The loop problem for monoids and semigroups

Abstract We propose a way of associating to each finitely generated monoid or semigroup a formal language, called itsloop problem. In the case of a group, the loop problem is essentially the same as theword problemin the sense of combinatorial group theory. Like the word problem for groups, the loop problem is regular if and only if the monoid is finite. We also study the case in which the loop problem iscontext-free, showing that a celebrated group-theoretic result of Muller and Schupp extends to describe completely simple semigroups with context-free loop problems. We consider right cancellative monoids, establishing connections between the loop problem and the structural theory of these semigroups by showing that the syntactic monoid of the loop problem is theinverse hullof the monoid.