Equivalence of Definitions of Lower Section

Theorem

Let $\struct {S, \preceq}$ be an ordered set.

Let $U \subseteq S$.

The following definitions of the concept of Lower Section are equivalent:

Definition 1

$L$ is a lower section in $S$ if and only if:

$\forall l \in L, s \in S: s \preceq l \implies s \in L$

Definition 2

$L$ is a lower section in $S$ if and only if:

$L^\preceq \subseteq L$

where $L^\preceq$ is the lower closure of $L$.

Definition 3

$L$ is a lower section in $S$ if and only if:

$L^\preceq = L$

where $L^\preceq$ is the lower closure of $L$.

Proof

We are required to show that the following are equivalent:

\((1)\)	$:$				\(\ds \forall l \in L: \forall s \in S: s \preceq l \implies s \in L \)
\((2)\)	$:$				\(\ds L^\preceq \subseteq L \)
\((3)\)	$:$				\(\ds L^\preceq = L \)

By the Duality Principle, it suffices to prove that:

$(1^*)$, $(2^*)$ and $(3^*)$ are equivalent

where these are the dual statements of $(1)$, $(2)$ and $(3)$, respectively.

By Dual Pairs, it can be seen that these dual statements are as follows:

\((1^*)\)	$:$				\(\ds \forall l \in L: \forall s \in S: l \preceq s \implies s \in L \)
\((2^*)\)	$:$				\(\ds L^\succeq \subseteq L \)
\((3^*)\)	$:$				\(\ds L^\succeq = L \)

Their equivalence is proved on Equivalence of Definitions of Upper Section.

$\blacksquare$