Definition:Finite Meet Preserving Mapping

Definition

Let $L_1 = \struct{S_1, \wedge_1, \preceq_1}$ and $L_2 = \struct{S_2, \wedge_2, \preceq_2}$ be meet semilattices with greatest elements $\top_1$ and $\top_2$ respectively.

Let $\phi: S_1 \to S_2$ be a mapping between the underlying sets of $L_1$ and $L_2$.

Then:

$\phi$ is finite meet preserving, denoted $\phi:L_1 \to L_2$

if and only if:

$\forall$ finite $A \subseteq S_1 : \map \phi {\inf A} = \inf \set{\map \phi a : a \in A}$

Also see

• Definition:Finite Join Preserving Mapping

• Definition:Arbitrary Meet Preserving Mapping

Sources

• 1982: Peter T. Johnstone: Stone Spaces: Chapter II: Introduction to Locales, $\S1.1$ Definition (a)