Every Pseudoprime Element is Prime implies Lattice is Arithmetic

From ProofWiki
Jump to navigation Jump to search

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