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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 and associative binary operations ("join") and ("meet") that are connected by the absorption law is called a lattice; in this case, both operations are necessarily idempotent (i.e. *a* *a* = *a* and *a* *a* = *a*).

Examples of lattices include Heyting algebras and Boolean algebras,^{[1]} in particular sets of sets with *union* (∪) and *intersection* (∩) operators, 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 Boolean algebra (structure)#Axiomatics for a proof of the absorption laws from the distributivity, identity, and boundary laws.

- Brian A. Davey; Hilary Ann Priestley (2002).
*Introduction to Lattices and Order*(2nd ed.). Cambridge University Press. ISBN 0-521-78451-4. LCCN 2001043910. - "Absorption laws",
*Encyclopedia of Mathematics*, EMS Press, 2001 [1994] - Weisstein, Eric W. "Absorption Law".
*MathWorld*.