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In [[algebra]], the '''absorption law''' or '''absorption identity''' is an [[identity (mathematics)|identity]] linking a pair of binary operations.
 
Two [[binary operations]], say ¤ and *, are said to be connected by the absorption law if:
 
:''a'' ¤ (''a'' * ''b'') = ''a'' * (''a'' ¤ ''b'') = ''a''.
 
A [[set (mathematics)|set]] equipped with two [[commutative]], [[associative]] and idempotent binary operations ∨ ("join") and ∧ ("meet") that are connected by the absorption law
:''a'' ∨ (''a'' ∧ ''b'') = ''a'' ∧ (''a'' ∨ ''b'') = ''a''
is called a [[lattice (order)|lattice]].
 
Examples of lattices include [[Boolean algebra (structure)|Boolean algebra]]s, the set of sets with the ''union'' and ''intersection'' operators, [[Heyting algebra]]s, and ordered sets with ''min'' and ''max'' operations.
 
In [[classical logic]], and in particular in [[Boolean algebra (logic)|Boolean algebra]], the operations [[logical disjunction|OR]] and [[logical conjunction|AND]], which are also denoted by <math>\scriptstyle \lor</math> and <math>\scriptstyle \land</math>, 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 ring]], ''e.g.'' the [[field (mathematics)|field]] of [[real number]]s, [[relevance logic]]s, [[linear logic]]s, and [[substructural logic]]s. In the last case, there is no [[one-to-one correspondence]] between the [[free variable]]s of the defining pair of identities.
 
== References ==
* {{cite book
|author=Davey, B. A., and Priestley, H. A.
|year=2002
|title='''Introduction to Lattices and Order'''
|edition=second
|publisher=Cambridge University Press
|isbn=0-521-78451-4
}}
* {{springer|title=Absorption laws|id=p/a010440}}
*{{MathWorld|title=Absorption Law|urlname=AbsorptionLaw}}
 
[[Category:Abstract algebra]]
[[Category:Boolean algebra]]
[[Category:Theorems in propositional logic]]
[[Category:Lattice theory]]

Latest revision as of 08:18, 22 November 2014

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