What is the smallest possible cardinality of a non-finitely based magma?

Question

I was told that every magma $(S,*)$ whose base set $S$ has 2 elements has a finite basis of identities. The natural question is, what is the smallest possible cardinality of a non-finitely based magma? I would be very interested to see a 3-element set and a binary operation on it which is not finitely based.

Answer 1

V.L. Murskii in 1965 found an example of a 3 element magma that is not finitely based.

Here is its multiplication table:

$$\begin{array}{c|ccc} & 0 & 1 & 2 \\ \hline 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1\\ 2 & 0 & 2 & 2\\ \end{array}$$