A clot, a positive cone and a normal subobject of a monoid are defined as the zero-class of an internal reflexive relation, of a preorder or of a congruence, respectively. For a category \(\mathcal R\), a normal monomorphism with respect to \(\mathcal R\) as well as the normalization functor \(N\colon {\mathcal R} \to \) Mono(C), where C is a certain category, are defined. It is proved that if C is a pointed, well-povered and finitely complete category with arbitrary intersections and \(\mathcal R\) a pullback stable full subcategory of the one of internal relations on C closed under arbitrary intersections, then the normalization functor \(N\colon {\mathcal R} \to \) Mono(C) has a left adjoint \(F\colon\) Mono(C)\(\to \mathcal R\). Several equivalent conditions for a monoid monomorphism \(m\colon M\to A\) to be a normal subobject (a positive cone, a clot, respectively) are derived (here \(m(M)\) is identified with \(M\)). For example, a monoid monomorphism \(m\colon M\to A\) is a normal subobject (a positive cone, a clot, respectively) if and only if for every \(x,y\in A\) and every \(u\in M\), \(xy\in M \iff xuy\in M\) (\(xy\in M \Rightarrow xuy\in M; xy=1\Rightarrow xuy\in M\), respectively). Eleven examples were constructed to illustrate different aspects of the topics covered.