Comments (6)
I was about to post the same thing. I think there are some (to me) simpler counterexamples than the one you posted, for example the natural numbers with the maximum operation. Under this operation (let's denote it with @), we have 2@5=5=3@5
, but certainly 2!=3
. If you want three operands you can add a 1
.
By the way monoids where you can do the claimed manipulation are called cancelable. Every cancelable monoid can be naturally extended to a group, which is really nice. That group is called the Grothendieck group. Noncancellable monoids are important in the study of semifield theory.
from category-theory-illustrated.
Thanks, corrected.
from category-theory-illustrated.
The new explanation is still not what associativity means. It mentions two equation rewriting conditions (the latter of which is a consequence of the former) that are true by basic logical assumptions.
Associativity means that it doesn't matter in what order we calculate pairwise combinations of terms in an expression. Alternatively, it means we can insert matching parentheses wherever we want and always get the same thing back. By definition, it just means that x (y z) = (x y) z
for any three elements, and can be extended to x (y (z w)) = x ((y z) w) = (x y) (z w) = ((x y) z) w) = ...
, and so on for any number of elements by induction.
from category-theory-illustrated.
I disagree, I think the new explanation looks good. The first part says that associativity means that the order in which we apply the operations doesn't matter, which is correct. The second part "when a rule is associative..." explains the consequences of associativity and mentions that we can replace individual elements by products without worry, which is also correct.
from category-theory-illustrated.
What @vincentrolfs, plus I wrote a detailed explanation of what associativity is in the previous chapter https://boris-marinov.github.io/category-theory-illustrated/02_category/
from category-theory-illustrated.
Feel free to open other issues for either this or something else, I really appreciate your feedback.
from category-theory-illustrated.
Related Issues (19)
- OR and AND diagram labels are reversed HOT 2
- poset used without being defined HOT 1
- Chapter 02 - `isomorphism.svg` image is broken HOT 3
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- Please may you make a libre license official? HOT 4
- Singlton Example HOT 4
- Grammar: it's > its HOT 3
- For or: p → p ∨ q and p → p ∨ q HOT 1
- Isomorphism vs bijection HOT 3
- Is the arrow direction wrong here? HOT 1
- Image of is letter function - incorrect mappings HOT 1
- A few typos HOT 1
- minor typos HOT 2
- About ePub format output HOT 5
- small error HOT 1
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