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:
“The symbolic view of things is a consequence of long absorption in images. Is sign language the real language of Paradise?”
—Hugo Ball (1886–1927)
“The image cannot be dispossessed of a primordial freshness, which idea can never claim. An idea is derivative and tamed. The image is in the natural or wild state, and it has to be discovered there, not put there, obeying its own law and none of ours. We think we can lay hold of image and take it captive, but the docile captive is not the real image but only the idea, which is the image with its character beaten out of it.”
—John Crowe Ransom (1888–1974)