Definition:Join (Order Theory)
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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 $b \preceq s \implies c \preceq s$
Also known as
Some sources refer to this as the union of $a$ and $b$.
Also see
- Results about the join operation can be found here.
Sources
- 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 7$
- 1982: Peter T. Johnstone: Stone Spaces ... (previous) ... (next): Chapter $\text I$: Preliminaries, Definition $1.2$