Absorption Laws (Lattice Theory)

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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 $\left({S, \vee, \preceq}\right)$ be a join semilattice.

Let $\wedge$ denote meet.


Then $\wedge$ absorbs $\vee$.

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

$a \wedge \left({a \vee b}\right) = a$