Definition:Monomorphism (Abstract Algebra)

Definition
A homomorphism which is an injection is descibed as monic, or called a monomorphism.

Monomorphism on an Ordered Structure
A (structure) monomorphism from an ordered structure $\left({S, \circ, \preceq}\right)$ to another $\left({T, *, \preccurlyeq}\right)$ is a mapping $\phi: S \to T$ that is both:


 * A monomorphism, i.e. an injective homomorphism, from the structure $\left({S, \circ}\right)$ to the structure $\left({T, *}\right)$


 * An order monomorphism from the poset $\left({S, \preceq}\right)$ to the poset $\left({T, \preccurlyeq}\right)$.

Category Theory
In a category $\mathcal C$, a monomorphism is a morphism $\alpha \in \operatorname{mor}\mathcal C$ such that $\alpha\beta = \alpha\gamma$ implies $\beta = \gamma$ for all morphisms $\beta,\gamma \in \operatorname{mor}\mathcal C$ for which the composition is defined.

Linguistic Note
The word monomorphism comes from the Greek morphe (μορφή) meaning form or structure, with the prefix mono- meaning single.

Thus monomorphism means single (similar) structure.

Ordered Structure definition

 * : $\S 15$