Editing Distributive property (section)

== Generalizations ==
{{anchor|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 in [[order theory]] 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 {{em|one}} binary operation, such as the according definitions and their relations are given in the article [[distributivity (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 <math>\,=\,</math> by either <math>\,\leq\,</math> or <math>\,\geq.</math> 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]].

In [[category theory]], if <math>(S, \mu, \nu)</math> and <math>\left(S^{\prime}, \mu^{\prime}, \nu^{\prime}\right)</math> are [[Monad (category theory)|monad]]s on a [[Category (mathematics)|category]] <math>C,</math> a '''distributive law''' <math>S . S^{\prime} \to S^{\prime} . S</math> is a [[natural transformation]] <math>\lambda : S . S^{\prime} \to S^{\prime} . S</math> such that <math>\left(S^{\prime}, \lambda\right)</math> is a [[lax map of monads]] <math>S \to S</math> and <math>(S, \lambda)</math> is a [[colax map of monads]] <math>S^{\prime} \to S^{\prime}.</math> This is exactly the data needed to define a monad structure on <math>S^{\prime} . S</math>: the multiplication map is <math>S^{\prime} \mu . \mu^{\prime} S^2 . S^{\prime} \lambda S</math> and the unit map is <math>\eta^{\prime} S . \eta.</math> See: [[distributive law between monads]].

A [[generalized distributive law]] has also been proposed in the area of [[information theory]].

=== Antidistributivity ===

The ubiquitous [[Identity (mathematics)|identity]] that relates inverses to the binary operation in any [[Group (mathematics)|group]], namely <math>(x y)^{-1} = y^{-1} x^{-1},</math> which is taken as an axiom in the more general context of a [[semigroup with involution]], has sometimes been called an '''antidistributive property''' (of inversion as a [[unary operation]]).<ref name="BrinkKahl1997">{{cite book|author1=Chris Brink|author2=Wolfram Kahl|author3=Gunther Schmidt|title=Relational Methods in Computer Science|url=https://archive.org/details/relationalmethod00jips|url-access=limited|date=1997|publisher=Springer|isbn=978-3-211-82971-4|page=[https://archive.org/details/relationalmethod00jips/page/n16 4]}}</ref>

In the context of a [[near-ring]], 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 <math>a</math> reverses the order of addition when multiplied to the right: <math>(x + y) a = y a + x a.</math><ref>{{cite book|author1=Celestina Cotti Ferrero|author2=Giovanni Ferrero|title=Nearrings: Some Developments Linked to Semigroups and Groups|year=2002|publisher=Kluwer Academic Publishers|isbn=978-1-4613-0267-4|pages=62 and 67}}</ref>

In the study of [[propositional logic]] and [[Boolean algebra]], the term '''antidistributive law''' is sometimes used to denote the interchange between conjunction and disjunction when implication factors over them:<ref name="Hehner1993">{{cite book|author=Eric C.R. Hehner|author-link=Eric Hehner|title=A Practical Theory of Programming|year=1993|publisher=Springer Science & Business Media|isbn=978-1-4419-8596-5|page=230}}</ref>
<math display="block">(a \lor b) \Rightarrow c \equiv (a \Rightarrow c) \land (b \Rightarrow c)</math>
<math display="block">(a \land b) \Rightarrow c \equiv (a \Rightarrow c) \lor (b \Rightarrow c).</math>

These two [[Tautology (logic)|tautologies]] are a direct consequence of the duality in [[De Morgan's laws]].