Timeline for Is there an identity between the associative identity and the constant identity?
Current License: CC BY-SA 4.0
14 events
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| Jul 20, 2023 at 21:38 | comment | added | YCor | @TerryTao Ah, indeed, I focussed on constancy of the law (instead of the triple law). I now see the point of your comment. | |
| Jul 20, 2023 at 15:59 | comment | added | Terry Tao | @YCor All of the laws you list are not obeyed by the magma I described in my comment, and so are not consequences of the triple constant law (which is one of the four requirements of the "improved" version of the OP question. Of course, you can eliminate each of these laws by other means on a case-by-case basis, as you do in your comment, if you prefer. | |
| Jul 20, 2023 at 7:45 | comment | added | YCor | If $t\neq t'$ have both length $<3$, I don't understand @TerryTao's conclusion (one needs a non-associative law satisfying the identity). If length is 1,1, this implies constant. If length is 1,2, then this is not implied by constant law. If length is 2,2, we have (modulo flip) one of 1 $xy=zt$, 2 $xy=xz$, 3 $xy=zx$, 4 $xy=yx$, 5 $x^2=yz$, 6 $x^2=y^2$, 7 $x^2=xy$. Each of 1, 3, 5 implies constant. Each of 2, 4, 6, 7 (which is equivalent to 2) has obvious non-associative models. For instance if $f$ is a non-idempotent self-map on a set, the law $xy=f(x)$ satisfies 2 and is non-associative. | |
| Jul 18, 2023 at 16:50 | comment | added | Pace Nielsen | @TerryTao Excellent! I upvoted your answer. | |
| Jul 18, 2023 at 16:38 | comment | added | Terry Tao | OK, it turns out that a small perturbation of the associative axiom of this form works. See my other answer. | |
| Jul 18, 2023 at 16:08 | comment | added | Terry Tao | Actually, your first magma already rules out the case where $t'$ has repeated variables and $t$ is of the indicated form ($x(yz)$ with $x,y,z$ not necessarily distinct). Similarly if $t'$ has repeated variables. So the only remaining cases are of the form $x(yz) = (pq) r$ where $x,y,z$ are distinct and $p,q,r$ are distinct, but there can be repetitions between $x,y,z$ and $p,q,r$. | |
| Jul 18, 2023 at 15:47 | comment | added | Terry Tao | If one similarly takes the free magma on three generators $a,b,c$ and replaces any element outside of $\{a,b,c,ab,bc,(ab)c, a(bc)\}$ by zero, one obtains a non-associative magma that can be used to rule out the case where $t$ and $t'$ both involve repeated variables. | |
| Jul 18, 2023 at 15:43 | comment | added | Terry Tao | To handle the case where $t,t'$ both have two letters or less, consider the free magma on four generators $a,b,c,d$, and then replace any expression of length greater than two by zero, thus giving a new magma $\{0,a,b,c,d,ab,ac,ad,ba,bc,bd,ca,cb,cd,da,db,dc\}$ with constant triples. This magma will not obey the identity $t=t'$ unless $t,t'$ are identical strings, but then the identity is tautological and will not imply associativity. | |
| Jul 18, 2023 at 13:38 | comment | added | Pace Nielsen | Yes, I meant 3 letters, not three products. That's now fixed. Also, I'll handle that extra possibility you mentioned shortly. | |
| Jul 18, 2023 at 13:37 | history | edited | Pace Nielsen | CC BY-SA 4.0 |
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| Jul 18, 2023 at 13:32 | comment | added | Emil Jeřábek | I don't see the conclusion of the third paragraph. Another possibility is that both sides of the equation have strictly less than 3 variables occurrences. | |
| Jul 18, 2023 at 13:29 | comment | added | Emil Jeřábek | When you write "3 products", you actually mean "2 products (of 3 variable occurrences)", right? | |
| Jul 18, 2023 at 13:01 | history | edited | Pace Nielsen | CC BY-SA 4.0 |
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| Jul 18, 2023 at 12:36 | history | answered | Pace Nielsen | CC BY-SA 4.0 |