There is a definition of a 'monotone machine' in Li & Vitany's Book, and another one which is for example stated in this paper via c.e. (computably enumerable) sets. I can't see why these definitions should be equivalent.
Furthermore in Li & Vitany's book, there is a statement which says that monotone machines compute p.r. (primitive recursive) functions f with $f(x) \le f(y)$ when $x \le y$ (lexicographical ordering).
I don't see how such a function can be computed by a monotone machine.
Maybe someone can explain this to me or refer to a paper / book which descripes these connections.
The first definition: A monotone machine has a one-way read-only input tape, a one-way write-only output tape and work tapes. The input is read one bit at a time, and the output written one bit at a time, with the one-way nature of the output tape allowing no revisions. For such a machine M, we have M(x) terminates with output y, where x is a finite string and y is a possibly infinite string, if either M halts after reading the bits of x from the input tape and y is what is on the output tape at that point, or M computes forever after reading exactly the bits of x, and y is on the output tape on the limit.
The second one: A monotone machine L is a c.e. (computably enumerable) set of pairs of finite strings such that if , is in L and $x \le x'$ then $y$ and $y'$ are comparable (i.e. $y \le y'$ or $y' \le y$).