Is there a term for permutations where the elements are optionally included?

Question

The permutation for abc would be:

abc acb bac bca cab cba

But if the elements are optional:

"" a b c ab ba ac ca bc cb abc acb bac bca cab cba

Is there a term for when we include the elements optionally?

Answer 1

In semigroup theory there is the notion of partial transformations on a set $X$ which are, roughly, self-maps of $X$ defined only a subset of $X$. This allows you to (again, roughly), speak of all functions into $X$ whose domain is any subset of $X$, and to speak of them all at once. What you have are the partial injections on $\{a,b,c\}$. Alot of this is central to the theory of inverse semigroups.