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

Any two binary operations, say $ and %, are subject to the absorption law if:

a $ (a % b) = a % (a $ b) = a.

The operations $ and % are said to form a dual pair.

Let there be some set closed under two binary operations. If those operations commute, associate, and satisfy the absorption law, the resulting abstract algebra is a lattice, in which case the two operations are called meet and join. Since commutativity and associativity often characterize other algebraic structures, absorption is the defining property of lattice theory. Since Boolean algebras and Heyting algebras are lattices, they too obey the absorption law.

Since classical logic is a model of Boolean algebra, and the same is true of intuitionistic logic and Heyting algebras, the absorption law holds for the truth functors $\backslash vee$ and $\backslash wedge$, denoting OR and AND, respectively. Hence:

$a\; \backslash vee\; (a\; \backslash wedge\; b)\; =\; a\; \backslash wedge\; (a\; \backslash vee\; b)\; =\; a$

where = is understood to be logical equivalence over formulae.

The absorption law does not hold for 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.

Abstract algebra | Boolean algebra | Lattice theory

Lei de absorção