Timeline for Why are so few operations with arity bigger than 2?
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
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| Jan 26 at 11:08 | history | made wiki | Post Made Community Wiki by Stefan Kohl♦ | ||
| Feb 8, 2012 at 21:13 | comment | added | Gerhard Paseman | Quid, it may be my general algebra bias showing, but as you change domains when you move from factors to their Cartesian product ,, so too do you change domains when you move from an algebra to a subalgebra or a homomorphic image. It's Johan's problem that he is working with 0,2,4 instead of 1,3,5. Gerhard "One Plus One Equals Five" Paseman, 2012.02.08 | |
| Feb 7, 2012 at 12:26 | comment | added | user9072 | @Gerhard: why should the cart product not be closed (even in the present sense). On the product (the set) one has a different operation (of course defined via the origunal ones). So, somewhat opposed to the original example, there is no reason why the empty prod with respect to that operation should not be something else. Yet, conversely at least in my opinion/def, the rings one gets when setting one of the coordinates to 0 are not subgrings of the product ring; but the 'diagonal' is a subring. | |
| Feb 7, 2012 at 8:24 | comment | added | Johan Wästlund | +1 to Gerhard :) but quid is right, I was thinking of the empty product. | |
| Feb 7, 2012 at 7:38 | comment | added | Gerhard Paseman | Hmm. If you take the Cartesian product of two rings with unit, then that set is also not multiplicatively closed either? I think the empty product is a product of convenience, and should be whatever identity is available. Gerhard "Use Product Three Times Fast" Paseman, 2012.02.06 | |
| Feb 7, 2012 at 0:13 | comment | added | user9072 | @Vectornaut: I'm not certain what is meant, but I guess in view of the answer the comment is to, the point is the empty product (in the the ring of integers mod 6, so 1), is not an element of the set {0,2,4}. And thus this set is not multiplicatively closed (under the inherited opration) as 'surely' it should also contain the value of empty product. I am a bit torn as I agree that {0,2,4} is not a subring as as a subring it should (or at least this is the def I prefer) have the same neutral element, but have issues with saying it is not mult closed; as the term is useful in this way. | |
| Feb 6, 2012 at 22:45 | comment | added | Vectornaut | @Johan Wästlund: I don't get it. 2×2=4, 2×4=2, and 4×4=4, so how is {0,2,4} not closed under multiplication? | |
| Feb 6, 2012 at 16:03 | comment | added | Gerhard Paseman | Johan, is it because 4*4 is really 16? Gerhard "Ask Me About System Design" Paseman, 2012.02.06 | |
| Feb 6, 2012 at 11:05 | comment | added | Martin Brandenburg | Mariano has mentioned the connection to PROPs and operads, but I think it's more basic: The forgetul functor from monoids to sets is monadic. | |
| Feb 6, 2012 at 9:58 | comment | added | Marcos Cossarini | It's a rng! I view the concept of ring as inspired in the example of the endomorphism ring of an abelian group, and the concept of rng as inspired in the example of the example of... an ideal in a ring. | |
| Feb 6, 2012 at 9:47 | history | edited | Marcos Cossarini | CC BY-SA 3.0 |
corrected typo
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| Feb 6, 2012 at 9:42 | comment | added | Johan Wästlund | Excellent answer! I recall how disturbed I was by the abstract algebra textbook example of $\{0,2,4\}$ as a subring of the ring of integers modulo 6. I wanted to shout BUT THE DAMNED THING IS NOT CLOSED UNDER MULTIPLICATION but had to restrain myself since I was the teacher. Twenty years later I still haven't fully recovered from part (b) of the exercise, which was to show that this "ring" has a unit. NO IT DOESN'T. AND THEREFORE IT'S NOT A RING. | |
| Feb 5, 2012 at 21:58 | comment | added | Mariano Suárez-Álvarez | This is precisely the idea behind the notions of PROPs and operads, in fact. | |
| Feb 5, 2012 at 20:40 | history | answered | Marcos Cossarini | CC BY-SA 3.0 |