Re: [Haskell-cafe] Re: categories and monoids
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.
So a clearer reframing might be: “Ring is like Field, but without
multiplicative inverse”.
On Wed, Mar 18, 2009 at 7:17 AM, Kalman Noel
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.
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. Best wishes, Wolfgang
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)
I've been thinking a lot about these issues lately, mainly because I have been working with a toy compiler that uses multiple inheritance heavily, and which uses adjectives explicitly for context sensitive mixins. 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. In that setting 'generalized' as applied in the scenarios mentioned is just a cotrait. Its not wrong, its just not the more common notion of refinement you are used to when seeing adjectives applied to mathematical primitives. Whether traits or cotraits are applied when generating a new idea tends to be a function of primacy. Sure, perhaps every field should be viewed as a specialization of term for non-associative-field adding associativity, but often we don't find out that these weakened notions are even meaningful under after the more constrained topic has gained wide adoption. Neither notion is necessarily more correct than the other. Abstract algebra can be taught bottom up from groups to fields and beyond or 'top down' in the more traditional manner by progressively weakening the definition of a field. Eventually you wind up having to go both ways. After all if you started 'bottom up' from groups, you've probably got to go back and work down through monoids and semigroups to magmas if you want to be pedantic later. ;) -Edward Kmett On Thu, Mar 19, 2009 at 6:43 AM, Wolfgang Jeltsch < g9ks157k@acme.softbase.org> wrote:
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.
Best wishes, Wolfgang _______________________________________________ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
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
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@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 _______________________________________________ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
Wolfgang Jeltsch
Is ___semi___ and adjective at all? In German, we say ___halb___ instead of ___semi___ and the semi ring becomes a Halbring.
"Halbring" as in "halber Ring", isn't it? Synonymous with "partly a ring" (which uses an adverb)... In german, you can tack almost any word type using any case onto a noun, consider e.g. _Mit_fahrer, or Schifffahrt_s_gesellschaft. The terminology that I learnt is quite clear: If it changes the meaning of a noun, it's an adjective, which isn't so much a word-class but a function defined by context: The same roots are used for adverbs, too. Horrible memories consisting of grammar tests start to creep up. I'd rather stop now, if you don't mind. -- (c) this sig last receiving data processing entity. Inspect headers for copyright history. All rights reserved. Copying, hiring, renting, performance and/or quoting of this signature prohibited.
participants (6)
-
Achim Schneider -
Conal Elliott -
Edward Kmett -
Gregg Reynolds -
Kalman Noel -
Wolfgang Jeltsch