Absorption law

This article is about the algebraic identity. For relation between chemical material and attenuation of light, see Beer–Lambert 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.

Examples

Lattices

A set equipped with two commutative and associative binary operations ${\displaystyle \scriptstyle \lor }$  ("join") and ${\displaystyle \scriptstyle \land }$  ("meet") that are connected by the absorption law is called a lattice; in this case, both operations are necessarily idempotent (i.e. a ${\displaystyle \scriptstyle \lor }$  a = a and a ${\displaystyle \scriptstyle \land }$  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.

Logic

In classical logic, and in particular Boolean algebra, the operations OR and AND, which are also denoted by ${\displaystyle \scriptstyle \lor }$  and ${\displaystyle \scriptstyle \land }$ , satisfy the lattice axioms, including the absorption law. The same is true for intuitionistic logic.

Non-examples

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

• Absorption (logic)

References

1. 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.