  In the category of monoids we characterize monomorphisms that are normal, in
an appropriate sense, to internal reflexive relations, preorders or equivalence
relations. The zero-classes of such internal relations are first described in
terms of convenient syntactic relations associated to them and then through the
adjunctions associated with the corresponding normalization functors. The
largest categorical equivalences induced by these adjunctions provide an
equivalence between the categories of relations generated by their zero-classes
and the ones of monomorphisms that we suggest to call {normal with respect to}
the internal relations considered. This idea, although being transverse to the
literature in the field, has not in our opinion been presented and explored in
full generality. The existence of adjoints to the normalization functors
permits developing a theory of normal monomorphisms, thus extending many
results from groups and protomodular categories to monoids and unital
categories.
