Twin pairs of equations

Stream: Equational

[Bruno Le Floch]

Looking through the commentary folder, I see mention of twinning: equations A and B are twinned if they imply/are implied by the same set of equations (up to order 4). Two pairs were found,

• equations 124 (x=y◇((y◇x)◇x)) and 1119 (x=y◇((y◇(y◇x))◇x)).
• equations 476 (x=y◇(x◇(y◇(y◇x)))) and 503 (x=y◇(y◇(x◇(y◇x)))),

Twinning is quite visible in the linear models $x\diamond y=ax+by$. Each of these four equations admits two types of linear models: $(a,b)=(0,b)$ with $b^2=1$ (for the first pair) or $b^4=1$ (for the second pair), and $(a,b)=(1-\psi,\psi)$ with $\psi$ some algebraic number that is different for all four equations (root of $\psi^2-\psi-1$, $\psi^3-\psi-1$, $\psi^3+\psi^2+1$, $\psi^3+\psi+1$ for 124, 1119, 476, 503, respectively).

• For multi-variable consequences, the rich set of linear models, and especially the complexity of $\psi$, makes it hard for the linear models to obey any low-order equation (variables must appear several times in the equation, in specific ways). Consequences of the equations are of high order, and we don't see them until order 5.

• When restricting to expressions of a single variable, the value of $\psi$ is unimportant (since these linear models are idempotent), so this difference between twins disappears. This leads twinned equations to share some one-variable consequences. In fact, the free 124-magma and free 1119-magma with one generator are the same (see below). I have no idea if 476 and 503 have the same one-generator free magma.

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All consequences of Equations 124 or 1119 (see the Equation Explorer for 124) are single-variable equations, except for 3319 and 3915, which are simply equation 8 (x=x◇(x◇x)) multiplied on both sides by y. To see that their free magma with a single generator is the same two-element magma $\{x,x^2\}$ with $\_\diamond x = x^2$ and $\_\diamond x^2 = x$, one simply has to show that both equations imply equations 8, 151, 359. That's easy for 124, harder for 1119.

Equation 5936 x = y ◇ (y ◇ (x ◇ ((y ◇ x) ◇ x))) is a consequence of 124, but is not satisfied by linear models of 1119. Thus, twinning must stop there. (I didn't check if that is the earliest order 5 equation with these properties.)

I find the other direction unclear, namely equations that imply 124 or 1119. Only very basic equations show up: the singleton, the right projection, and (an equation equivalent to) x=(y◇(z◇x)). The last two "correspond" to the linear models with $(a,b)=(0,\pm 1)$. There is presumably no short equation (other than 124 and 1119) whose linear model is $(a,b)=(1-\psi,\psi)$ with the above $\psi$, and that could be a partial explanation.

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For the second pair 476/503, there is quite little data to explain: the equations only imply one non-trivial equation 411 x=x◇(x◇(x◇(x◇x))), which is the obvious specialization setting all variables equal, and are implied by (equivalence classes of) equations of the form $L_{u_1}\circ\dots\circ L_{u_k} = \mathrm{id}$ for $k=1,2,4$, which is obvious from the form of the equations $x=L_? L_? L_? L_? x$.

Terence Tao proposed an explanation for the second pair: 476 and 503 magmas that are additionally assumed to be left-cancellative are mapped to each other by changing the operation to $x\tilde\diamond y = L_x^{-1}(y)$. Checking that 476 is swapped with 503 is routine. However, note that the equations implied by 476 and 503 would then generally differ by the same transformation. I think that this inversion is still helpful to see why there could be two equations with some "simple" non-idempotent linear models $(a,b)=(0,\pm i)$ and $(a,b)=(0,-1)$ and yet different complicated idempotent linear models $(a,b)=(1-\psi,\psi)$ with different $\psi$, which made them have very few multi-variable consequences.

[Terence Tao]

Thanks for the analysis! I have nothing mathematical to add, but I updated the commentary with a link to this thread.

[Bruno Le Floch]

Thanks for adding to the commentary. My explanation (same or similar one-generator free magma, and few low-order multivariate consequences) is still very incomplete, but I won't investigate further.

The linear models don't fully explain why the four equations have very few low-order multi-variable consequences: all linear models of all four equations obey equation 375 (x ◇ y = (x ◇ x) ◇ y), but that equation is not actually implied by any of 124, 1119, 476, 503.

• It especially remains mysterious why equation 124 has so few low-order consequences. Its linear models all obey quite a few multi-variable equations of order up to 4, as listed by the script twinning.py. (I filtered out all equations that simplify when using one-variable equations.) Nevertheless, we know that none of these (except 124) are really consequences of equation 124, thanks to counter-models that are not linear.

124 x = y ◇ ((y ◇ x) ◇ x)
375 x ◇ y = (x ◇ x) ◇ y
1109 x = y ◇ ((y ◇ (x ◇ x)) ◇ x)
1322 x = y ◇ (((y ◇ y) ◇ x) ◇ x)
1728 x = (y ◇ y) ◇ ((y ◇ x) ◇ x)
1924 x = (y ◇ (y ◇ x)) ◇ (y ◇ x)

• For equations 476 and 503, it is not even clear why the one-variable consequences are similar, and how far (in higher orders) this similarity extends. The fact that 476 implies 3862 x ◇ x = (x ◇ (x ◇ x)) ◇ x for finite magmas but 503 does not is especially unexplained.