Partially Ordered Set

up:: 020 MOC Mathematics

A set $X$ is partially ordered if it has a partial order $\le$. That is, this relation is

• Reflexive: $\forall x\in X:x\le x$
• Transitive: $\forall x,y,z\in X:(x\le y)\wedge (y\le z)⟹x\le z$
• Antisymmetric: $\forall x,y\in X:(x\le y)\wedge (y\le x)⟹x=y$

Examples

• Given any set $X$, one can induce a partial order in its power set $\mathcal{P}(X)$ by the subset $\subseteq$ relation. Note that it is not total in general.

Related

• A set $X$ is a Totally Ordered Set if any two elements in $X$ can be compared by this order relation.
• An element $x_0\in X$ is said to be maximal if $\forall x\in X:x\ne x_0⟹x\le x_0$.
  • An example of a maximal element is a union of subsets $X_i\subseteq {\bigcup }_i{X}_i$
• An element $m\in A\subseteq X$ is an upper bound of $A$ if all elements in $A$ are bounded by $m$ ─ $\forall a\in A:a\le m$

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References

• Um Curso de Álgebra Linear, Flávio Ulhoa Coelho & Mary Lilian Lourenço. Editora EDUSP.