A function that is the response function of a generalized sequential machine, and therefore generalizes the notion of sequential function. Without constraining the machine to have a finite state-set, generalized sequentiality is equivalent to the following property of initial subwords preservation:

for all u,v in I^*, f(uv) has the form f(u)w for some w in O^*, where I^* and O^* are the sets of all input and output strings.