Timeline for A standard name for the algebraic structure on a projective line?
Current License: CC BY-SA 4.0
17 events
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| Apr 9 at 22:35 | history | edited | Taras Banakh | CC BY-SA 4.0 |
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| Apr 9 at 22:06 | history | edited | Taras Banakh | CC BY-SA 4.0 |
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| Apr 9 at 21:37 | comment | added | Taras Banakh | @GeraldEdgar Why we should tend to define $0\cdot \infty=1$ in the projective line? Because when we calculate the cross-ratio, we use exactly these equalities: $(x-\infty)/(y-\infty)=1$, which is equivalent to $\infty+x=\infty$ for $x\ne\infty$, $1/\infty=0$ and $\infty\cdot 0=1$. | |
| Apr 9 at 21:32 | comment | added | Taras Banakh | @GeraldEdgar This example is just the ring $\mathbb Z_4=\{0,1,2,3\}$ in which $2$ is proclaimed to be $\infty$ and the multiplicative structure is a bit changed in order to guarantee that $0\cdot \infty=1$ and $x\cdot\infty=\infty$ for nonzero $x$. | |
| Apr 9 at 21:29 | comment | added | Taras Banakh | @GeraldEdgar It is of course standard to leave $0\cdot \infty$ and $\infty+\infty$ undefined, but nonetheless, I finally have written down a proof that my 10 axioms do imply that $0\cdot\infty=1$ and $\infty+\infty=0$ in the set $L$ has cardinality $|L|>4$. For cardinality $4$ I have a strange example of the algebraic structure satisfying my 10 axioms and such that $1+\infty\ne \infty$. | |
| Apr 9 at 17:45 | comment | added | Gerald Edgar | I am familiar with the Riemann sphere (which is the complex projective line). In that case we leave $0\cdot\infty$ undefined; so technically $\cdot$ is not an operation. Similarly we leave $\infty + \infty$ undefined, so $+$ is not an operation. | |
| Apr 9 at 17:15 | history | edited | Taras Banakh | CC BY-SA 4.0 |
Added info about wheel.
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| Apr 9 at 17:11 | comment | added | Taras Banakh | @BenMcKay My motivation is geometri: fix three points on a projective line in a Pappian projective plane and call them $0,1,\infty$. With those fixed points we can define natural binary operations of addition and multiplication so that the above 9 axioms are satisfied. | |
| Apr 9 at 15:50 | comment | added | Ben McKay | In classical projective geometry over a field, the projective automorphism group acts transitively on the points defined over that field, so no point has a special role. But your notion of projective line does not have transtive automorphism group. So your geometry is very different. Is there some motivation for your definition? | |
| Apr 9 at 11:27 | history | edited | YCor |
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| Apr 9 at 10:27 | comment | added | Taras Banakh | @YCor $0\cdot \infty=1$ is not an artefact. It follows from the axiom 8. Also the axiom 4 implies that $\infty+\infty=0$. | |
| Apr 9 at 9:13 | comment | added | YCor | OK, maybe you need to check that all axioms are fulfilled then. Although the choice $0\infty=1$ seem to be an artefact and probably it should better not be defined. | |
| Apr 9 at 8:54 | comment | added | Taras Banakh | @YCor But you used a distributivity law (with multiplication by $\infty$), which is not allows by the axioms. | |
| Apr 9 at 8:29 | comment | added | YCor | But then $1=0\cdot\infty=(0+0)\cdot\infty=1+1$. My point is that these should be undefined. | |
| Apr 9 at 8:04 | comment | added | Taras Banakh | @YCor This is a very good question. At the moment I am trying to deduce from the axioms that $0\cdot\infty=1=\infty\cdot 0$ and $\infty+\infty=0$. The equality $0\cdot \infty=1$ indeed can be deduced from the axioms, but with $\infty+\infty=0$ I have troubles. | |
| Apr 9 at 7:15 | comment | added | YCor | How do you define $0\cdot\infty$, $\infty\cdot 0$, and $\infty+\infty$? | |
| Apr 9 at 6:08 | history | asked | Taras Banakh | CC BY-SA 4.0 |