Consider a set $X$ which contains strings of $n$ bits. Each string starts with zero and it is non-zero string. For example n=3
$$ 001\\ 010\\ 011 $$ and for $n=4$
$$ 0 0 0 1\\ 0 0 1 0\\ 0 1 0 0\\ 0 0 1 1\\ 0 1 0 1\\ 0 1 1 0\\ 0 1 1 1 $$
Is there a proper name for this? especially in boolean functions i.e. constant or dictator functions. Also, is there a general name when the domain is not binary i.e., $\{0,1,..z\}$ ?
On
The set you're describing doesn't have a formal name that is common in the mathematical community. As some of the comments have indicated you could describe it in several ways, but ultimately if you want to name it formally, go for it.
As for the last part of your question, the set $\mathbb{Z}_n = \{0, 1, \ldots, n-1\}$ is often times referred to as either the least residues mod $n$ or just $\mathbb{Z}$ mod $n$. This set represents the set of possible remainders when an integer is divided by $n$.
I have an answer to your question. Firstly let me define what is an Initial Segment is.
Let, $(X,\le)$ be a Well-ordered set and $A(\ne\emptyset)\subset X$ and $b\in A$, then the set $A_b=\{x\in A:x\le b,x\ne b\}$ is called an Initial Segment of $A$. Geometrically you can visualize it as cutting a line to segments.
Now in your problem for some $n\in \Bbb N$ the set $X$ consists of Binary representation of all the Positive integers which start with $1$ and end with $2^{n-1}-1$ (whose binary representation is $011111...1,(n-1)$ ones).
So I can redefine your $X$ as $X_n=\{y\in\Bbb N:y<2^n-1\}, $for all-natural numbers $n$, which I would like to call the initial segment of $\Bbb N$ at $2^n-1$. So, the given set $$X=\Bbb N_n,\forall n\in\Bbb N$$ $\Bbb N =\{1,2,3,4,5,6,...\}$ Set of Natural Numbers.
Hope it works.