- Distributivity
In

mathematics , and in particular inabstract algebra ,**distributivity**is a property ofbinary operation s that generalises the**distributive law**fromelementary algebra .For example:: 2 • (1 + 3) = (2 • 1) + (2 • 3).In the left-hand side of the above equation, the 2 multiplies the sum of 1 and 3; on the right-hand side, it multiplies the 1 and the 3 individually, with the results added afterwards.Because these give the same final answer (8), we say that multiplication by 2 "distributes" over addition of 1 and 3.Since we could have put anyreal number s in place of 2, 1, and 3 above, and still have obtained a true equation, we say thatmultiplication of real numbers "distributes" overaddition of real numbers.**Definition**Given a set "S" and two

binary operation s • and + on "S", we say that the operation •* is "left-distributive" over + if,

given any elements "x", "y", and "z" of "S",::"x" • ("y" + "z") = ("x" • "y") + ("x" • "z");

* is "right-distributive" over + if,given any elements "x", "y", and "z" of "S":::("y" + "z") • "x" = ("y" • "x") + ("z" • "x");

* is "distributive" over + if it is both left- and right-distributive.Notice that when • is

commutative , then the three above conditions are logically equivalent.**Examples**# Multiplication of

number s is distributive over addition of numbers, for a broad class of different kinds of numbers ranging fromnatural number s tocomplex number s andcardinal number s.

# Multiplication ofordinal number s, in contrast, is only left-distributive, not right-distributive.

# Matrix multiplication is distributive overmatrix addition , even though it's not commutative.

# The union of sets is distributive over intersection, and intersection is distributive over union. Also, intersection is distributive over thesymmetric difference .

#Logical disjunction ("or") is distributive overlogical conjunction ("and"), and conjunction is distributive over disjunction. Also, conjunction is distributive overexclusive disjunction ("xor").

# Forreal number s (or for anytotally ordered set ), the maximum operation is distributive over the minimum operation, and vice versa: max("a",min("b","c")) = min(max("a","b"),max("a","c")) and min("a",max("b","c")) = max(min("a","b"),min("a","c")).

# Forinteger s, thegreatest common divisor is distributive over theleast common multiple , and vice versa: gcd("a",lcm("b","c")) = lcm(gcd("a","b"),gcd("a","c")) and lcm("a",gcd("b","c")) = gcd(lcm("a","b"),lcm("a","c")).

# For real numbers, addition distributes over the maximum operation, and also over the minimum operation: "a" + max("b","c") = max("a"+"b","a"+"c") and "a" + min("b","c") = min("a"+"b","a"+"c").**Distributivity and rounding**In practice, the distributive property of multiplication (and division) over addition is lost around the limits of

arithmetic precision . For example, the identity ⅓+⅓+⅓ = (1+1+1)/3 appears to fail if conducted indecimal arithmetic ; however manysignificant digit s are used, the calculation will take the form 0.33333+0.33333+0.33333 = 0.99999 ≠ 1. Even where fractional numbers are representable exactly, errors will be introduced if rounding too far; for example, buying two books each priced at £14.99 before a tax of 17.5% in two separate transactions will actually save £0.01 over buying them together: £14.99×1.175 = £17.61 to the nearest £0.01, giving a total expenditure of £35.22, but £29.98×1.175 = £35.23. Methods such as banker's rounding may help in some cases, as may increasing the precision used, but ultimately some calculation errors are inevitable.**Distributivity in rings**Distributivity is most commonly found in rings and

distributive lattice s.A ring has two binary operations (commonly called "+" and "*"), and one of the requirements of a ring is that * must distribute over +.Most kinds of numbers (example 1) and matrices (example 3) form rings.A lattice is another kind of

algebraic structure with two binary operations, ^ and v.If either of these operations (say ^) distributes over the other (v), then v must also distribute over ^, and the lattice is called distributive. See also the article ondistributivity (order theory) .Examples 4 and 5 are Boolean algebras, which can be interpreted either as a special kind of ring (a

Boolean ring ) or a special kind of distributive lattice (aBoolean lattice ). Each interpretation is responsible for different distributive laws in the Boolean algebra. Examples 6 and 7 are distributive lattices which are not Boolean algebras.Rings and distributive lattices are both special kinds of rigs, certain generalisations of rings.Those numbers in example 1 that don't form rings at least form rigs.

Near-rig s are a further generalisation of rigs that are left-distributive but not right-distributive; example 2 is a near-rig.**Generalizations of distributivity**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 in

order theory one finds numerous important variants of distributivity, some of which include infinitary operations, such as theinfinite distributive law ; others being defined in the presence of only "one" binary operation, such as theimplication operator ofHeyting algebra s. Details of the according definitions and their relations are given in the articledistributivity (order theory) . 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 = by either ≤ or ≥. 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 intervals.In

category theory , if "(S, μ, η)" and "(S', μ', η')" are monads on a category "C", a**distributive law**"S.S' → S'.S" is anatural transformation "λ : S.S' → S'.S" such that ("S' ", λ) is alax map of monads "S → S" and ("S", λ) is acolax map of monads "S' → S' ". This is exactly the data needed to define a monad structure on "S'.S": the multiplication map is "S'μ.μ'S².S'λS" and the unit map is "η'S.η". See:distributive law between monads .**External links*** [

*http://www.cut-the-knot.org/Curriculum/Arithmetic/DistributiveLaw.shtml A demonstration of the Distributive Law*] for integer arithmetic (fromcut-the-knot )

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