Adjoining universal inverses to families of elements of free monoids

Let X ⟨ X ⟩ be the free monoid on a generating set X , and suppose one adjoins to X ⟨ X ⟩ universal 2-sided inverses to a finite set S of its elements. We note an elementary algorithm which yields a normal form for elements of the resulting monoid M . In particular, either M will be the free group on X , or, more generally, the free product as monoids of the free group on a subset X₀⊆ X X 0 ⊆ X with the free monoid on X X₀, X \ X 0 , or M will contain $$1\!$$ 1 -sided invertible elements that are not 2-sided invertible. If S is allowed to be infinite, we show that the corresponding normal form still exists, though it cannot necessarily be computed algorithmically. We note work by others on the related topic of “special monoids”, monoids presented by finitely many generators and finitely many relations of the form $$w=1.$$ w = 1 .