Definition:Join (Order Theory)

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

Definition

Let $\struct {S, \preceq}$ be an ordered set.

Let $a, b \in S$.

Let their supremum $\sup \set {a, b}$ exist in $S$.

Then the join of $a$ and $b$ is defined as:

$a \vee b = \sup \set {a, b}$

Expanding the definition of supremum, one sees that $c = a \vee b$ if and only if:

$(1): \quad a \preceq c$ and $b \preceq c$

$(2): \quad \forall s \in S: a \preceq s$ and $b \preceq s \implies c \preceq s$

Some sources refer to this as the union of $a$ and $b$.