Prime is Pseudoprime (Order Theory)

Theorem

Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be an up-complete lattice.

Let $p \in S$ be a prime element.

Then $p$ is pseudoprime.

Proof

By Lower Closure is Prime Ideal for Prime Element:

$p^\preceq$ is prime ideal.

By Supremum of Lower Closure of Element:

$ \sup \left({ p^\preceq }\right) = p$

Hence $p$ is pseudoprime.

$\blacksquare$

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_7:34