Join Prime Element is Dual of Meet Prime Element

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Theorem

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

Let $z \in S$.

The following are dual statements:

$z$ is the meet prime element of the meet semilattice $\struct {S, \wedge, \preceq}$

$z$ is the join prime element of the join semilattice $\struct {S, \vee, \preceq}$

Proof

By definition of meet prime element:

$z$ is the meet prime element of the meet semilattice $\struct {S, \wedge, \preceq}$

if and only if:

$\forall x, y \in S: \paren{x \wedge y \preceq z} \implies \leftparen{x \preceq z}$ or $\rightparen{y \preceq z}$

By the duality principle, the dual of this statement is:

$\forall x, y \in S: \paren{z \preceq x \vee y} \implies \leftparen{z \preceq z}$ or $\rightparen{z \preceq y}$

By definition of join prime element:

$z$ is the join prime element of the join semilattice $\struct {S, \vee, \preceq}$

$\blacksquare$

Also see

• Duality Principle (Order Theory)