is an entire relation on the integers, making it an true inverse operation to the group multiplication

With multiplication given as a map $\mathbb{Z} \times \mathbb{Z} \to \mathbb{Z}$, the link to inverse suggests a map the other way.

• CommentRowNumber12.
• CommentAuthorTim_Porter
• CommentTimeDec 29th 2016

Is this not a good example of the ’maths made difficult’ syndrome? Subtraction on the monoid of natural numbers is a partial operation. It is an excellent example of such, but a complicated definition involving ’pred’ does not relate directly to the monoid structure on $\mathbb{N}$. The point is that $n-m$ is the unique solution to $m+x=n$ when a solution exists. Subtraction illustrates ’partial opertaion’ not the other way around as far as most people would be concerned.

• CommentRowNumber13.
• CommentAuthorKeithEPeterson
• CommentTimeDec 29th 2016
• (edited Dec 29th 2016)

This is what happens when you do math at 5am.

I knew if I messed it up, it would be checked over and fixed. That was the hack. ;)

• CommentRowNumber14.
• CommentAuthorTodd_Trimble
• CommentTimeDec 29th 2016

Oddly enough, I just finished an edit (on my to-do list for some weeks) at integer which alludes to integer subtraction, although not by name.