  We investigate the notion of soficity for monoids. A group is sofic as a
group if and only if it is sofic as a monoid. All finite monoids, all
commutative monoids, all free monoids, all cancellative one-sided amenable
monoids, all multiplicative monoids of matrices over a field, and all monoids
obtained by adjoining an identity element to a semigroup without identity
element are sofic. On the other hand, although the question of the existence of
a non-sofic group remains open, we prove that the bicyclic monoid is not sofic.
This shows that there exist finitely presented amenable inverse monoids that
are non-sofic.
