Equivalence of Definitions of Bounded Lattice

Theorem
The following definition of bounded lattice are equivalent:

Proof
From Supremum of Empty Set is Smallest Element, we have that:


 * $\bot := \sup \varnothing$

satisfies $\bot \preceq a$ for all $a \in S$.

By Ordering in terms of Join, this is equivalent to:


 * $\forall a \in S: a \vee \bot = a = \bot \vee a$

where the last equality follows as Join is Commutative.

Thus $\bot$ is an identity element for $\vee$, and conversely.

That $\inf \varnothing$ is an identity element for $\wedge$, and conversely, follows by the Duality Principle.

Hence the result.

Also see

 * Bounded Lattice