I can't think of any way for 1 * 1 to be concatenation.
I think the point GP was making is that a monoid has something to do with a function f that takes two arguments a and b where there is a single value for b such that f(a, special_value) == a. I think.
As someone who doesn't really understand FP concepts I would probably call those functions "functions that can sensibly be used in a call to array.reduce"
It doesn't need to have a perfect metaphor for all possible cases, just something you can picture in an instant: 'behaves like a concating variable length box sequences'. It's a mnemonic / learning aid, not a formal spec. But I can stretch it :)
+: concat marbles sequence
* : concat prime factors sequence
++: concat box sequence
For * , it can also be reduced to + via logarithms. I just don't have a good metaphor for '+ over reals' this morning.
“append”* is the name chosen for the binary operator in the Haskell standard library’s monoid abstraction and “concat” seems like the same genre, so I would expect people would be fine with the intuition for “concat” as well.
Given that this is a thread on functional jargon, it might be interesting to note list metaphors as being useful for thinking about monoid might derive from lists being free monoids (in many contexts). This means that all functions from a collection elements to a monoid can be thought of as functions from that collection into a list with the appropriate type of elements and then a fold (also called a reduction) over that list. This knowledge can be super useful for thinking about program structure e.g. conceptually mapreduce is a general tool for large-scale distributed computation over monoids.
*it’s actually called “mappend”[0], short for monoid append, but I think the point still stands (and there is, in fact, a function called “mconcat”[1] but folds of a list of monoidal values using “mappend” to combine them.)
> "functions that can sensibly be used in a call to array.reduce"
You can use any function you want in a fold in Haskell. You're conflating two typeclasses, Monoids and Foldables, and restricting yourself to the function known in Haskell as foldMap.
> As someone who doesn't really understand FP concepts I would probably call those functions "functions that can sensibly be used in a call to array.reduce"
That implies that they can be reduced. Maybe "reductors"?
That said, I am on the "name abstractions after the math" side of things. It makes it much clearer what guarantees I have when working with the interface, on either side of it.
I think the point GP was making is that a monoid has something to do with a function f that takes two arguments a and b where there is a single value for b such that f(a, special_value) == a. I think.
As someone who doesn't really understand FP concepts I would probably call those functions "functions that can sensibly be used in a call to array.reduce"