Characterization of Pseudoprime Element when Way Below Relation is Multiplicative
Theorem
Let $L = \struct {S, \vee, \wedge, \preceq}$ be a bounded below continuous lattice such that
- $\ll$ is multiplicative relation
where $\ll$ denotes the way below relation of $L$.
Let $p \in S$.
Then $p$ is pseudoprime element if and only if
- $\forall a, b \in S: a \wedge b \ll p \implies a \preceq p \lor b \preceq p$
Proof
Sufficient Condition
Let $p$ be pseudoprime element.
Let $a, b \in S$ such that
- $a \wedge b \ll p$
By definition of meet:
- $\inf \left\{ {a, b}\right\} \ll p$
By Characterization of Pseudoprime Element by Finite Infima:
- $\exists c \in \left\{ {a, b}\right\}: c \preceq p$
Thus
- $a \preceq p$ or $b \preceq p$
$\Box$
Necessary Condition
Suppose
- $\forall a, b \in S: a \wedge b \ll p \implies a \preceq p \lor b \preceq p$
Aiming for a contradiction, suppose:
- $p$ is not a pseudoprime element.
By Prime is Pseudoprime (Order Theory):
- $p$ is not a prime element.
By definition of prime element:
- $\exists x, y \in S: x \wedge y \preceq p$ and $x \npreceq p$ and $y \npreceq p$
By definition of continuous:
- $\forall z \in S: z^\ll$ is directed.
and
- $L$ satisfies axiom of approximation.
By Axiom of Approximation in Up-Complete Semilattice:
- $\exists u \in S: u \ll x \land u \npreceq p$
and
- $\exists v \in S: v \ll y \land v \npreceq p$
By Way Below Relation is Auxiliary Relation:
- $\ll$ is auxiliary relation.
By Multiplicative Auxiliary Relation iff Congruent:
- $u \wedge v \ll x \wedge y$
By definition of transitivity:
- $u \wedge v \ll p$
By assumption:
- $u \preceq p$ or $v \preceq p$
This contradicts $u \npreceq p$ and $v \npreceq p$.
$\blacksquare$
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_7:44