Absorption law

From Academic Kids

In algebra, the absorption law is an identity between two binary operations, say $ and %. It is valid and fundamental in Boolean algebra, and lattice theory.

The absorption law states that

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

The interest arises because of the cases where $ and % are meet and join in order theory. There it is easy to see that the law should hold. In particular, for the binary operators ∧ and ∨, which are defined respectively as the logical AND and OR, the following applies:

a ∧ (ab) = a ∨ (ab) = a.

where = is understood to be logical equivalence over formulae.

Since Boolean algebras are the general form of models for classical logic, and Heyting algebras for intuitionistic logic, the absorption rule also expresses logical equivalences between those logics. Note, however, in the domain of substructural logic, the absorption rule, since it is not linear (there being no 1-1 correspondence between the free variables of the two equations), it does not express logical equivalences in linear logic, nor does it in relevance logic.

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