Join Semilattice has Smallest Element iff has Identity

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Theorem

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

Let $s \in S$.

Then:

$s$ is the smallest element of $S$

if and only if

$s$ is the identity in $\struct{S, \vee}$.

Proof

By definition of smallest element:

$s$ is the smallest element of $S$

if and only if:

$\forall t \in S : s \preceq t$

We have:

\(\ds \forall t \in S: \, \)

\(\ds s \preceq t\)

\(\iff\)

\(\ds t = s \vee t\)

Successor is Supremum

\(\ds \)

\(\iff\)

\(\ds t = t \vee s\)

Semilattice Axiom $\text {SL} 2$: Commutativity

By definition of identity:

$s$ is the identity in $\struct{S, \vee}$

if and only if

$\forall t \in S :s \vee t = t = t \vee s$

The result follows.

$\blacksquare$