In algebra, the absorption law is an identity linking a pair of binary operations. This article is about the branch of mathematics. ... In mathematics, the term identity has several important uses: An identity is an equality that remains true regardless of the values of any variables that appear within it, to distinguish it from an equality which is true under more particular conditions. ... In mathematics, a binary operation is a calculation involving two input quantities, in other words, an operation whose arity is two. ...

Any two binary operations, say $ and %, are subject to the absorption law if: In mathematics, a binary operation, or binary operator, is a calculation involving two input quantities and one kind of a specific operation. ...

a $ (a % b) = a % (a $ b) = a.

The operations $ and % are said to form a dual pair. In functional analysis and related areas of mathematics a dual pair or dual system is a pair of vector spaces with an associated bilinear form. ...

Let there be some set closed under two binary operations. If those operations are commutative, associative, and satisfy the absorption law, the resulting abstract algebra is a lattice, in which case the two operations are sometimes called meet, join; other possibilities include and,or. Since commutativity and associativity are often properties of other algebraic structures (for example, addition and multiplication of real numbers), absorption is the defining property of a lattice. Since Boolean algebras and Heyting algebras are lattices, they too obey the absorption law. In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ... In mathematics, associativity is a property that a binary operation can have. ... Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ... The name lattice is suggested by the form of the Hasse diagram depicting it. ... See lattice for other mathematical as well as non-mathematical meanings of the term. ... Look up join in Wiktionary, the free dictionary. ... Look up and in Wiktionary, the free dictionary. ... OR and/or or may stand for: Look up OR, Or, or in Wiktionary, the free dictionary. ... The name lattice is suggested by the form of the Hasse diagram depicting it. ... In abstract algebra, a Boolean algebra is an algebraic structure (a collection of elements and operations on them obeying defining axioms) that captures essential properties of both set operations and logic operations. ... In mathematics, Heyting algebras are special partially ordered sets that constitute a generalization of Boolean algebras. ...

Since classical logic is a model of Boolean algebra, and the same is true of intuitionistic logic and Heyting algebras, the absorption law holds for operations and , denoting OR and AND, respectively. Hence: Classical logic identifies a class of formal logics that have been most intensively studied and most widely used. ... Boolean algebra is the finitary algebra of two values. ... Intuitionistic logic, or constructivist logic, is the logic used in mathematical intuitionism and other forms of mathematical constructivism. ... In mathematics, Heyting algebras are special partially ordered sets that constitute a generalization of Boolean algebras. ... OR logic gate. ... AND Logic Gate In logic and mathematics, logical conjunction (usual symbol and) is a two-place logical operation that results in a value of true if both of its operands are true, otherwise a value of false. ...

where = is understood to be logical equivalence over formulae. In logic, statements p and q are logically equivalent if they have the same logical content. ... A formula is a concise way of expressing information symbolically (as in a mathematical or chemical formula) or a general relationship between quantities. ...

The absorption law does not 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. Relevance logic, also called relevant logic, is any of a family of non-classical substructural logics that impose certain restrictions on implication. ... In mathematical logic, linear logic is a type of substructural logic that denies the structural rules of weakening and contraction. ... In 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. ... In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one-to-one and onto. ... In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a free variable is a notation for a place or places in an expression, into which some definite substitution may take place, or with respect to which some operation (summation or quantification, to give two...