Every Pseudoprime Element is Prime implies Lattice is Arithmetic
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Theorem
Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a bounded below algebraic distributive lattice.
Assume that
- for every element $p$ of $S$ if $p$ is pseudoprime element, then $p$ is prime element.
Then $L$ is arithmetic.
Proof
By If Every Element Pseudoprime is Prime then Way Below Relation is Multiplicative:
- $\ll$ is a multiplicative relation.
where $\ll$ denotes the way below relation.
Thus by Arithmetic iff Way Below Relation is Multiplicative in Algebraic Lattice:
- the result holds.
$\blacksquare$
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_8:22