Proposition 1.4.3. Assuming that:

$L$ is a bounded lattice
$\leq$ is the order induced by the operations in $L$ ( $a \leq b$ if $a \land b = a$ )

Then

\leq

is a partial order with least element

⊥

, greatest element

⊤

, and for any

a, b \in L

, we have

a \land b = \inf {a, b}

and

a \land b = \sup {a, b}

. Conversely, every partial order with all finite infs and sups is a bounded lattice.