Definition:Ordered Structure Monomorphism
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Definition
Let $\left({S, \circ, \preceq}\right)$ and $\left({T, *, \preccurlyeq}\right)$ be ordered structures.
An ordered structure monomorphism from $\left({S, \circ, \preceq}\right)$ to $\left({T, *, \preccurlyeq}\right)$ 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 $\left({T, \preccurlyeq}\right)$.
Ordered Semigroup Monomorphism
Let $\left({S, \circ, \preceq}\right)$ and $\left({T, *, \preccurlyeq}\right)$ be ordered semigroups.
An ordered semigroup monomorphism from $\left({S, \circ, \preceq}\right)$ to $\left({T, *, \preccurlyeq}\right)$ is a mapping $\phi: S \to T$ that is both:
- A (semigroup) monomorphism from the semigroup $\left({S, \circ}\right)$ to the semigroup $\left({T, *}\right)$
- An order monomorphism from the poset $\left({S, \preceq}\right)$ to the poset $\left({T, \preccurlyeq}\right)$.
Ordered Group Monomorphism
Let $\left({S, \circ, \preceq}\right)$ and $\left({T, *, \preccurlyeq}\right)$ be ordered groups.
An ordered group monomorphism from $\left({S, \circ, \preceq}\right)$ to $\left({T, *, \preccurlyeq}\right)$ is a mapping $\phi: S \to T$ that is both:
- A group monomorphism from the group $\left({S, \circ}\right)$ to the group $\left({T, *}\right)$
- An order monomorphism from the poset $\left({S, \preceq}\right)$ to the poset $\left({T, \preccurlyeq}\right)$.
Ordered Ring Monomorphism
Let $\left({S, +, \circ, \preceq}\right)$ and $\left({T, \oplus, *, \preccurlyeq}\right)$ be ordered rings.
An ordered ring monomorphism from $\left({S, +, \circ, \preceq}\right)$ to $\left({T, \oplus, *, \preccurlyeq}\right)$ is a mapping $\phi: S \to T$ that is both:
- An ordered group monomorphism from the ordered group $\left({S,+, \preceq}\right)$ to the ordered group $\left({T, \oplus, \preccurlyeq}\right)$
- A semigroup monomorphism from the semigroup $\left({S, \circ}\right)$ to the semigroup $\left({T, *}\right)$.
Also see
Sources
- Seth Warner: Modern Algebra (1965): $\S 15$
