Suppose I have a first order theory over some signature $\Sigma$ with constant symbols.
Is there a name for the theory I obtain by replacing the constant symbols with unary function symbols along with axioms that make them behave like constants?
For example, if I look at the theory of groups over signature $(2,1,0)$, there is a corresponding theory with signature $(2,1,1)$, where the new function symbol acts as if it sends everything to the (now not explicitly specified) identity element. Without being too specific about the formalism, has this structure been studied independently in any form?