Definition:Finite Join Preserving Mapping

Definition

Let $L_1 = \struct{S_1, \vee_1, \preceq_1}$ and $L_2 = \struct{S_2, \vee_2, \preceq_2}$ be join semilattices with smallest elements $\bot_1$ and $\bot_2$ 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 join preserving, denoted $\phi:L_1 \to L_2$

if and only if:

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

Also see

• Definition:Finite Meet Preserving Mapping

• Definition:Arbitrary Join Preserving Mapping

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