Todd and Coxeter's method for enumerating cosets of finitely generated
subgroups in finitely presented groups (abbreviated by T
C here)
is one famous method from combinatorial
group theory for studying the subgroup problem.
Since prefix string rewriting is also an appropriate method to study this problem,
prefix string rewriting methods have been compared to T
C.
We recall and compare two of them briefly, one by Kuhn and Madlener [
4] and one by Sims [
15].
A new approach using prefix string rewriting in free groups is derived from the algebraic method
presented by Reinert, Mora and Madlener in [
14] which directly emulates T
C.
It is extended to free monoids and an algebraic characterization for the ``cosets'' enumerated
in this setting is provided.
Keywords.
coset enumeration,
subgroup problem,
prefix string rewriting,
Gröbner bases in monoid and group rings.
