  We present an algorithmic approach to the conjugacy problems in monoids and
semigroups, using rewriting systems. There is a class of monoids and semigroups
that satisfy the condition that the transposi- tion problem and the left and
right conjugacy problem are equivalent. The free monoid and the completely
simple semigroups belong to this class. We give a solution to the conjugacy
problem for monoids and semigroups in this class that are presented by a
complete rewriting system that satisfies some additional conditions.
