  We explore new interactions between finite model theory and a number of
classical streams of universal algebra and semigroup theory. A key result is an
example of a finite algebra whose variety is not finitely axiomatisable in
first order logic, but which has first order definable finite membership
problem. This algebra witnesses the simultaneous failure of the {\L}os-Tarski
Theorem, the SP-preservation theorem and Birkhoff's HSP-preservation theorem at
the finite level as well as providing a negative solution to a first order
formulation of the long-standing Eilenberg Sch\"utzenberger problem. The
example also shows that a pseudovariety without any finite pseudo-identity
basis may be finitely axiomatisable in first order logic. Other results include
the undecidability of deciding first order definability of the pseudovariety of
a finite algebra and a mapping from any fixed template constraint satisfaction
problem to a first order equivalent variety membership problem, thereby
providing examples of variety membership problems complete in each of the
classes $\texttt{L}$, $\texttt{NL}$, $\texttt{Mod}_p(\texttt{L})$,
$\texttt{P}$, and infinitely many others (depending on complexity-theoretic
assumptions).
