Absorption Laws (Lattice Theory)

This proof is about Absorption Laws in the context of Lattice Theory. For other uses, see Absorption Laws.

Join Absorbs Meet

Let $\struct {S, \wedge, \preceq}$ be a meet semilattice.

Let $\vee$ denote join.

Then $\vee$ absorbs $\wedge$.

That is, for all $a, b \in S$:

$a \vee \paren {a \wedge b} = a$

Meet Absorbs Join

Let $\struct {S, \vee, \preceq}$ be a join semilattice.

Let $\wedge$ denote meet.

Then $\wedge$ absorbs $\vee$.

That is, for all $a, b \in S$:

$a \wedge \paren {a \vee b} = a$