Algorithmic properties of some fragments of concatenation theory

The paper considers two fragments of the word theory with concatenation. The first fragment has two relations denoting that one of the words is a prefix (respectively, a suffix) of another one. It is proved that this theory is algorithmically equivalent to elementary arithmetic and, therefore, undecidable. The second fragment has a countable set of power operations. It is proved that this theory admits effective quantifier elimination and is, therefore, decidable.