Star free language vs. regular language
Schützenberger (1965) gave an algebraic characterization of the star-free languages: a regular language is star-free if and only if its syntactic monoid is aperiodic. Contrary to the logical characterization (star-free = FO[&lt;]), this algebraic characterization gives an algorithm to decide whether a given regular language is star-free (the regular language can be given by a finite automaton, a regular expression or a regular grammar). Using the logical characterization (due to McNaughton and Papert) one can then decide the following problem: given a WMSO formula, is there a FO formula describing the same language? 

M.-P. Schützenberger, On finite monoids having only trivial subgroups, Information and Control 8 (1965), 190-194.

R.~McNaughton and S.~Papert, Counter-free automata, The M.I.T. Press, Cambridge, Mass.-London, 1971.

A full proof of Schützenberger's theorem can be found in various textbooks or survey papers. For an elementary presentation of the corresponding algorithm (without a proof), see 

J.-É. Pin, Finite semigroups and recognizable languages : an introduction, in NATO Advanced Study Institute Semigroups, Formal Languages and Groups, J. Fountain (éd.), 1-32, Kluwer academic publishers, (1995).