On finitely recognizable semigroups

We analyze some algebraic properties of semigroups whose finite subsets are recognizable (finitely recognizable semigroups). We show that these semigroups are stable and their subgroups are of finite order. As a consequence of this we prove an interesting decomposition theorem for finitely recognizable semigroups which are finitely generated. Moreover we give more equivalent characterizations of these semigroups under the additional hypothesis that they have aJ-depth function; we show, in particular, that this class of semigroups coincides with the class of finiteJ-above semigroups. Finally we prove that finitely recognizable semigroups which are finitely presented in a finitely based variety have a solvable word problem.