[Haskell-cafe] Re: categories and monoids

[Wolfgang Jeltsch]

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