Axiomatizability of equational theories with no less than n equations.

Question

For every positive integer $n$, is it the case that there are equational theories of a single binary operation $*$ that can be axiomatized with $n$ equations, but not with $n-1$ equations? And also, are there equational theories that can't be finitely axiomatized?

Answer 1

For your first question, there are $C_n$ many ways to parenthesize $n+1$ many terms, where $C_n$ is the $n$th Catalan Number. Then for any number of axioms you want, you can find $N$ big enough so that you can parenthesize $N$ terms in more than $2n$ ways. Then by setting pairs of these equations to be equal to each other, you can find $n$ equations which are all necessary. I admit I have not worked through the details, though.

For your second question, it is "well known" that the theory of the Lyndon Groupoid is not finitely axiomatizable. The Lyndon Groupoid has the following multiplication table (thanks to Keith Kearnes for the correction):

\begin{array} [c]{c|ccccccc} L & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 4 & 0 & 4 & 5 & 6 & 0 & 0 & 0 \\ 5 & 0 & 5 & 5 & 5 & 0 & 0 & 0 \\ 6 & 0 & 6 & 6 & 6 & 0 & 0 & 0 \\ \end{array}

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I hope this helps ^_^

Answer 2

1. Yes, in a sort of trivial way: you can give the theory $n$ unrelated operations and have one equation imposing a nontrivial condition on each operation. A more interesting question, I guess, would be if the same thing could be done with a fixed number of operations; probably the answer is still yes but it would take a bit more cunning.

2. Yes, again in a sort of trivial way: you can give the theory countably many unrelated operations and countably many equations imposing a nontrivial condition on each...