In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice.

* It is useful to define gcd ( 0, 0 ) = 0 and lcm ( 0, 0 ) = 0 because then the natural numbers become a complete distributive lattice with gcd as meet and lcm as join operation.

This partially ordered set is always a distributive lattice.

If either of these operations ( say ∧) distributes over the other (∨), then ∨ must also distribute over ∧, and the lattice is called distributive.

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 ( a Boolean lattice ).

This also includes the notion of a completely distributive lattice.

* Boolean algebra: a complemented distributive lattice.

* Heyting algebra: a bounded distributive lattice with an added binary operation, relative pseudo-complement, denoted by infix " ' ", and governed by the axioms x ' x = 1, x ( x ' y )

* A bounded distributive lattice with an involution satisfying De Morgan's laws ( i. e. a De Morgan algebra ), additionally satisfying the inequality x ∧− x ≤ y ∨− y.

Every Boolean algebra is a Heyting algebra when a → b is defined as usual as ¬ a ∨ b, as is every complete distributive lattice when a → b is taken to be the supremum of the set of all c for which a ∧ c ≤ b. The open sets of a topological space form a complete distributive lattice and hence a Heyting algebra.

In the finite case every nonempty distributive lattice, in particular every nonempty finite chain, is automatically bounded and complete and hence a Heyting algebra.

Indeed, these lattices of sets describe the scenery completely: every distributive lattice is – up to isomorphism – given as such a lattice of sets.

As in the case of arbitrary lattices, one can choose to consider a distributive lattice L either as a structure of order theory or of universal algebra.

A lattice ( L ,, ) is distributive if the following additional identity holds for all x, y, and z in L:

A morphism of distributive lattices is just a lattice homomorphism as given in the article on lattices, i. e. a function that is compatible with the two lattice operations.

Because such a morphism of lattices preserves the lattice structure, it will consequently also preserve the distributivity ( and thus be a morphism of distributive lattices ).

* The Lindenbaum algebra of most logics that support conjunction and disjunction is a distributive lattice, i. e. " and " distributes over " or " and vice versa.

* Every Boolean algebra is a distributive lattice.

A near-field is an algebraic structure similar to a division ring, except that it has only one of the two distributive laws.

Another form is a distributive egalitarianism in which the wealth created by labor is organized and controlled in some equal manner.

For example, Jex & Britt mention research that indicates that interactional justice is a better predictor than procedural justice, which is in turn a better predictor than distributive justice.

The main distinction is between theories that argue the basis of just deserts is held equally by everyone, and therefore derive egalitarian accounts of distributive justice — and theories that argue the basis of just deserts is unequally distributed on the basis of, for instance, hard work, and therefore derive accounts of distributive justice by which some should have more than others.

In his A Theory of Justice, John Rawls used a social contract argument to show that justice, and especially distributive justice, is a form of fairness: an impartial distribution of goods.

Robert Nozick's influential critique of Rawls argues that distributive justice is not a matter of the whole distribution matching an ideal pattern, but of each individual entitlement having the right kind of history.

Moreover, although NA reduces gains in integrative tasks, it is a better strategy than PA in distributive tasks ( such as zero-sum ).

Two terms with the same variables raised to the same powers are called " similar terms " or " like terms ", and they can be combined, using the distributive law, into a single term whose coefficient is the sum of the coefficients of the terms that were combined.

To work out the product of two polynomials into a sum of terms, the distributive law is repeatedly applied, which results in each term of one polynomial being multiplied by every term of the other.

Then to define multiplication, it suffices by the distributive law to describe the product of any two such terms, which is given by the rule

For example, in " Anyone who thinks they have been affected should contact their doctor ", they and their are within the scope of the universal, distributive quantifier anyone, and can be interpreted as referring to an unspecified individual or to people in general ( notwithstanding the fact that " anyone " is strictly grammatically singular ).

It is most clearly evident in the special case of distributive constructions,

Hence, the Shakespeare quote above is semantically distributive, because there's not a man who ... is logically equivalent to every man does not ....

Since distributive constructions apply an idea relevant to each individual in the group, rather than to the group as a whole, they are most often conceived of as singular, and a singular pronoun is used.

The fact that singular forms are, nonetheless, more natural in distributive constructions is inadvertently demonstrated by websites that, not having access to the original languages in these cases, assume singular interpretations of they in what are actually translations of plurals.

* A complemented distributive lattice.

Every complemented distributive lattice has a unique orthocomplementation and is in fact a Boolean algebra.

He proved decidability of the elementary theory of the field of p-adic numbers ( independently proven by J. Ax and S. Kochen ), undecidability of the elementary theory of finite symmetric groups, decidability of the elementary theory of relatively complemented distributive lattices.

* is distributive over + if it is left-and right-distributive.

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 a natural transformation λ: S. S ' → S '. S such that ( S ', λ ) is a lax map of monads S → S and ( S, λ ) is a colax map of monads S ' → S '.

Some theories of procedural justice hold that fair procedure leads to equitable outcomes, even if the requirements of distributive or restorative justice are not met.

* left distributive if it satisfies the identity x * yz

* right distributive if it satisfies the identity yz * x =

* autodistributive if it is both left and right distributive,

The cube root operation is associative with exponentiation and distributive with multiplication and division if considering only real numbers, but not always if considering complex numbers, for example:

For example, L is distributive if and only if the following holds for all elements x, y, z in L:

Similarly, L is distributive if and only if

A lattice is distributive if and only if none of its sublattices is isomorphic to M < sub > 3 </ sub > or N < sub > 5 </ sub >; a sublattice is a subset that is closed under the meet and join operations of the original lattice.

For example, an element of a distributive lattice is meet-prime if and only if it is meet-irreducible, though the latter is in general a weaker property.

The introduction already hinted at the most important characterization for distributive lattices: a lattice is distributive if and only if it is isomorphic to a lattice of sets ( closed under set union and intersection ).

A predicate F is distributive if, whenever some things are F, each one of them is F. But in standard logic, every ( monadic ) predicate is distributive ( in standard logic, these " predicates " would be represented by relations ).

* A group is locally cyclic if and only if its lattice of subgroups is distributive.