Equivalence of Definitions of Bottom

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Theorem

Let $\left({S, \vee, \wedge, \preceq}\right)$ be a lattice.

Let $\bot$ be a bottom of $\left({S, \vee, \wedge, \preceq}\right)$.

The following definitions of the concept of Bottom in the context of Lattice Theory are equivalent:

Definition 1

Let $S$ admit a smallest element $\bot$.


Then $\bot$ is called the bottom of $S$.

Definition 2

Let $\vee$ have an identity element $\bot$.


Then $\bot$ is called the bottom of $S$.


Proof

By definition, $\bot$ is the smallest element of $S$ if and only if for all $a \in S$:

$\bot \preceq a$

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

$a \vee \bot = a$

If this equality holds for all $a \in S$, then by definition $\bot$ is an identity for $\vee$.


The result follows.

$\blacksquare$


Also see