Definition:Convex Set (Order Theory)

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This page is about Convex Set in the context of Order Theory. For other uses, see Convex Set.

Definition

Definition 1

A subset $A$ of an ordered set $\struct {S, \preceq}$ is convex (in $S$) if and only if:

$\forall x, y \in A: \forall z \in S: x \preceq z \preceq y \implies z \in A$


Definition 2

A subset $A$ of an ordered set $\struct {S, \preceq}$ is convex (in $S$) if and only if:

$\forall x, y \in A: \forall z \in S: x \prec z \prec y \implies z \in A$


Specific Instances

Convex Subset of Natural Numbers

Let $A \subseteq \N$ be a subset of the set of natural numbers.

$A$ is defined as being convex (in $\N$) if and only if:

$\forall x, y, z \in \N: \paren {x, z \in A \land x \le y \le z} \implies y \in A$


Also see

  • Results about convex sets can be found here.