On Parallel Deletions Applied to a Word

We consider sets arising from a single word by parallel deletion of subwords belonging to a given language. The issues dealt with are rather basic in language theory and combinatorics of words. We prove that every finite set is a parallel deletion set but a strict hierarchy results from kbounded parallel deletions. We also discuss decidability, the parallel deletion number associated to a word and a certain collapse set of a language, as well as point out some open problems.