Has this model theory definition been studied in the literature?

Question

Let $S$ be a non-empty set. I define the "total structure" of $S$ to be the following structure. First, there is a distinct symbol for every constant $s$ of $S$. Next, there is a distinct symbol for every $n$-ary function of $S$. And last, there is also a distinct symbol for every $n$-ary relation on $S$. So, for example, on the set $\{0,1\}$, the total structure of $\{0,1\}$ includes the constants $0$ and $1$, all unary, binary, ternary, etc relations on $\{0,1\}$, and all unary, binary, ternary, etc operations on $\{0,1\}$. Has this definition been mentioned in the mathematical literature?

Answer 1

I don't think this construction has a name. I have seen applied sometime to obtain (trivially) that all types over $S$ are definable.

Ofcouse if $S$ is infinite no elementary extension of it is "total".