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 $y \preceq p$
- $p \preceq x$ and $p \preceq y$
Thus by definition of antisymmetry:
- $p = x$ or $p = y$
$\blacksquare$
Sources
- 1980: G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M.W. Mislove and D.S. Scott: A Compendium of Continuous Lattices
- Mizar article WAYBEL_6:24