Home > Absorption law
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 ∧ (a ∨ b) = a ∨ (a ∧ b) = 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 logicIn mathematical logic, in particular in connection with proof theory, a number of substructural logics have been introduced, as systems of propositional calculus that are weaker than the conventional one. They differ in having fewer structural rules avail, the absorption rule, since it is not linear (there being no 1-1 correspondence between the free variables of the two euqations), it does not express logical equivalences in linear logicIn mathematical logic, linear logic is a type of substructural logic that denies the structural rules of weakening and contraction''. The interpretation is of hypotheses as resources every hypothesis must be consumed exactly once in a proof. This differs, nor does it in relevance logic.
Abstract algebraAbstract algebra Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groups, rings and fields. The term "abstract algebra" is used to distinguish the field from " elementary algebra" or "high school algebr
Boolean algebra
Lattice theory
