Hi, we often come across certain constructions in algebra that make use of some "formal" sum or "formal" linear combination or "formal" string of elements. Because this term is never defined, I have always been a little uncomfortable when it comes up. For a specific example, consider the construction of the free group on a set X. We begin by defining a "word" in X to be a formal string of elements in X. How do we make this a little more precise? Can we think of a word as an equivalence class of functions into X and concatenation as gluing these functions together? If so, how does that work?