free group

Given a set S of "free generators" of a free group, let S⁻¹ be the set of inverses of the generators, which are in one-to-one correspondence with the generators (the two sets are disjoint), then let (S∪S⁻¹)^* be the Kleene closure of the union of those two sets. For any string w in the Kleene closure let r(w) be its reduced form, obtained by cutting out any occurrences of the form xx⁻¹ or x⁻¹x where x∈S. Noting that r(r(w)) = r(w) for any string w, define an equivalence relation ∼ such that u∼v if and only if r(u)=r(v). Then let the underlying set of the free group generated by S be the quotient set (S∪S⁻¹)^*/∼ and let its operator be concatenation followed by reduction.