Definition:Ordered Field Isomorphism
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Definition
Let $\struct {S, +, \circ, \preceq}$ and $\struct {T, \oplus, *, \preccurlyeq}$ be ordered fields.
An ordered field isomorphism from $\struct {S, +, \circ, \preceq}$ to $\struct {T, \oplus, *, \preccurlyeq}$ is a mapping $\phi: S \to T$ that is both:
- $(1): \quad$ An ordered group isomorphism from the ordered group $\struct {S, +, \preceq}$ to the ordered group $\struct {T, \oplus, \preccurlyeq}$
- $(2): \quad$ A group isomorphism from the group $\struct {S_{\ne 0}, \circ}$ to the semigroup $\struct {T_{\ne 0}, *}$
where $S_{\ne 0}$ and $T_{\ne 0}$ denote the sets $S$ and $T$ without the zeros of $S$ and $T$ respectively.
Also see
Linguistic Note
The word isomorphism derives from the Greek morphe (μορφή) meaning form or structure, with the prefix iso- meaning equal.
Thus isomorphism means equal structure.
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): $\S 23$