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

Let $\left({S, \preceq}\right)$ be an ordered set.
 * 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 $S$.
 * 1) Duals of Isomorphic Ordered Sets are Isomorphic
 * Let $\left({S, \mathop{\preccurlyeq_1} }\right)$ and $\left({T, \mathop{\preccurlyeq_2} }\right)$ be ordered sets.
 * Let $\left({S, \mathop{\succcurlyeq_1} }\right)$ and $\left({T, \mathop{\succcurlyeq_2} }\right)$ be the dual ordered sets of $\left({S, \mathop{\preccurlyeq_1} }\right)$ and $\left({T, \mathop{\preccurlyeq_2} }\right)$ respectively.
 * Let $f: \left({S, \mathop{\preccurlyeq_1} }\right) \to \left({T, \mathop{\preccurlyeq_2} }\right)$ be an order isomorphism.
 * Then $f: \left({S, \mathop{\succcurlyeq_1} }\right) \to \left({T, \mathop{\succcurlyeq_2} }\right)$ is also an order isomorphism.
 * 1) Supremum is Unique

Let $T$ be a non-empty subset of $S$.

Then $T$ has at most one supremum in $S$. Let $\left({S, \preceq}\right)$ be an ordered set.
 * 1) Infimum is Unique

Let $T$ be a non-empty subset of $S$.

Then $T$ has at most one infimum in $S$. Let $S$ be a set.
 * 1) Power Set is Complete Lattice

Let $\left({\mathcal P \left({S}\right), \subseteq}\right)$ be the relational structure defined on $\mathcal P \left({S}\right)$ by the relation $\subseteq$.

Then $\left({\mathcal P \left({S}\right), \subseteq}\right)$ is a complete lattice. Let $\left({S, \preceq}\right)$ be a totally ordered set.
 * 1) Subset of Toset is Toset

Let $T \subseteq S$.

Then $\left({T, \preceq \restriction_T}\right)$ is also a totally ordered set. Every totally ordered set is a lattice. Let $\preccurlyeq$ be a total ordering.
 * 1) Totally Ordered Set is Lattice
 * 1) Dual of Total Ordering is Total Ordering

Then its dual ordering $\succcurlyeq$ is also a total ordering. Let $\left({S, \preceq}\right)$ be an ordered set.
 * 1) Trichotomy Law (Ordering)

Then $\preceq$ is a total ordering :
 * $\forall a, b \in S: \left({a \prec b}\right) \lor \left({a = b}\right) \lor \left({a \succ b}\right)$

A mapping that is strictly increasing is an increasing mapping. A mapping that is strictly increasing is an decreasing mapping. A mapping that is strictly monotone is a monotone mapping. Let $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$ be ordered sets.
 * 1) Strictly Increasing Mapping is Increasing
 * 1) Strictly Decreasing Mapping is Decreasing
 * 1) Strictly Monotone Mapping is Monotone
 * 1) Order Embedding into Image is Isomorphism

Let $S'$ be the image of a mapping $\phi: \left({S, \preceq_1}\right) \to \left({T, \preceq_2}\right)$.

Then:
 * $\phi$ is an order embedding from $\left({S, \preceq_1}\right)$ into $\left({T, \preceq_2}\right)$


 * $\phi$ is an order isomorphism from $\left({S, \preceq_1}\right)$ into $\left({S', \preceq_2 \restriction_{S'}}\right)$.
 * $\phi$ is an order isomorphism from $\left({S, \preceq_1}\right)$ into $\left({S', \preceq_2 \restriction_{S'}}\right)$.

Let $\left({S, \preceq_1}\right)$ be a totally ordered set.
 * 1) Strictly Monotone Mapping with Totally Ordered Domain is Injective

Let $\left({T, \preceq_2}\right)$ be an ordered set.

Let $\phi: \left({S, \preceq_1}\right) \to \left({T, \preceq_2}\right)$ be a strictly monotone mapping.

Then $\phi$ is injective. Let $\left({S, \preceq_1}\right)$ be a totally ordered set and let $\left({T, \preceq_2}\right)$ be an ordered set.
 * 1) Mapping from Totally Ordered Set is Order Embedding iff Strictly Increasing

Let $\phi: S \to T$ be a mapping.

Then $\phi$ is an order embedding iff $\phi$ is strictly increasing. Every finite totally ordered set is well-ordered.
 * 1) Finite Totally Ordered Set is Well-Ordered