I am have recently started to learn about model theory, so this might be a stupid question. To learn model theory, I am reading David Marker's Model Theory. This is the situation in the proof of 3.1.4:
We have a model $M$ of a theory $T$. Moreover, we have a set of new constant symbols $d_1,\dots,d_m$. Then, Marker says "Let $A$ be the substructure of $M$ generated by $d_1,\dots,d_m$".
I don't understand how a substructure can be generated by a set of constant symbols as they are part of the Language $L$ not of the domain of $A$. Where is my lack of understanding?
Am I right that substructures are normally generated by giving the domain? Or is it indeed possible to alter the language to generate a substructure?