There are many examples of non-finitely based varieties. In a finite signature, is there an example of such variety with a known explicit set of identities? 



Example of a non-finitely based variety with explicit set of defining identities
You can look at "Bases for Equational Theories of Semigroups" by P Perkins, J Algebra 11, 298-314 (1968). Theorem 2: the identities


$xyzw=xzyw$ and $yx^ky=xyx^{k-2}yx$ for $k=2,3,\dots$


define a non-finitely based variety of semigroups.


