Is there a name for this construction?

Question

Consider a set $S$, which we call the alphabet. What is the name for the least set $T$, such that $S$ is a subset of $T$, all finite sequences from $S$ are in $T$, all finite sequences of sequences from $S$ are in $T$, etc. For example, if we let $S$ be the natural numbers, then some of the elements of $T$ would be, $0$, $(0,1)$, $(1,(1,10),2)$, $((1,10),(2,3))$, etc. Is there a name for this construction? It is not the Kleene closure.

Answer 1

In formal language theory (your tag) there is no common name for this. Not even the different "layers" are treated.

• Your layer zero, the alphabet, is, of course, there.
• Your first layer, sequences of elements of the alphabet are words or strings. Note, however, that there is usually no formal distinction made between symbols and strings of length one. For you, $0$ and $(0)$ are distinct already in notation.
• The next layer sequences of sequences corresponds to (formal) languages. However, languages have no multiplicities and no order of their elements. While there are automata and grammars for multisets of symbols, I am not aware of devices for multisets of strings instead of languages.
• Finally, there are families/classes of languages in different contexts. But again, these are just sets, no order, no multiplicity.
• For higher layers I cannot think of any reference.

So, concluding, not even the single layers appear, let alone a mix of them all. Maybe in some different field there is something like what you are looking for. You could start by asking in general set theory.