Definition:Monomorphism (Abstract Algebra)

A monomorphism is a homomorphism which is an injection.

Group Monomorphism
If both $$\left({G, \circ}\right)$$ and $$\left({H, *}\right)$$ are groups, then a monomorphism $$\phi: \left({G, \circ}\right) \to \left({H, *}\right)$$ is called a group monomorphism.

Ring Monomorphism
If both $$\left({R, +, \circ}\right)$$ and $$\left({S, \oplus, *}\right)$$ are rings, then a monomorphism $$\phi: \left({R, +, \circ}\right) \to \left({S, \oplus, *}\right)$$ is called a ring 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 $$\left({T; \preccurlyeq}\right)$$.