Bounded prefix concatenation operation and finite bases with respect to the superposition

The paper is concerned with word functions over the alphabet $\{1,2\}$. Given arbitrary one-place functions $f_1,\ldots,f_l$, the class BPC$[f_1,\ldots,f_l]$ is defined as the closure of the set of simplest word functions and the functions $f_1,\ldots,f_l$ under the operations of superposition and bounded prefix concatenation. The class BPC$[f_1,\ldots,f_l]$ is shown to have a finite basis with respect to the superposition.