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Two binary operations, say ¤ and *, are said to be connected by the absorption law if:
- a ¤ (a * b) = a * (a ¤ b) = a.
- a ∨ (a ∧ b) = a ∧ (a ∨ b) = a
is called a lattice.
In classical logic, and in particular in Boolean algebra, the operations OR and AND, which are also denoted by and , satisfy the lattice axioms, including the absorption law. The same is true for intuitionistic logic.
The absorption law does not hold in many other algebraic structures, such as commutative ring, e.g. the field of real numbers, 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.
- Davey, B. A., and Priestley, H. A. (2002). Introduction to Lattices and Order (second ed.). Cambridge University Press. ISBN 0-521-78451-4.
- Hazewinkel, Michiel, ed. (2001), "Absorption laws", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
- Weisstein, Eric W., "Absorption Law", MathWorld.