Equational classes generated by 3-element magmas

Stream: Equational

[Stanley Burris]

A computer study of key properties of equational classes generated by 3-element magmas, properties studied by universal algebraists, appeared in a 1996 paper by Joel Berman and Stanley Burris in
A computer study of 3-element groupoids.
In Logic and Algebra, (The Proceedings of the Magari Conference),
edited by A. Ursini and P. Agliano (1996), 379 - 430, Marcel Dekker, Inc.

A PDF preprint can be found as item 58 on https://math.uwaterloo.ca/~snburris/htdocs/MYWORKS/publ.html

[Terence Tao]

Thanks for the reference! In our project, I guess computational power had advanced sufficiently since 1996 that we were able to brute force the consequences of 3-element and even 4-element magmas without needing to first classify them up to equivalence first, but if compute was more of a bottleneck then this work would certainly have been very helpful. In any event, I will add a reference to this work to our draft paper.

[Bruno Le Floch]

For the study of which equations have full spectrum, the 2-element magmas were very useful as many of them were such that their direct powers had submagmas of all sizes. The 3-element magmas were also quite useful, but a bit too diverse to make such sweeping statements. I'll see what I can learn from that reference, at least in terms of relevant terminology. For instance it might be possible to say that "any equation that is obeyed by this 3-element magma has full spectrum".

[Stanley Burris]

In our paper of 1996 we did not look at the spectrum of the equational class V(A) generated by a 3-element magma A, nor do I know any great results to apply here. After playing around with this a little I am now wondering : does V(A) has full spectrum imply that some 2-element B in V(A) is such that V(B) has full spectrum. If so then your work on 2-element magmas would cover the cases of equations with full spectra obtained from examining 3-element magmas.

[Bruno Le Floch]

If I understood correctly and did not mess up my calculations, [[0, 0, 1], [0, 2, 1], [1, 1, 1]] satisfies your requirements. It has a 2-element submagma {1,2} isomorphic to NAND that does not give full spectrum, but in the k-th power of the 3-element magma one finds a 2k+1 element submagma (then a direct product with some copies of NAND gives any number of elements). The submagma consists of elements of the form (0,\dots,0,x,\dots,x) with any number of zeros to begin with and otherwise all entries equal. For instance for k=3 it is {(0,0,0),(0,0,1),(0,0,2),(0,1,1),(0,2,2),(1,1,1),(2,2,2)}.

I was somewhat lucky: programmatically I was only able to reduce from 3330 three-element magmas to 397 candidates, and thankfully the first one was ok. The next two in my list, [[0, 0, 1], [0, 2, 1], [1, 1, 2]] and [[0, 0, 1], [0, 2, 1], [2, 1, 1]], do not seem to have full spectrum, but I don't know how to prove it. Maybe if we figure this out I can filter the list further.

[Stanley Burris]

Indeed you have found an excellent example. which I will call A, to refute my query. In B&B (my magma catalog with Berman) your example is 562, one of the representatives of the clone equivalence classes. Since it generates a congruence distributive variety V(A), all subdirectly irreducibles in the variety must be quotients of subalgebras of A. Thus the only 2-element algebra in V(A) is the 2-element subalgebra of A. and as you observe, it does not generate a variety with full spectrum. The natural question now is whether or not there is a magma equation that holds in A but not in any 2-element magma with full spectrum.

The next two magmas in your list have B&B catalog numbers 563 and 571. Since 571 is quasi-primal then all finite models of the variety it generates are products of it and its two-element subalgebra, so the spectrum of V(571) is the set of numbers of the form 2^m x 3^n. Definitely not full. 563 looks like a challenge to analyze. It is distinguished in the B&B catalog by having an enormously long Mal'cev term. I have long been inactive in universal algebra, but my guess is that analyzing the spectra of varieties generated by 3-element magmas, even determining which are full, could be a very challenging project.

[Stanley Burris]

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[Stanley Burris]

I didn't see any mention of the famous results of V.L. Murskii --- my apologies if I am repeating items that have already been covered.

First there is his result from 1965 that the equational theory of the 3-element magma whose table is [ [0 0 0], [0 0 1], [0 2 2] ] does not have a finite basis (is not finitely axiomatizable).( All 2-element magmas have equational theories that are finitely based.)

Second, and more dramatic, is his result from 1975 that says almost all finite algebras are quasi-primal. I have not studied this result but my understanding is that it applies to magmas. Quasi-primal algebras are probably the most successful generalization of the 2-element Boolean algebra. If A is quasi-primal then it is hereditarily simple (all subalgebras are simple), its equational theory is finitely based, and the finite members of the variety V(A) that it generates are precisely the finite products of subalgebras of A, consequently its spectrum is far from being full. The infinite members of V(A) also have a powerful structure theorem, namely they are "Boolean products" of the simples.

For details on the universal algebra involved, but not Murskii's results, see the online 2012 free PDF copy of "A Course in Universal Algebra", originally published in 1981 by Springer Verlag. The 2012 version has several corrections to the original text, but does not make an attempt to bring the reader up to date. It does give the biblio info on the two papers of Murskii mentioned here.

[Terence Tao]

Thanks! I'll incorporate these references into the draft paper soon. EDIT: equational#1159

[Stanley Burris]

My colleague Joel Berman just informed me that the result is stronger than I stated above, namely almost all finite magma are idemprimal, so in addition to being quasi-primal they have no non-trivial proper submagmas and have only the trivial automorphism. For such a magma A the only finite members of V(A) are powers of A.

For references Joel said Cliff Bergman’s 2012 book Universal Algebra contains a very detailed proof of The probability that a random finite binar is idemprimal is 1. And another place where Murskii’s "almost all finite algebras”  results are proved in detail is in Volume III of Algebras, Lattices Varieties by Freese, McKenzie, McNulty and Taylor. Section 9.7 of that book it titled “Properties of ‘Almost all Finite Algebras’ ”.

A quick search on idemprimal led me to a 2020 Master's Thesis online by Florian Aichinger titled Probability in Universal Algebra that covers the topic following Bergman's book, proving the result for idemprimal groupoids.

[Кирилл Фролов]

There is a typo in commit https://github.com/teorth/equational_theories/commit/133c881ef09685cdeff94ca97a9ca3ec1dbe72bf
It duplicates line It was also shown in \cite{murskii-2} that ``almost all'' magmas $M$ (in a certain precise sense) are

[Terence Tao]

Thanks, I fixed it now.

[Shreyas Srinivas]

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