Laws with finite free magmata

Stream: Equational

[Fox Room]

I don't know if simply proposing a question without providing any insight is unwelcome here, but I would like to know what (nontrivial) laws have finite free magmata, if any. This was inspired by learning the rather unintuitive (to me, anyways) fact that the free band on n generators is finite. While a band is characterized by two axioms (idempotent law and associative law) I am very curious if there are other single axiomata with this "free finiteness" property, or perhaps a single axiom characterization of a band.

Of course, the free idempotent magma (I've taken to calling idempotent magmata 'quasibands' which fits nicely with the extant terminology of 'quasisemilattice' for the version with commutativity, as a semilattice is a commutative band) with 1 generator has 1 element, but this borders on being a trivial fact xP
In any case, laws implying idempotence seem like good candidates.

Presumably free-finite for all orders: 13, 24, 62, 256, 314, 366
Free-finite for at least one order: 106, 332

[Alex Meiburg]

There is at least the law 1571, x = (y ◇ z) ◇ (y ◇ (x ◇ z)). It describes Abelian groups of exponent 2. That fits your criterion. (For any n, there are other, much longer laws to characterize Abelian groups of exponent n.)

[Fox Room]

Free-finite for all orders, with (tentative) explicit formula. Note that all of these are in particular semigroups (associative) unless noted otherwise:

• 13, 24: 2n (not semigroups)
• 4, 5, 41: n (are semigroups, and in fact 4 and 5 are bands)
• 62, 256: 3n (not semigroups)
• 336, 374: 2n
• 463, 3051: 4n (not semigroups)
• 3385, 4197: 2^n + n
• 3349, 4141: 3n (per Bruno Le Floch; not semigroups)
• 3591, 3940: 2^n * n + n
• 3634, 3933: n^2 + n
• 3740: n^2 * 2^n + n
• 3744: n^2 + n
• 4513, 4564: 2n^2 + n
• 4517, 4537: n^2 + 2n
• 4526, 4568: n^2 + n + 1
• 4520: 2n^2 + n
• 26302: 4n^2 (per Bruno Le Floch; semigroup status idk)

Free-finite for all orders (idk formula, this is 1571 as above, so Abelian groups and thus semigroups): 895, 2789

Free-finite for at least order one (trivial; satisfy idempotence and cube-associativity): 9, 28, 48, 102, 103, 228, 238, 270, 412, 418, 428, 618, 824, 827, 829, 1027, 1032, 1033, 1037, 1232, 1233, 1235, 1236, 1237, 2337, 2347, 2381, 2398, 2415, 2499, 2513, 2550, 2584, 2716, 2787, 2804, 2973, 3102, 3108, 3122
Free-finite for at least order one (trivial; satisfy idempotence but not cube-associativity): 96, 297
Free-finite for at least order one (not trivial; do not satisfy idempotence): 106, 224, 332, 387
The latter pair above does satisfy flexibility.

There appears not to be an equation of order 4 equivalent to being a band, although as noted above, equations 4 and 5 do determine bands.

Some additional equations satisfying 151, a (two-sided) weaker form of idempotence and thus may be of note, but none are even cube-associative so then again perhaps not: 66, 124, 127, 138, 152, 153, 156, 159, 160, 161, 162, 166, 167, 168, 206, 1119, 1353, 1448, 1636, 1687, 1697, 1849, 1904, 2260

[Bruno Le Floch]

Are you relying on the explicit descriptions of the free magmas that I wrote in the commentary files? For the laws 4, 13, 41, 62, 336, 463, 3385, 3591, 3634, 3740, 3744, 4513, 4517, 4520, 4526 (and their duals) they agree with your counts. If you have time it would be great to also confirm the descriptions I gave for the free magmas, and not just their cardinal. There are two more laws with finite free magma for all orders: Law 3349, 4141: 3n and Law 26302 (natural central groupoid): 4n^2.

[Fox Room]

Bruno Le Floch said:

Are you relying on the explicit descriptions of the free magmas that I wrote in the commentary files? For the laws 4, 13, 41, 62, 336, 463, 3385, 3591, 3634, 3740, 3744, 4513, 4517, 4520, 4526 (and their duals) they agree with your counts. If you have time it would be great to also confirm the descriptions I gave for the free magmas, and not just their cardinal. There are two more laws with finite free magma for all orders: Law 3349, 4141: 3n and Law 26302 (natural central groupoid): 4n^2.

Maybe? I was using the equation explorer, which has notes for many equations, and perhaps they are sourced from you, but I did not look directly. In any case thanks for the contribution!

EDIT: yes, it looks like the comments on the equation explorer are sourced from you. I simply missed the dual pair of order <5 you pointed out, and 26302 is of too high order to be included in the equation explorer.

[Bruno Le Floch]

In an upcoming PR I will add commentary on the various laws you just listed, as well as commentary around law 14 (semisymmetric quasigroup) and many related laws. If you have energy, here are some ideas.

1. Law 124 implies law 8 and 359 hence its free one-generator magma has two elements. Are there any other laws implying (8 and 359), or (307 and 359), or similar combinations? If you have a list I can add commentaries on them.

2. It would be very useful if you manage to confirm (or even formalize?) some of the free magma descriptions. If you come across some equations for which you think the free magma might be finite, or even some which deserve other comments I'm also interested.

3. I couldn't figure out how to systematize the search of finite free magmas with an ATP. But for any given law, we can search with mace4 (or other vampire?) for finite models with some elements being distinct. For instance, for law 66 I searched for models with (0*0)*0=1. 0*1=2. 0*2=3. 0*3=4. 0*4=5. etc up to some point, and saw there are very large magmas generated by a single element, so probably its free one-generator magma is infinite.

4. Linear magmas x◇y=ax+by+c with x,y,c in an abelian group and a,b endomorphisms are a convenient class of examples. For law 66, take an integer $m$ and consider $M=\mathbb{Z}/(m^2+m+1)\mathbb{Z}$ with a linear magma operation with $a=m^2$, $b=m$, and $c=1$. I think it obeys law 66 and with some work 0 generates the whole magma. Since m is arbitrarily large, the free magma with one generator has to be infinite.

[Fox Room]

Bruno Le Floch said:

1. Law 124 implies law 8 and 359 hence its free one-generator magma has two elements. Are there any other laws implying (8 and 359), or (307 and 359), or similar combinations? If you have a list I can add commentaries on them.

Oh, good point, I'll have a look.
Bruno Le Floch said:

It would be very useful if you manage to confirm (or even formalize?) some of the free magma descriptions. If you come across some equations for which you think the free magma might be finite, or even some which deserve other comments I'm also interested.

I certainly agree, but I have no idea how to do these things. I'm not actually a mathematician, I'm only good for analyzing mathematical data that other people have produced, remembering terminology from obscure sources, or playing around with Mace4. I may have some interesting comments about implications between combinations of properties in my notes though (found via Prover9, or conjectured via Mace4).
Bruno Le Floch said:

I couldn't figure out how to systematize the search of finite free magmas with an ATP. But for any given law, we can search with mace4 (or other vampire?) for finite models with some elements being distinct. For instance, for law 66 I searched for models with (0*0)*0=1. 0*1=2. 0*2=3. 0*3=4. 0*4=5. etc up to some point, and saw there are very large magmas generated by a single element, so probably its free one-generator magma is infinite.

I was actually thinking about something like this, but I didn't have any real ideas on how to narrow down a search to models that are likely free. I don't quite follow the logic you have here either.
Bruno Le Floch said:

Linear magmas x◇y=ax+by+c with x,y,c in an abelian group and a,b endomorphisms are a convenient class of examples. For law 66, take an integer $m$ and consider $M=\mathbb{Z}/(m^2+m+1)\mathbb{Z}$ with a linear magma operation with $a=m^2$, $b=m$, and $c=1$. I think it obeys law 66 and with some work 0 generates the whole magma. Since m is arbitrarily large, the free magma with one generator has to be infinite.

That's very interesting, but again I don't quite follow. Where did this operation even come from?

[Bruno Le Floch]

If a magma M (obeying some law) is generated by a single element 0, then the magma is smaller than the free single generator magma. So if we can exhibit arbitrarily large such M, the free magma must be infinite.

I see two ways to construct large M:

• With mace4 and 0*0=1. 0*1=2. 0*2=3. 0*3=4. etc, or variants thereof when mace4 fails to find models. Here I made sure to pick relations that imply that {0,1,2,3,4} etc is in the submagma generated by 0. If mace4 continues finding models, it's a hint that the free magma is infinite.

• With a linear model on the set $\mathbb{Z}/k\mathbb{Z}$, with the magma operation x◇y=ax+by+c where $a,b,c\in\mathbb{Z}/k\mathbb{Z}$ (there are slight generalizations). To make it obey the law under consideration, the parameters a,b,c must obey some simple polynomial equations, obtained by plugging the expression of x◇y into the law, expanding the lhs and rhs, and asking for the coefficients of each variable x,y,z,… on both sides to be equal modulo k. Rather than fixing k and finding a,b, it's much easier to go the other way, fixing a,b in a way consistent with the equations and finding modulo what value of k the equations hold. This is what I did for law 66 to find $(a,b,c,k)=(m^2,m,1,m^2+m+1)$. Finding such a model is not quite enough, it has to have a large submagma generated by one element (not necessarily $0$).

I'm slowly going through your list of equation comments

[Fox Room]

Bruno Le Floch said:

1. Law 124 implies law 8 and 359 hence its free one-generator magma has two elements. Are there any other laws implying (8 and 359), or (307 and 359), or similar combinations? If you have a list I can add commentaries on them.

So, we have four pairs: 8 and 359, 307 and 359, 8 and 23, or 23 and 307. However, we actually need either 151 or 3659 to additionally hold.

41, 106, 124, 206, 224, 332, 335, 336, 374, 384, 434, 887, 895, 1043, 1119, 1636, 1687, 1849, 1904, 2536, 3145, 3566, 3583, 3587, 3600, 3634, 3715, 3716, 3718, 3735, 3736, 3744, 3755, 3790, 3919, 3926, 3929, 3930, and 3933 satisfy these constraints.

[Fox Room]

Unfortunately, I'm pretty sure it's not possible to get anything else as simple as this when restricted to 4 operations, because something like (x * (x * x)) * (x * (x * x)) = x is already 5.
(Well, you could enforce something like (x * (x * x)) * y = x (204), but in any case it gets quite tedious to check.)

[Fox Room]

I imagine most of these diverge to infinity at even 2 generators anyways.

[Fox Room]

I really feel like there should be a tool like Mace4, but instead of finding a finite model from your axioms, with settings for constraining the order, it finds subsets of the free magma satisfying your axioms, with settings for how many generators you want to consider and how many "generations" you want to consider. If anything has a small enough "greatest generation" it would be verifiably finite. This seems completely within the realm of possibility, but I don't think it exists and I haven't the slightest clue how to do it.

There's a relevant OEIS entry, actually. And another.

[Alex Meiburg]

Possibly part of the reason it doesn't exist is that, depending on what exactly you want to do, you can run into undecidability questions very quickly.

I imagine what you're saying is (if you take, say, 2 generators), you have the expressions "x" and "y", then {x*x,x*y,y*x,y*y}, then all their left-three-way products and the right-three-way products, and so on, and each of these is a new generation; and each element, you try to check if it's equal to one of the expressions you already have. And if the whole thing closes at some point (all elements in a new generation are equal to elements in earlier generations) then you're done, you have your proof of finiteness.

But just deciding whether two elements are equal, given a minimal set of equations, is undecidable. So just this "subroutine" can't be guaranteed to terminate.

So this means you need to simultaneously search across adding more generations, and trying to prove elements are equal by equations with more and more steps in hope you find a relation.

Or, you adopt a different strategy, like: take more and more generations, and only quotient out by single applications of an equation. That is, if I have equations that x*x = x*(y*(x*x)) and x*(y*(x*x)) = y, then I won't get x*x = y until I include x*(y*(x*x)) in my generation. Once I do, I take the transitive closure of the equations, but only with my current elements. This strategy is much simpler to implement, and obviously complete. But I think it would be very painfully inefficient in practice: many interesting relations have consequences that only (when you explicitly chain the equations together) occur at fairly long words.

[Alex Meiburg]

At one point I suggested (and there was a tiny bit of interest in) describing all 4694 free magmas. That would certainly make this task, of finding which free magmas are finite or not, very easy then. :)

[Fox Room]

Ah, crap, you're right.

[Fox Room]

Describing all 4694 free magmata would be awesome! Seems like quite a lot of nontrivial work I have no input for though...

[Bruno Le Floch]

Thanks for the list @Alex Meiburg. Do we have a list of the confluent laws anywhere? That would be a start.

[Alex Meiburg]

I don't know of any such list. It sounded like maybe @Amir Livne Bar-on would know.

[Alex Meiburg]

I think an easy place to start would be any laws that imply associativity, since it's possible to always think of them as Lists quotiented by some other relation, which to me becomes much easier to think about than messy expression trees.

[Bruno Le Floch]

I already noticed last week that free magmas for laws that imply associativity are easy so almost all of them are described in commentary (maybe only in my upcoming PR?). I'll make sure to complete the list. I just messed up slightly by writing "power set" of the set of generators instead of "finite power set". E.g., the free 895 magma is the free abelian group of exponent 2 over that set S of generators, and the elements of that free magma are finite subsets of S. EDIT: now it's been merged so the comments are available on the Equation Explorer.

[Bruno Le Floch]

The one-generator free magma has the following sizes (I omit dual equations and equivalents, I stop at order 3):

• cardinal 1 (idempotence) for laws [2,3,4,9,48,102,103];
• cardinal 2 for laws [13,41,106,124];
• cardinal 3 for law [62, finite-65, 72];
• infinite for laws [1,8,10,11,16,38,40,43,47,138].

For the following equations the size is at least 7 (probably infinite): law 14, 49, 50, 52, 53, 55, 56, 58, 63, 66, 73, 75, 99, 100, 101, 104, 105, 107, 108, 109, 111, 115, 117, 118, 125, 127, 151, 152, 153, 159, 160, 161, 162, 166, 167, 168

The Asterix law 65 is special: the only finite magmas generated by a single generator have size 3 or 1. I think the free magma with one generator is infinite, but its largest finite quotient has three elements.

[Fox Room]

how did you establish the infinite ones?

[Bruno Le Floch]

Thanks for asking, it made me realize my logic was completely wrong because I misremembered the notion of simple magma. But results here are still correct.

Key idea: any magma M that is generated by one element x is a quotient of the free magma F with one generator s, by mapping F→M, s↦x. Thus |M|≤|F|. So if we find arbitrarily large magmas generated by one element, the free magma is infinite. This deals with the laws [1,8,10,11,16,38,40,43,47,138]. But I failed for equation 14, so I'm removing it in the list above.

Details for laws whose free magma is infinite.

• Laws 1, 8, 11 are obeyed by the models on Nat with x*x=0, x*0=x, all other products arbitrary. Taking for instance 0*x=x+1 for x>0 we find that 1 generates the whole magma. This works modulo any n too.
• Law 10 is obeyed by a model on Nat given by 0◇0=1 and for x>0, x◇0=x, x◇(x-1)=x, x◇x=x+1, x◇(x+1)=x, and all other products vanish (so x◇y=0 for y>x+1 or 0<y<x-1), and that magma is generated by 0. A finite model on [0,n-1] is obtained by setting (n-1)◇(n-1)=0 instead.
• Law 16: the model x*y = (x+1 if y=x else x if y=x+1 else y) on Nat is generated by 0. This works modulo any n too.
• Law 38: x◇y=x+1 on Nat is generated by 0. This works modulo any n too.
• Law 40 (unipotence): x◇y=(0 if x=y else x+1) on Nat is generated by 1. This works modulo any n too.
• Law 43 (commutativity): strictly positive integers with addition are generated by 1. This works modulo any n too.
• Law 47: on [0,n-1], 0◇0=1, 0◇1=2, (n-1)◇0=1, (n-1)◇1=n-1, (n-1)◇(n-1)=0, and x◇x=x+1 and 0◇x=x for x∈[1,n-2], and all other products vanish. This easily extends to Nat.
• Law 138 is obeyed by x◇y=y+(-1)^x on all integers, which is generated by 0. This works modulo any n too.

I insist in the Details about the possibility to reduce the model to a finite model because for the Asterix equation 65 the free magma is (probably) infinite but its only finite quotients are of size 1 (trivial) and 3, so in any finite magma any element generates at most a three element submagma.

[Bruno Le Floch]

• Laws [117,127,138,151,159,166]: The model for 138 also works for the equations implied by it so they have infinite free magmas.
• Laws [160, 161]: are equivalent to lower-numbered equations (127 and 138) so should not be listed.
• Laws [151, 152, 166, 167, 168]: Knuth (D. E. Knuth, "Notes on central groupoids," Journal of Combinatorial Theory 8 (1970), 376-390. [Especially page 389.]) described the free central groupoid (law 168) in a way that makes it easy to see that all powers a*(a*(a*…)) are distinct. The free magma of 168 is infinite, hence also for the consequences law 151, 152, 166, 167.
• Laws [14,53,63,73,115,118,125]: have linear models (over $\mathbb{C}$ or large finite fields) in which 1 generates an infinite submagma
• Laws [49,50,52,55,56,58,75,99,100,101,104,105,107,108,109,111,152,153,162] have models (generated by a single generator) that appear easily generalized to arbitrarily large sizes and to infinite size.

Linear models

• Law 14: $x\diamond y=-x-y$ in any ring, for instance $\mathbb{Z}/n\mathbb{Z}$. By repeatedly squaring, the element $1$ generates all powers of $-2$, an arbitrarily large submagma as $n\to+\infty$.
• Laws 73 and 118 and 125 have a model over $\mathbb{C}$ (actually $\mathbb{Z}[\omega_3]$ where $\omega_3$ is a primitive third root of unity), with $x\diamond y=\omega_3(x+y)$, in which $1$ generates an infinite submagma (that includes for instance powers of $2\omega_3$). The model also works over any finite field with a third root of unity, but showing that 1 generates a large submagma is harder.
• Law 53 has a model over $\mathbb{Z}[\omega]$ defined by x◇y=(1+ω)x+ωy, with ω primitive third root of unity.
• Law 63 has x◇y=-x+ωy again over $\mathbb{Z}[\omega]$.
• Law 115 has a linear model on $\mathbb{C}$ with $x\diamond y=-b^2 x+b y$ with $b$ a complex root of $b^3-b-1$. The submagma generated by $1$ contains all powers of $b-b^2$ (which is not a root of unity) hence is infinite. The model also works over any finite field with a root of $b^3-b-1$ but showing that 1 generates a large submagma is harder.

laws and models that seem generalizable to arbitrarily large sizes

law 49: x=x*(x*(y*x))
[[1, 2, 0, 4, 0, 4, 0, 4, 0, 0],
 [1, 2, 0, 1, 0, 1, 0, 1, 0, 0],
 [1, 2, 3, 1, 0, 1, 0, 1, 0, 1],
 [3, 2, 3, 4, 0, 1, 0, 1, 0, 0],
 [1, 4, 0, 1, 5, 1, 0, 1, 0, 0],
 [5, 2, 5, 1, 0, 6, 0, 1, 0, 0],
 [1, 6, 0, 1, 0, 1, 7, 1, 0, 0],
 [7, 2, 7, 1, 0, 1, 0, 8, 0, 0],
 [1, 8, 0, 1, 0, 1, 0, 1, 9, 1],
 [3, 2, 9, 9, 0, 1, 0, 1, 0, 0]]

law 50: x=x*(x*(y*y))
[[1, 2, 0, 0, 0, 0, 0, 0, 0, 0],
 [2, 1, 0, 0, 0, 0, 0, 0, 0, 0],
 [3, 3, 1, 2, 0, 0, 0, 0, 0, 0],
 [4, 2, 3, 1, 0, 0, 0, 0, 0, 0],
 [5, 2, 4, 0, 1, 0, 0, 0, 0, 0],
 [6, 2, 5, 0, 0, 1, 0, 0, 0, 0],
 [7, 2, 6, 0, 0, 0, 1, 0, 0, 0],
 [8, 2, 7, 0, 0, 0, 0, 1, 0, 0],
 [9, 2, 8, 0, 0, 0, 0, 0, 1, 0],
 [9, 0, 0, 0, 0, 0, 0, 0, 0, 1]]

law 52: x=x*(y*(x*x))
[[1, 2, 0, 1, 2, 3, 4, 5, 6, 0],
 [1, 2, 0, 1, 2, 3, 4, 5, 6, 0],
 [1, 2, 3, 1, 2, 3, 4, 5, 6, 0],
 [1, 2, 3, 4, 2, 3, 4, 5, 6, 0],
 [1, 2, 0, 4, 5, 3, 4, 5, 6, 0],
 [1, 2, 0, 1, 5, 6, 4, 5, 6, 0],
 [1, 2, 0, 1, 2, 6, 7, 5, 6, 0],
 [1, 2, 0, 1, 2, 3, 7, 8, 6, 7],
 [8, 2, 0, 1, 2, 3, 4, 8, 9, 0],
 [1, 9, 0, 1, 2, 3, 4, 5, 9, 0]]

law 55: x=x*(y*(y*x))
[[1, 2, 0, 0, 0, 0, 0, 0, 0, 0],
 [1, 2, 0, 1, 1, 1, 1, 1, 1, 1],
 [2, 2, 3, 0, 2, 2, 2, 2, 2, 2],
 [3, 3, 3, 4, 0, 3, 3, 3, 3, 3],
 [4, 4, 4, 4, 5, 0, 4, 4, 4, 4],
 [5, 5, 5, 5, 5, 6, 0, 5, 5, 5],
 [6, 6, 6, 6, 6, 6, 7, 0, 6, 6],
 [7, 7, 7, 7, 7, 7, 7, 8, 0, 7],
 [8, 8, 8, 8, 8, 8, 8, 8, 9, 0],
 [1, 9, 9, 9, 9, 9, 9, 9, 9, 0]]

law 56: x=x*(y*(y*y))
[[1, 9, 0, 0, 0, 0, 0, 0, 0, 0],
 [0, 2, 9, 0, 0, 0, 0, 0, 0, 1],
 [0, 0, 3, 9, 0, 0, 0, 0, 0, 2],
 [0, 0, 0, 4, 9, 0, 0, 0, 0, 3],
 [0, 0, 0, 0, 5, 9, 0, 0, 0, 4],
 [0, 0, 0, 0, 0, 6, 9, 0, 0, 5],
 [0, 0, 0, 0, 0, 0, 7, 9, 0, 6],
 [0, 0, 0, 0, 0, 0, 0, 8, 9, 7],
 [9, 0, 0, 0, 0, 0, 0, 0, 0, 8],
 [0, 0, 0, 0, 0, 0, 0, 0, 0, 9]]

law 58: x=x*(y*(z*x))
[[1, 2, 0, 4, 0, 6, 0, 8, 0],
 [1, 2, 0, 2, 0, 2, 0, 2, 0],
 [3, 2, 0, 2, 0, 2, 0, 2, 0],
 [3, 2, 0, 2, 0, 2, 0, 2, 0],
 [5, 4, 0, 4, 0, 4, 0, 4, 0],
 [5, 2, 0, 2, 0, 2, 0, 2, 0],
 [7, 6, 0, 6, 0, 6, 0, 6, 0],
 [7, 2, 0, 2, 0, 2, 0, 2, 0],
 [1, 8, 0, 8, 0, 8, 0, 8, 0]]

law 75: x=y*(y*(y*x))
[[1, 2, 0, 3, 4, 5, 6, 7, 8, 9],
 [2, 0, 1, 3, 4, 5, 6, 7, 8, 9],
 [3, 0, 2, 1, 4, 5, 6, 7, 8, 9],
 [4, 0, 2, 3, 1, 5, 6, 7, 8, 9],
 [5, 0, 2, 3, 4, 1, 6, 7, 8, 9],
 [6, 0, 2, 3, 4, 5, 1, 7, 8, 9],
 [7, 0, 2, 3, 4, 5, 6, 1, 8, 9],
 [8, 0, 2, 3, 4, 5, 6, 7, 1, 9],
 [9, 0, 2, 3, 4, 5, 6, 7, 8, 1],
 [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]]

law 104: x=x*((y*x)*x)
[[1, 0, 0, 0, 0, 0, 0, 0, 0, 0],
 [1, 2, 0, 0, 0, 0, 0, 0, 0, 0],
 [2, 0, 3, 0, 0, 0, 0, 0, 0, 0],
 [3, 0, 0, 4, 0, 0, 0, 0, 0, 0],
 [4, 0, 0, 0, 5, 0, 0, 0, 0, 0],
 [5, 0, 0, 0, 0, 6, 0, 0, 0, 0],
 [6, 0, 0, 0, 0, 0, 7, 0, 0, 0],
 [7, 0, 0, 0, 0, 0, 0, 8, 0, 0],
 [8, 0, 0, 0, 0, 0, 0, 0, 9, 0],
 [9, 0, 0, 0, 0, 0, 0, 0, 0, 0]]

law 111 (x=x*((y*z)*y)) implies 99, 100, 101, 105, 107, 108, 109
[[1, 0, 0, 0, 0, 0, 0, 0, 0, 0],
 [2, 1, 1, 1, 1, 1, 1, 1, 1, 1],
 [3, 2, 2, 2, 2, 2, 2, 2, 2, 2],
 [4, 3, 3, 3, 3, 3, 3, 3, 3, 3],
 [5, 4, 4, 4, 4, 4, 4, 4, 4, 4],
 [6, 5, 5, 5, 5, 5, 5, 5, 5, 5],
 [7, 6, 6, 6, 6, 6, 6, 6, 6, 6],
 [8, 7, 7, 7, 7, 7, 7, 7, 7, 7],
 [9, 8, 8, 8, 8, 8, 8, 8, 8, 8],
 [1, 9, 9, 9, 9, 9, 9, 9, 9, 9]]

law 152: x=(x*x)*(x*y)
[[1, 2, 3, 4, 5, 6, 7, 8, 9, 0],
 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
 [1, 2, 2, 1, 1, 1, 1, 1, 1, 1],
 [1, 3, 1, 3, 1, 1, 1, 1, 1, 1],
 [1, 4, 1, 1, 4, 1, 1, 1, 1, 1],
 [1, 5, 1, 1, 1, 5, 1, 1, 1, 1],
 [1, 6, 1, 1, 1, 1, 6, 1, 1, 1],
 [1, 7, 1, 1, 1, 1, 1, 7, 1, 1],
 [1, 8, 1, 1, 1, 1, 1, 1, 8, 1],
 [1, 9, 1, 1, 1, 1, 1, 1, 1, 9]]

law 153: x=(x*x)*(y*x)
[[1, 0, 0, 0, 0, 0, 0, 2, 3, 4],
 [2, 0, 0, 0, 0, 0, 0, 2, 3, 4],
 [3, 0, 7, 0, 0, 0, 0, 2, 3, 4],
 [4, 0, 0, 8, 0, 0, 0, 2, 3, 4],
 [5, 0, 0, 0, 9, 0, 0, 2, 3, 4],
 [6, 0, 0, 0, 0, 6, 0, 2, 3, 4],
 [5, 0, 0, 0, 0, 0, 5, 2, 3, 4],
 [2, 0, 0, 0, 0, 0, 0, 2, 3, 4],
 [3, 0, 0, 0, 0, 0, 0, 2, 3, 4],
 [4, 0, 0, 0, 0, 0, 0, 2, 3, 4]]

law 162: x=(x*y)*(z*x)
[[5, 6, 7, 8, 9, 6, 5, 8, 7, 9],
 [5, 6, 7, 8, 9, 6, 5, 8, 7, 9],
 [5, 6, 7, 8, 9, 6, 5, 8, 7, 9],
 [5, 6, 7, 8, 9, 6, 5, 8, 7, 9],
 [5, 6, 7, 8, 9, 6, 5, 8, 7, 9],
 [0, 2, 1, 4, 3, 0, 1, 2, 3, 4],
 [0, 2, 1, 4, 3, 0, 1, 2, 3, 4],
 [0, 2, 1, 4, 3, 0, 1, 2, 3, 4],
 [0, 2, 1, 4, 3, 0, 1, 2, 3, 4],
 [0, 2, 1, 4, 3, 0, 1, 2, 3, 4]]

[Bruno Le Floch]

Collecting all this we arrive at the following for all laws of order up to 3:

• cardinal 1 (idempotence) for laws [2,3,4,9,48,102,103];
• cardinal 2 for laws [13,41,106,124];
• cardinal 3 for law [62, 72];
• infinite for laws [1,8,10,11,14,16,38,40,43,47,49,50,52,53,55,56,58,63,75,99,100,101,104,105,107,108,109,111,115,117,118,125,127,138,151,152,153,159,162,166,167,168].
• unknown (probably infinite) for 65, 66, explicitly

Equation 65: x = y ◇ (x ◇ (y ◇ x))
Equation 66: x = y ◇ (x ◇ (y ◇ y))

[Fox Room]

Oh okay, awesome.

Key idea: any magma M that is generated by one element x is a quotient of the free magma F with one generator s, by mapping F→M, s↦x. Thus |M|≤|F|. So if we find arbitrarily large magmas generated by one element, the free magma is infinite.

Ohh, yeah okay I should have realized.

[Bruno Le Floch]

Just like the Asterix law 65 x=y◇(x◇(y◇x)), law 713 x=y◇(y◇((y◇x)◇x)) has a free magma whose only finite quotients are of size 1 and 3. These "finite free magmas" are the same: x◇y=y+1 on Z/3Z. I wonder how the two laws are related, maybe one is definable in terms of the other?

[Bruno Le Floch]

Ah, the explanation is that law 65 (Asterix) and 713 are equivalent for finite magmas (or left quasigroups) to law 1491 (Obelix) and 11228, respectively, and that the free magma of either of these laws is Z/3Z with x◇y=y+1.