# Ordering Induced by Join Semilattice

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## Theorem

Let $\struct {S, \vee, \preceq}$ be a join semilattice.

By Join Semilattice is Semilattice, $\struct {S, \vee}$ is a semilattice.

By Semilattice Induces Ordering, $\struct {S, \vee}$ induces an ordering $\preceq'$ on $S$, by:

- $a \preceq' b$ if and only if $a \vee b = b$

for all $a, b \in S$.

The ordering $\preceq'$ coincides with the original ordering $\preceq$.

## Proof

It is to be shown that, for all $a, b \in S$:

- $a \preceq b$ if and only if $b = \sup \set {a, b}$

by definition of join.

Here $\sup$ denotes supremum.

Since any upper bound $c$ of $\set {a, b}$ must satisfy:

- $b \preceq c$

it suffices to verify that:

- $a \preceq b$ if and only if $b$ is an upper bound for $\set {a, b}$

Since $\preceq$ is reflexive, we know that:

- $b \preceq b$

and therefore said equivalence is established.

We conclude that $\preceq'$ and $\preceq$ coincide.

$\blacksquare$

This article is complete as far as it goes, but it could do with expansion.We need to start using a construct for duality. (Global) Dual is Ordering Induced by Meet SemilatticeYou can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Expand}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |