How to define a certain language

Question

Suppose we are given a binary function symbol $*$ and a countably infinite set of variables and also a pair of parentheses. In most logic textbooks, they have a rule that says that $(A * B)$ is well-formed whenever $A$ and $B$ are. However, they then say that we can omit the outermost parentheses as an abbreviation. But how does one formally define that language, where outermost parentheses are not allowed? I apologize if this question is similar to one I asked before.

Answer 1

To formally define this, you need to split the definition in two levels: the outermost that does not contain parenthesis, and the inner that does.

First you define expressions $E'$ with parentheses:

$E' := A | (E' \ast E')$.

(Expressions $E'$ are either variables, $A$, or "(expression $\ast$ expression)")

Then, you nest that definition inside another one, $E$, that just adds an extra level of expression-construction without parenthesis.

Expressions without outer parentheses:

$E := A | E' \ast E'$.

So now, you can use expressions of the form described by $E$, which does not have outermost parentheses, but consists of sub-expressions ($E'$) that do contain parentheses.