User:Ascii/ProofWiki Sampling Notes for Theorems/Order Theory


 * 1) Identity Mapping is Order Isomorphism
 * Let $\left({S, \preceq}\right)$ be an ordered set.
 * The identity mapping $I_S$ is an order isomorphism from $\left({S, \preceq}\right)$ to itself.
 * 1) Equivalence of Definitions of Order Isomorphism
 * 2) Composite of Order Isomorphisms is Order Isomorphism
 * Let $\left({S_1, \preceq_1}\right)$, $\left({S_2, \preceq_2}\right)$ and $\left({S_3, \preceq_3}\right)$ be ordered sets.
 * Let $\phi: \left({S_1, \preceq_1}\right) \to \left({S_2, \preceq_2}\right)$ and $\psi: \left({S_2, \preceq_2}\right) \to \left({S_3, \preceq_3}\right)$ be order isomorphisms.
 * Then $\psi \circ \phi: \left({S_1, \preceq_1}\right) \to \left({S_3, \preceq_3}\right)$ is also an order isomorphism.
 * 1) Dual Ordering is Ordering
 * Let $\left({S, \preceq}\right)$ be an ordered set and $\succeq$ denote the dual ordering of $\preceq$.
 * Then $\succeq$ is an ordering on $S$.