Equalizers and kernels in categories of monoids

In dealing with monoids, the natural notion of kernel of a monoid morphism (Formula presented.) between two monoids M and N is that of the congruence (Formula presented.) on M defined, for every (Formula presented.), by (Formula presented.) if (Formula presented.). In this paper, we study kernels and equalizers of monoid morphisms in the categorical sense. We consider the case of the categories of all monoids, commutative monoids, cancellative commutative monoids, reduced Krull monoids, inverse monoids and free monoids. In all these categories, the kernel of (Formula presented.) is simply the embedding of the submonoid (Formula presented.) into M, but a complete characterization of kernels in these categories is not always trivial, and leads to interesting related notions.