Remarks on recognizable subsets and local rank

Given a monoid $(M,\varepsilon,\cdot )$ it is shown that a subset $A\subseteq M$ is recognizable in the sense of automata theory if and only if the $\varphi$-rank of $x=x$ is zero in the first-order theory $\operatorname{Th}(M,\varepsilon ,\cdot ,A)$, where $\varphi (x;u)$ is the formula $xu\in A$. In the case where $M$ is a finitely generated free monoid on a finite alphabet $\Sigma$, this gives a model-theoretic characterization of the regular languages over $\Sigma$. If $A$ is a regular language over $\Sigma$ then the $\varphi$-multiplicity of $x=x$ is the state complexity of $A$. Similar results holds for $\varphi' (x;u,v)$ given by $uxv\in A$, with the $\varphi'$-multiplicity now equal to the size of the syntactic monoid of $A$.