On Thu, Mar 19, 2009 at 5:43 AM, Wolfgang Jeltsch
Am Mittwoch, 18. März 2009 15:17 schrieben Sie:
Wolfgang Jeltsch schrieb:
Okay. Well, a monoid with many objects isn’t a monoid anymore since a monoid has only one object. It’s the same as with: “A ring is a field whose multiplication has no inverse.” One usually knows what is meant with this but it’s actually wrong. Wrong for two reasons: First, because the multiplication of a field has an inverse. Second, because the multiplication of a ring is not forced to have no inverse but may have one.
“A ring is like a field, but without a multiplicative inverse” is, in my eyes, an acceptable formulation. We just have to agree that “without” here refers to the definition, rather than to the definitum.
Note that you said: “A ring is *like* a field.”, not “A ring is a field.” which was the formulation, I criticized above.
"Alternatively, the fundamental notion of category theory is that of a monoid ... a category itself can be regarded as a sort of generalized monoid." -- Saunders MacLane, "Categories for the Working Mathematician" (preface)