Equalizers and kernels in categories of monoids.

In dealing with monoids, the natural notion of kernel of a monoid morphism $$f:M\rightarrow N$$ between two monoids M and N is that of the congruence $$\sim _f$$ on M defined, for every $$m,m'\in M$$ , by $$m\sim _fm'$$ if $$f(m)=f(m')$$ . 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 $$f:M\rightarrow N$$ is simply the embedding of the submonoid $$f^{-1}(1_N)$$ into M, but a complete characterization of kernels in these categories is not always trivial, and leads to interesting related notions. [ABSTRACT FROM AUTHOR]