Prime is Pseudoprime (Order Theory)
Jump to navigation
Jump to search
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