Let $L$ be a first order language. An $L$-structure $M$ consists of a nonempty set, also denoted by $M$, together with the following data:
- for each constant $c$ of $L$, an element $c^M$ of $M$;
- for each $n$-place function symbol $f$ of $L$, a function $f^M:M^n\to M$;
- for each $n$-place relation symbol $R$ of $L$, a subset $R^M\subseteq M^n$.
Given an $L$-structure $M$, we consider the language $L_M := L\cup \{c_m\mid m\in M\}$ by adding constants for each element in $M$. If we stipulate that the interpretation in $M$ of each new constant $c_m$ is the element $m$ itself, then $M$ is also an $L_M$-structure.
Now, I'm my notes, we often restrict a model of a $L_M$-theory (therefore an $L_M$-structure) by "omitting the interpretations of the new constant corresponding to $m\in M$" to obtain an $L$-structure that remains a model.
Other instances of this restriction process happen when there is some $L_M$-sentences that holds in an $L$-theory $T$. We then 'restrict' this $L_M$-sentence by substituting variables for all extra constant which were initially not present in $L$, and the resulting $L$-sentence is then still true in $T$.
I would like to have a rigorous explanation for these processes. Sure, we can omit interpretations of constants, but what happens with formulas that contained these constants then? There's a conceptual bridge that I need to cross here, and I would really appreciate a clear explanation.
Thanks.
This answer only addresses half your question, namely the part about structures. The paragraph beginning "Other instances" is somewhat unclear to me; once clarified, I'll address it here too.
It helps to be a bit more precise in the definition of "structure" in the first place: if $J$ is a language, a $J$-structure $\mathcal{M}$ is a pair $(M,\mathfrak{I})$ where $M$ is a set and $\mathfrak{I}$ is a function. Specifically, we have $$\mathfrak{I}:J\rightarrow M\sqcup\bigsqcup_{n\in\mathbb{N}}\mathcal{P}(M^n)\sqcup\bigsqcup_{n\in\mathbb{N}}Func(M^n,M)$$ (where "$Func(A,B)$" is the set of functions from $A$ to $B$) such that $\mathfrak{F}(c)\in M$ for each constant symbol $c\in J$, $\mathfrak{F}(R)\in\mathcal{P}(M^n)$ for each $n$-ary relation symbol $R\in J$, and $\mathfrak{I}(f)\in Func(M^n,M)$ for each $n$-ary function symbol in $J$.
Now suppose $\mathcal{M}=(M,\mathfrak{I})$ is a $J$-structure and $L$ is a sublanguage of $J$ (think $J=L_M$). The reduct of $\mathcal{M}$ to $L$ is gotten by restricting the function $\mathfrak{I}$: specifically, we let $$\mathcal{M}\vert_L=(M,\mathfrak{I}\vert_L)$$ where $$\mathfrak{I}\vert_L: L\rightarrow M\sqcup\bigsqcup_{n\in\mathbb{N}}\mathcal{P}(M^n)\sqcup\bigsqcup_{n\in\mathbb{N}}Func(M^n,M): x\mapsto \mathfrak{I}(x).$$ This is completely rigorous - language like "forgetting the interpretation of the extra symbols" is merely about restricting the domain of a certain function.
The "$L_M$-to-$L$" construction is just a special case of the above idea. It may help to first consider something like the underlying (additive) group of a ring, which is another special case of the above idea which occurs naturally outside logic.