# Absorption law

In algebra, the **absorption law** or **absorption identity** is an identity linking a pair of binary operations.

Two binary operations, ¤ and ⁂, are said to be connected by the absorption law if:

*a*¤ (*a*⁂*b*) =*a*⁂ (*a*¤*b*) =*a*.

A set equipped with two commutative, associative and idempotent binary operations ("join") and ("meet") that are connected by the absorption law is called a lattice.

Examples of lattices include Boolean algebras, the set of sets with *union* and *intersection* operators, Heyting algebras, and ordered sets with *min* and *max* operations.

In classical logic, and in particular Boolean algebra, the operations OR and AND, which are also denoted by and , satisfy the lattice axioms, including the absorption law. The same is true for intuitionistic logic.

The absorption law does not hold in many other algebraic structures, such as commutative rings, *e.g.* the field of real numbers, relevance logics, linear logics, and substructural logics. In the last case, there is no one-to-one correspondence between the free variables of the defining pair of identities.

## See also

## References

- Brian A. Davey; Hilary Ann Priestley (2002).
*Introduction to Lattices and Order*(2nd ed.). Cambridge University Press. ISBN 0-521-78451-4. LCCN 2001043910. - Hazewinkel, Michiel, ed. (2001) [1994], "Absorption laws",
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4 - Weisstein, Eric W. "Absorption Law".
*MathWorld*.