In algebra

Algebra

Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...

, the absorption law or absorption identity is an identity

Identity (mathematics)

In mathematics, the term identity has several different important meanings:*An identity is a relation which is tautologically true. This means that whatever the number or value may be, the answer stays the same. For example, algebraically, this occurs if an equation is satisfied for all values of...

 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

Lattice (order)

In mathematics, a lattice is a partially ordered set in which any two elements have a unique supremum and an infimum . Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities...

.
Examples of lattices include Boolean algebras and Heyting algebra

Heyting algebra

In mathematics, a Heyting algebra, named after Arend Heyting, is a bounded lattice equipped with a binary operation a→b of implication such that ∧a ≤ b, and moreover a→b is the greatest such in the sense that if c∧a ≤ b then c ≤ a→b...

s.

In classical logic

Classical logic

Classical logic identifies a class of formal logics that have been most intensively studied and most widely used. The class is sometimes called standard logic as well...

, and in particular in Boolean algebra, the operations OR

Logical disjunction

In logic and mathematics, a two-place logical connective or, is a logical disjunction, also known as inclusive disjunction or alternation, that results in true whenever one or more of its operands are true. E.g. in this context, "A or B" is true if A is true, or if B is true, or if both A and B are...

 and AND

Logical conjunction

In logic and mathematics, a two-place logical operator and, also known as logical conjunction, results in true if both of its operands are true, otherwise the value of false....

, which are also denoted by and , also satisfy the lattice axioms, including the absorption law. The same is true for intuitionistic logic

Intuitionistic logic

Intuitionistic logic, or constructive logic, is a symbolic logic system differing from classical logic in its definition of the meaning of a statement being true. In classical logic, all well-formed statements are assumed to be either true or false, even if we do not have a proof of either...

.

The commutative and associative laws also hold for addition and multiplication in commutative ring

Commutative ring

In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....

s, e.g. in the field

Field (mathematics)

In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...

 of real number

Real number

In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

s. 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 logic

Relevance logic

Relevance logic, also called relevant logic, is a kind of non-classical logic requiring the antecedent and consequent of implications be relevantly related. They may be viewed as a family of substructural or modal logics...

s, linear logic

Linear logic

Linear logic is a substructural logic proposed by Jean-Yves Girard as a refinement of classical and intuitionistic logic, joining the dualities of the former with many of the constructive properties of the latter...

s, and substructural logic

Substructural logic

In logic, a substructural logic is a logic lacking one of the usual structural rules , such as weakening, contraction or associativity...

s. In the last case, there is no one-to-one correspondence between the free variables of the defining pair of identities.