Order, lattices, ordered algebraic structures Proposition

Absorption law

Statement

Let $L$ be a lattice with join $\vee$ and meet $\wedge$ . Then $a \vee (a \wedge b) = a$ and $a \wedge (a \vee b) = a$ for all $a, b \in L$ .

Remarks

In a lattice defined as a partially ordered set, the proof is short. Since $a \wedge b \le a$ , the join of $a$ and $a \wedge b$ is $a$ . The second law is dual. In the algebraic definition of a lattice, the two absorption laws are axioms, together with associativity and commutativity. The absorption laws imply idempotence: $a \vee a = a \vee (a \wedge (a \vee a)) = a$ .

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Same as	Wikidata Q1154428 Wikipedia Absorption law

Absorption law

Statement

Remarks

Further reading

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