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