In algebra, the absorption law or absorption identity is an 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 equipped with two commutative and associative binary operations ∨ ("join") and ∧ ("meet") which are connected by the absorption law
- a ∨ (a ∧ b) = a ∧ (a ∨ b) = a
is called a lattice. Examples of lattices include Boolean algebras and Heyting algebras.
In classical logic, and in particular in Boolean algebra, the operations OR and AND, which are also denoted by and, also satisfy the lattice axioms, including the absorption law. The same is true for intuitionistic logic.
The commutative and associative laws also hold for addition and multiplication in commutative rings, e.g. in the field of real numbers. The absorption law is the critical property that is missing in this case, since in general a · (a + b) ≠ a and a + (a · b) ≠ a.
The absorption law also fails to 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.
Famous quotes containing the words absorption and/or law:
“These philosophers dwell on the inevitability and unchangeableness of laws, on the power of temperament and constitution, the three goon, or qualities, and the circumstances, or birth and affinity. The end is an immense consolation; eternal absorption in Brahma.”
—Henry David Thoreau (1817–1862)
“Will mankind never learn that policy is not morality,—that it never secures any moral right, but considers merely what is expedient? chooses the available candidate,—who is invariably the devil,—and what right have his constituents to be surprised, because the devil does not behave like an angel of light? What is wanted is men, not of policy, but of probity,—who recognize a higher law than the Constitution, or the decision of the majority.”
—Henry David Thoreau (1817–1862)