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

V.L. Murskii in 1965 found an example of a 3 element magma that is not finitely based.
Here is its multiplication table:
$$\begin{array}{c|ccc}
& 0 & 1 & 2 \\
\hline
0 & 0 & 0 & 0 \\
1 & 0 & 0 & 1\\
2 & 0 & 2 & 2\\
\end{array}$$
