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> "lo" <> "fa" "lofa" > 1 <> 2 Error found: in module $PSCI at :1:1 - 1:7 (line 1, column 1 - line 1, column 7) No type class instance was found for ...
#1: Initial revision
by
toraritte
·
2024-05-27T12:42:50Z (7 months ago)
Why is the `Data.Int` type not a `Semigroup` in PureScript but `String` is?
```
> "lo" <> "fa"
"lofa"
> 1 <> 2
Error found:
in module $PSCI
at :1:1 - 1:7 (line 1, column 1 - line 1, column 7)
No type class instance was found for
Data.Semigroup.Semigroup Int
```
Is this for purely technical reasons as there would have to be at least 2 [`Semigroup`][1] instances (for addition and multiplication), but type classes cannot be implemented multiple times for the same type?
Even if it could be done using [instance chaining][2], one would have to create an instance of the entire "stack" of algebraic abstractions ([`Monoid`][3], [`Group`][4], [`Commutative`][5]) for each operation, right?
In contrast, [`Semiring`][6] is conveniently defined as a set with 2 binary operations<sup><b>1</b></sup>, which fits the bill perfectly:
> A semiring (`S`, `+`, `•`) is a set `S` with two binary compositions such that (`S`, `+`) and (`S`, `•`) are semigroups and distributes over `+`.
> <sup>[source, pdf][7]</sup>
So, technically, [`Int`][9] **is** a semigroup under addition ([a commutative monoid even][10]), but PureScript is not capable to "infer" that [`Semigroup`][1]'s `append` (`<>`) and [`Semiring`][6]'s `add` (`+`), in effect, do the same thing, but, for obvious reasons, cannot be used interchangeably.<sup><b>2</b></sup>
Then again, I'm quite new to pure functional programming, PureScript, and abstract algebra, so my statements above may not make much sense - and thanks in advance for dispelling my confusion!
---
<sup>\[1]: Although, nLab states that "_[mathematicians disagree on the definition of a semiring][8]_"...</sup>
<sup>\[2]: Are there programming languages where this level of granularity is achieved? (That is, provided if this is even a thing and if I didn't totally misinterpret these mathematical ideas.)</sup>
[1]: https://pursuit.purescript.org/packages/purescript-prelude/6.0.1/docs/Data.Semigroup
[2]: https://stackoverflow.com/questions/50290695/how-to-use-instance-chain-to-solve-overlapping
[3]: https://pursuit.purescript.org/packages/purescript-prelude/6.0.1/docs/Data.Monoid#t:Monoid
[4]: https://pursuit.purescript.org/packages/purescript-group/4.1.1/docs/Data.Group
[5]: https://pursuit.purescript.org/packages/purescript-group/4.1.1/docs/Data.Semigroup.Commutative
[6]: https://pursuit.purescript.org/packages/purescript-prelude/6.0.1/docs/Data.Semiring
[7]: https://projecteuclid.org/journals/journal-of-science-of-the-hiroshima-university-series-ai-mathematics/volume-30/issue-2/On-semigroups-semirings-and-rings-of-quotients/10.32917/hmj/1206139104.pdf
[8]: https://ncatlab.org/nlab/show/semiring
[9]: https://pursuit.purescript.org/packages/purescript-integers/6.0.0/docs/Data.Int
[10]: https://pursuit.purescript.org/packages/purescript-prelude/6.0.1/docs/Data.Semiring#t:Semiring