[Haskell-cafe] Re: categories and monoids
Edward Kmett
ekmett at gmail.com
Fri Mar 20 08:51:25 EDT 2009
Yes, even in English semi- is a prefix, so it falls under the purview of
morphology, the borderline between syntax and phonetics where linguists on
either side of the divide shove things they don't want to think about, but
it was the nearest example to hand. =)
On the other hand, non-associative rings and non-associative fields, near
semi-rings, non-commutative rings, and so forth do exist, so alas
mathematical terminology is not perfectly additive.
As I mentioned in passing, abstract algebra in particular is rich in these,
because its historical derivation started from fields historically and
mathematicians worked down to find smaller but still useful structures, so
names gained traction early.
And since you called me on the abuse of a prefix, I should clarify my
previously qualified statement that 'most' adjectives (and prefixes) are
traits or cotraits in mathematics, you also have constructions like
'concrete category' which isn't a category, and acts more like the notion of
a 'fake gun' that linguists love to cite, its actually just a faithful
functor (usually to Set) from some category, and an abuse of terminology
because you are allowed to have multiple concrete categories for the same
'abstract' category.
-Edward Kmett
On Fri, Mar 20, 2009 at 5:09 AM, Wolfgang Jeltsch <
g9ks157k at acme.softbase.org> wrote:
> Am Donnerstag, 19. März 2009 13:58 schrieben Sie:
> > An easier idea to think about would be to categorize most adjectives
> > applied to mathematical constructs into traits and cotraits.
> >
> > A trait refines a notion and a cotrait broadens the definition.
> >
> > When talking about a commutative ring, commutativity is a trait, it
> narrows
> > the definition of the ring, adding a requirement of commutativity to the
> > multiplication operation.
> >
> > When talking about semi rings, semi is a cotrait. It broadens the
> > definition of a ring, removing the requirement that addition form a
> group,
> > weakening it to merely require a monoid.
>
> Is “semi” and adjective at all? In German, we say “halb” instead of “semi”
> and
> the semi ring becomes a Halbring. Note that “halb” and “ring” are written
> toghether which means that “Halbring” is a compound noun. (We always write
> compound nouns as a single word, e.g., “Apfelsaft” for “apple juice”). So
> at
> least in German (which shares common roots with English), the “halb” is not
> considered an adjectiv. “halb” means “half”, so a “Halbring” is just half
> of
> a ring – not a special ring but less than a ring.
>
> Best wishes,
> Wolfgang
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