In

A demonstration of the Distributive Law

for integer arithmetic (from

mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the distributive property of binary operation
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...

s generalizes the distributive law, which asserts that the equality
$$x\; \backslash cdot\; (y\; +\; z)\; =\; x\; \backslash cdot\; y\; +\; x\; \backslash cdot\; z$$
is always true in elementary algebra
Elementary algebra encompasses the basic concepts of algebra. It is often contrasted with arithmetic: arithmetic deals with specified numbers, whilst algebra introduces variable (mathematics), variables (quantities without fixed values).
This ...

.
For example, in elementary arithmetic
The operators in elementary arithmetic
Arithmetic () is an elementary part of mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are ...

, one has
$$2\; \backslash cdot\; (1\; +\; 3)\; =\; (2\; \backslash cdot\; 1)\; +\; (2\; \backslash cdot\; 3).$$
One says that multiplication
Multiplication (often denoted by the Multiplication sign, cross symbol , by the mid-line #Notation and terminology, dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Op ...

''distributes'' over addition
Addition (usually signified by the Plus and minus signs#Plus sign, plus symbol ) is one of the four basic Operation (mathematics), operations of arithmetic, the other three being subtraction, multiplication and Division (mathematics), division. ...

.
This basic property of numbers is part of the definition of most algebraic structure
In mathematics, an algebraic structure consists of a nonempty Set (mathematics), set ''A'' (called the underlying set, carrier set or domain), a collection of operation (mathematics), operations on ''A'' (typically binary operations such as addit ...

s that have two operations called addition and multiplication, such as complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...

s, polynomial
In mathematics, a polynomial is an expression (mathematics), expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addition, subtrac ...

s, matrices
Matrix most commonly refers to:
* The Matrix (franchise), ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within Th ...

, rings, and fields. It is also encountered in Boolean algebra
In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variable (mathematics), variables are the truth values ''true'' and ''false'', usually denote ...

and mathematical logic
Mathematical logic is the study of formal logic within mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantiti ...

, where each of the logical and (denoted $\backslash ,\backslash land\backslash ,$) and the logical or
In logic
Logic is the study of correct reasoning. It includes both Mathematical logic, formal and informal logic. Formal logic is the science of Validity (logic), deductively valid inferences or of logical truths. It is a formal science ...

(denoted $\backslash ,\backslash lor\backslash ,$) distributes over the other.
Definition

Given a set $S$ and twobinary operator
In mathematics, a binary operation or dyadic operation is a rule for combining two Element (mathematics), elements (called operands) to produce another element. More formally, a binary operation is an Operation (mathematics), operation of arity ...

s $\backslash ,*\backslash ,$ and $\backslash ,+\backslash ,$ on $S,$
*the operation $\backslash ,*\backslash ,$ is over (or with respect to) $\backslash ,+\backslash ,$ if, given any
In mathematical logic, a universal quantification is a type of Quantification (logic), quantifier, a logical constant which is interpretation (logic), interpreted as "given any" or "for all". It expresses that a predicate (mathematical logic), pr ...

elements $x,\; y,\; \backslash text\; z$ of $S,$
$$x\; *\; (y\; +\; z)\; =\; (x\; *\; y)\; +\; (x\; *\; z);$$
*the operation $\backslash ,*\backslash ,$ is over $\backslash ,+\backslash ,$ if, given any elements $x,\; y,\; \backslash text\; z$ of $S,$
$$(y\; +\; z)\; *\; x\; =\; (y\; *\; x)\; +\; (z\; *\; x);$$
*and the operation $\backslash ,*\backslash ,$ is over $\backslash ,+\backslash ,$ if it is left- and right-distributive.
When $\backslash ,*\backslash ,$ is commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...

, the three conditions above are logically equivalent
Logic is the study of correct reason
Reason is the capacity of Consciousness, consciously applying logic by Logical consequence, drawing conclusions from new or existing information, with the aim of seeking the truth. It is closely associat ...

.
Meaning

The operators used for examples in this section are those of the usualaddition
Addition (usually signified by the Plus and minus signs#Plus sign, plus symbol ) is one of the four basic Operation (mathematics), operations of arithmetic, the other three being subtraction, multiplication and Division (mathematics), division. ...

$\backslash ,+\backslash ,$ and multiplication
Multiplication (often denoted by the Multiplication sign, cross symbol , by the mid-line #Notation and terminology, dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Op ...

$\backslash ,\backslash cdot.\backslash ,$
If the operation denoted $\backslash cdot$ is not commutative, there is a distinction between left-distributivity and right-distributivity:
$$a\; \backslash cdot\; \backslash left(\; b\; \backslash pm\; c\; \backslash right)\; =\; a\; \backslash cdot\; b\; \backslash pm\; a\; \backslash cdot\; c\; \backslash qquad\; \backslash text$$
$$(a\; \backslash pm\; b)\; \backslash cdot\; c\; =\; a\; \backslash cdot\; c\; \backslash pm\; b\; \backslash cdot\; c\; \backslash qquad\; \backslash text.$$
In either case, the distributive property can be described in words as:
To multiply a sum (or difference) by a factor, each summand (or minuend
Subtraction is an arithmetic operation
Arithmetic () is an elementary part of mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they a ...

and subtrahend) is multiplied by this factor and the resulting products are added (or subtracted).
If the operation outside the parentheses (in this case, the multiplication) is commutative, then left-distributivity implies right-distributivity and vice versa, and one talks simply of .
One example of an operation that is "only" right-distributive is division, which is not commutative:
$$(a\; \backslash pm\; b)\; \backslash div\; c\; =\; a\; \backslash div\; c\; \backslash pm\; b\; \backslash div\; c.$$
In this case, left-distributivity does not apply:
$$a\; \backslash div(b\; \backslash pm\; c)\; \backslash neq\; a\; \backslash div\; b\; \backslash pm\; a\; \backslash div\; c$$
The distributive laws are among the axioms for rings (like the ring of integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language of ...

s) and fields (like the field of rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...

s). Here multiplication is distributive over addition, but addition is not distributive over multiplication. Examples of structures with two operations that are each distributive over the other are Boolean algebras such as the algebra of sets
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...

or the switching algebra
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...

.
Multiplying sums can be put into words as follows: When a sum is multiplied by a sum, multiply each summand of a sum with each summand of the other sum (keeping track of signs) then add up all of the resulting products.
Examples

Real numbers

In the following examples, the use of the distributive law on the set of real numbers $\backslash R$ is illustrated. When multiplication is mentioned in elementary mathematics, it usually refers to this kind of multiplication. From the point of view of algebra, the real numbers form a field, which ensures the validity of the distributive law.Matrices

The distributive law is valid formatrix multiplication
In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix (mathematics), matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the n ...

. More precisely,
$$(A\; +\; B)\; \backslash cdot\; C\; =\; A\; \backslash cdot\; C\; +\; B\; \backslash cdot\; C$$
for all $l\; \backslash times\; m$-matrices $A,\; B$ and $m\; \backslash times\; n$-matrices $C,$ as well as
$$A\; \backslash cdot\; (B\; +\; C)\; =\; A\; \backslash cdot\; B\; +\; A\; \backslash cdot\; C$$
for all $l\; \backslash times\; m$-matrices $A$ and $m\; \backslash times\; n$-matrices $B,\; C.$
Because the commutative property does not hold for matrix multiplication, the second law does not follow from the first law. In this case, they are two different laws.
Other examples

*Multiplication
Multiplication (often denoted by the Multiplication sign, cross symbol , by the mid-line #Notation and terminology, dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Op ...

of ordinal number
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets.
A finite set can be enumerated by successively labeling each element with the least n ...

s, in contrast, is only left-distributive, not right-distributive.
* The cross product
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two Euclidean vector, vectors in a three-dimensional Orientation (vector space), oriented E ...

is left- and right-distributive over vector addition
In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has Magnitude (mathematics), magnitude (or euclidean norm, length) and Direction ( ...

, though not commutative.
* The union of sets is distributive over intersection
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented ...

, and intersection is distributive over union.
* Logical disjunction
In logic
Logic is the study of correct reasoning. It includes both Mathematical logic, formal and informal logic. Formal logic is the science of Validity (logic), deductively valid inferences or of logical truths. It is a formal science ...

("or") is distributive over logical conjunction
In logic
Logic is the study of correct reasoning. It includes both Mathematical logic, formal and informal logic. Formal logic is the science of Validity (logic), deductively valid inferences or of logical truths. It is a formal science ...

("and"), and vice versa.
* For real number
In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...

s (and for any totally ordered set
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...

), the maximum operation is distributive over the minimum operation, and vice versa: $$\backslash max(a,\; \backslash min(b,\; c))\; =\; \backslash min(\backslash max(a,\; b),\; \backslash max(a,\; c))\; \backslash quad\; \backslash text\; \backslash quad\; \backslash min(a,\; \backslash max(b,\; c))\; =\; \backslash max(\backslash min(a,\; b),\; \backslash min(a,\; c)).$$
* For integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language of ...

s, the greatest common divisor
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mo ...

is distributive over the least common multiple
In arithmetic and number theory, the least common multiple, lowest common multiple, or smallest common multiple of two integers ''a'' and ''b'', usually denoted by lcm(''a'', ''b''), is the smallest positive integer that is divisible by bot ...

, and vice versa: $$\backslash gcd(a,\; \backslash operatorname(b,\; c))\; =\; \backslash operatorname(\backslash gcd(a,\; b),\; \backslash gcd(a,\; c))\; \backslash quad\; \backslash text\; \backslash quad\; \backslash operatorname(a,\; \backslash gcd(b,\; c))\; =\; \backslash gcd(\backslash operatorname(a,\; b),\; \backslash operatorname(a,\; c)).$$
* For real numbers, addition distributes over the maximum operation, and also over the minimum operation: $$a\; +\; \backslash max(b,\; c)\; =\; \backslash max(a\; +\; b,\; a\; +\; c)\; \backslash quad\; \backslash text\; \backslash quad\; a\; +\; \backslash min(b,\; c)\; =\; \backslash min(a\; +\; b,\; a\; +\; c).$$
* For binomial multiplication, distribution is sometimes referred to as the FOIL Method (First terms $a\; c,$ Outer $a\; d,$ Inner $b\; c,$ and Last $b\; d$) such as: $(a\; +\; b)\; \backslash cdot\; (c\; +\; d)\; =\; a\; c\; +\; a\; d\; +\; b\; c\; +\; b\; d.$
* In all semirings, including the complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...

s, the quaternion
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...

s, polynomial
In mathematics, a polynomial is an expression (mathematics), expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addition, subtrac ...

s, and matrices
Matrix most commonly refers to:
* The Matrix (franchise), ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within Th ...

, multiplication distributes over addition: $u\; (v\; +\; w)\; =\; u\; v\; +\; u\; w,\; (u\; +\; v)w\; =\; u\; w\; +\; v\; w.$
* In all algebras over a field, including the octonion
In mathematics, the octonions are a normed division algebra over the real numbers, a kind of Hypercomplex number, hypercomplex Number#Classification, number system. The octonions are usually represented by the capital letter O, using boldface or ...

s and other non-associative algebra
A non-associative algebra (or distributive algebra) is an algebra over a field where the binary operation, binary multiplication operation is not assumed to be associative operation, associative. That is, an algebraic structure ''A'' is a non-ass ...

s, multiplication distributes over addition.
Propositional logic

Rule of replacement

In standard truth-functional propositional logic, in logical proofs uses two valid rules of replacement to expand individual occurrences of certainlogical connective
In Mathematical logic, logic, a logical connective (also called a logical operator, sentential connective, or sentential operator) is a logical constant. They can be used to connect logical formulas. For instance in the syntax (logic), syntax o ...

s, within some formula
In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the terminology, term ''formula'' in science refers to the Commensurability (philosophy o ...

, into separate applications of those connectives across subformulas of the given formula. The rules are
$$(P\; \backslash land\; (Q\; \backslash lor\; R))\; \backslash Leftrightarrow\; ((P\; \backslash land\; Q)\; \backslash lor\; (P\; \backslash land\; R))\; \backslash qquad\; \backslash text\; \backslash qquad\; (P\; \backslash lor\; (Q\; \backslash land\; R))\; \backslash Leftrightarrow\; ((P\; \backslash lor\; Q)\; \backslash land\; (P\; \backslash lor\; R))$$
where "$\backslash Leftrightarrow$", also written $\backslash ,\backslash equiv,\backslash ,$ is a metalogic
Metalogic is the study of the metatheory of logic. Whereas ''logic'' studies how formal system, logical systems can be used to construct Validity (logic), valid and soundness, sound arguments, metalogic studies the properties of logical systems.Har ...

al symbol
A symbol is a mark, sign, or word that indicates, signifies, or is understood as representing an idea, object, or relationship. Symbols allow people to go beyond what is known or seen by creating linkages between otherwise very different ...

representing "can be replaced in a proof with" or "is logically equivalent
Logic is the study of correct reason
Reason is the capacity of Consciousness, consciously applying logic by Logical consequence, drawing conclusions from new or existing information, with the aim of seeking the truth. It is closely associat ...

to".
Truth functional connectives

is a property of some logical connectives of truth-functionalpropositional logic
Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations b ...

. The following logical equivalences demonstrate that distributivity is a property of particular connectives. The following are truth-functional tautologies.
;Double distribution:
$$\backslash begin\; \&((P\; \backslash land\; Q)\; \&\&\backslash ;\backslash lor\; (R\; \backslash land\; S))\; \&\&\backslash ;\backslash Leftrightarrow\backslash ;\&\&\; (((P\; \backslash lor\; R)\; \backslash land\; (P\; \backslash lor\; S))\; \&\&\backslash ;\backslash land\; ((Q\; \backslash lor\; R)\; \backslash land\; (Q\; \backslash lor\; S)))\; \&\&\; \backslash \backslash \; \&((P\; \backslash lor\; Q)\; \&\&\backslash ;\backslash land\; (R\; \backslash lor\; S))\; \&\&\backslash ;\backslash Leftrightarrow\backslash ;\&\&\; (((P\; \backslash land\; R)\; \backslash lor\; (P\; \backslash land\; S))\; \&\&\backslash ;\backslash lor\; ((Q\; \backslash land\; R)\; \backslash lor\; (Q\; \backslash land\; S)))\; \&\&\; \backslash \backslash \; \backslash end$$
Distributivity and rounding

In approximate arithmetic, such asfloating-point arithmetic
In computing, floating-point arithmetic (FP) is arithmetic that represents real numbers approximately, using an Integer (computer science), integer with a fixed precision, called the significand, scaled by an integer exponent of a fixed base. ...

, the distributive property of multiplication (and division) over addition may fail because of the limitations of arithmetic precision
Significant figures (also known as the significant digits, ''precision'' or ''resolution'') of a number in positional notation
Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any ...

. For example, the identity $1/3\; +\; 1/3\; +\; 1/3\; =\; (1\; +\; 1\; +\; 1)\; /\; 3$ fails in decimal arithmetic, regardless of the number of significant digit
Significant figures (also known as the significant digits, ''precision'' or ''resolution'') of a number in positional notation
Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any ...

s. Methods such as banker's rounding may help in some cases, as may increasing the precision used, but ultimately some calculation errors are inevitable.
In rings and other structures

Distributivity is most commonly found insemiring
In abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), ...

s, notably the particular cases of rings and distributive lattice
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mod ...

s.
A semiring has two binary operations, commonly denoted $\backslash ,+\backslash ,$ and $\backslash ,*,$ and requires that $\backslash ,*\backslash ,$ must distribute over $\backslash ,+.$
A ring is a semiring with additive inverses.
A lattice is another kind of algebraic structure
In mathematics, an algebraic structure consists of a nonempty Set (mathematics), set ''A'' (called the underlying set, carrier set or domain), a collection of operation (mathematics), operations on ''A'' (typically binary operations such as addit ...

with two binary operations, $\backslash ,\backslash land\; \backslash text\; \backslash lor.$
If either of these operations distributes over the other (say $\backslash ,\backslash land\backslash ,$ distributes over $\backslash ,\backslash lor$), then the reverse also holds ($\backslash ,\backslash lor\backslash ,$ distributes over $\backslash ,\backslash land\backslash ,$), and the lattice is called distributive. See also .
A Boolean algebra
In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variable (mathematics), variables are the truth values ''true'' and ''false'', usually denote ...

can be interpreted either as a special kind of ring (a Boolean ring In mathematics, a Boolean ring ''R'' is a ring (mathematics), ring for which ''x''2 = ''x'' for all ''x'' in ''R'', that is, a ring that consists only of idempotent element (ring theory), idempotent elements. An example is the ring of modular arithm ...

) or a special kind of distributive lattice (a Boolean lattice). Each interpretation is responsible for different distributive laws in the Boolean algebra.
Similar structures without distributive laws are near-ring In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mode ...

s and near-fields instead of rings and division ring
In algebra
Algebra () is one of the areas of mathematics, broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost a ...

s. The operations are usually defined to be distributive on the right but not on the left.
Generalizations

In several mathematical areas, generalized distributivity laws are considered. This may involve the weakening of the above conditions or the extension to infinitary operations. Especially inorder theory
Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article intro ...

one finds numerous important variants of distributivity, some of which include infinitary operations, such as the infinite distributive law; others being defined in the presence of only binary operation, such as the according definitions and their relations are given in the article distributivity (order theory)
In the mathematics, mathematical area of order theory, there are various notions of the common concept of distributivity, applied to the formation of supremum, suprema and infimum, infima. Most of these apply to partially ordered sets that are at l ...

. This also includes the notion of a completely distributive lattice.
In the presence of an ordering relation, one can also weaken the above equalities by replacing $\backslash ,=\backslash ,$ by either $\backslash ,\backslash leq\backslash ,$ or $\backslash ,\backslash geq.$ Naturally, this will lead to meaningful concepts only in some situations. An application of this principle is the notion of sub-distributivity as explained in the article on interval arithmetic
Interval arithmetic (also known as interval mathematics, interval analysis, or interval computation) is a mathematical technique used to Floating point error mitigation, put bounds on rounding errors and measurement errors in numerical analysis ...

.
In category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, categ ...

, if $(S,\; \backslash mu,\; \backslash nu)$ and $\backslash left(S^,\; \backslash mu^,\; \backslash nu^\backslash right)$ are monad
Monad may refer to:
Philosophy
* Monad (philosophy), a term meaning "unit"
**Monism, the concept of "one essence" in the metaphysical and theological theory
** Monad (Gnosticism), the most primal aspect of God in Gnosticism
* ''Great Monad'', an ...

s on a category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
*Category of being
*Categories (Aristotle), ''Categories'' (Aristotle)
*Category (Kant)
...

$C,$ a distributive law $S\; .\; S^\; \backslash to\; S^\; .\; S$ is a natural transformation
In category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topo ...

$\backslash lambda\; :\; S\; .\; S^\; \backslash to\; S^\; .\; S$ such that $\backslash left(S^,\; \backslash lambda\backslash right)$ is a lax map of monads $S\; \backslash to\; S$ and $(S,\; \backslash lambda)$ is a colax map of monads $S^\; \backslash to\; S^.$ This is exactly the data needed to define a monad structure on $S^\; .\; S$: the multiplication map is $S^\; \backslash mu\; .\; \backslash mu^\; S^2\; .\; S^\; \backslash lambda\; S$ and the unit map is $\backslash eta^\; S\; .\; \backslash eta.$ See: distributive law between monads.
A generalized distributive law The generalized distributive law (GDL) is a generalization of the distributive property which gives rise to a general message passing algorithm. It is a synthesis of the work of many authors in the information theory, digital communications, signal ...

has also been proposed in the area of information theory
Information theory is the scientific study of the quantification (science), quantification, computer data storage, storage, and telecommunication, communication of information. The field was originally established by the works of Harry Nyquist a ...

.
Antidistributivity

The ubiquitous identity that relates inverses to the binary operation in anygroup
A group is a number
A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, ...

, namely $(x\; y)^\; =\; y^\; x^,$ which is taken as an axiom in the more general context of a semigroup with involution In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mode ...

, has sometimes been called an antidistributive property (of inversion as a unary operation
In mathematics, an unary operation is an Operation (mathematics), operation with only one operand, i.e. a single input. This is in contrast to binary operations, which use two operands. An example is any function (mathematics), function , where ...

).
In the context of a near-ring In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mode ...

, which removes the commutativity of the additively written group and assumes only one-sided distributivity, one can speak of (two-sided) distributive elements but also of antidistributive elements. The latter reverse the order of (the non-commutative) addition; assuming a left-nearring (i.e. one which all elements distribute when multiplied on the left), then an antidistributive element $a$ reverses the order of addition when multiplied to the right: $(x\; +\; y)\; a\; =\; y\; a\; +\; x\; a.$
In the study of propositional logic
Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations b ...

and Boolean algebra
In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variable (mathematics), variables are the truth values ''true'' and ''false'', usually denote ...

, the term antidistributive law is sometimes used to denote the interchange between conjunction and disjunction when implication factors over them:
$$(a\; \backslash lor\; b)\; \backslash Rightarrow\; c\; \backslash equiv\; (a\; \backslash Rightarrow\; c)\; \backslash land\; (b\; \backslash Rightarrow\; c)$$
$$(a\; \backslash land\; b)\; \backslash Rightarrow\; c\; \backslash equiv\; (a\; \backslash Rightarrow\; c)\; \backslash lor\; (b\; \backslash Rightarrow\; c).$$
These two tautologies are a direct consequence of the duality in De Morgan's laws
In propositional calculus, propositional logic and Boolean algebra, De Morgan's laws, also known as De Morgan's theorem, are a pair of transformation rules that are both Validity (logic), valid rule of inference, rules of inference. They are na ...

.
Notes

External links

{{Wiktionary, distributivityA demonstration of the Distributive Law

for integer arithmetic (from

cut-the-knot
Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet
The Soviet Union,. officially the Union of Soviet Socialist Republics. (USSR),. was a List of former transcontinental countries#Since 1700, transcontinental country that sp ...

)
Properties of binary operations
Elementary algebra
Rules of inference
Theorems in propositional logic