Top is Prime Element

Theorem

Let $L = \struct {S, \wedge, \preceq}$ be a bounded above meet semilattice.

Then $\top$ is a prime element

where $\top$ denotes the greatest element of $L$.

Proof

Let $x, y \in S$ such that

$x \wedge y \preceq \top$

Thus by definition of greatest element:

$x \preceq \top$ or $y \preceq \top$

$\blacksquare$

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_6:20