Prime Element is Meet Irreducible

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Theorem

Let $L = \struct {S, \vee, \wedge, \preceq}$ be a lattice.

Let $p \in S$.

Let $p$ be a prime element of $L$.


Then $p$ is meet irreducible in $L$.


Proof

Let $p$ be a prime element.

Let $x, y \in S$ such that

$p = x \wedge y$

By definition of reflexivity:

$x \wedge y \preceq p$

By definition of prime element:

$x \preceq p$ or $y \preceq p$

By Meet Precedes Operands:

$p \preceq x$ and $p \preceq y$

Thus by definition of antisymmetry:

$p = x$ or $p = y$

$\blacksquare$


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