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English: Illustration of distributive property ab+ac=a(b+c) with rectangles for positive numbers a,b and c. ... Simple example of distributive property with rectangles.
Examples of structures with two operations that are each distributive over the other are Boolean algebras such as the algebra of sets or the switching algebra. 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 ...
A simple example is the Fermat factorization method, which considers the sequence of numbers :=, for := ⌈ ⌉ +. If one of the x i {\displaystyle x_{i}} equals a perfect square b 2 {\displaystyle b^{2}} , then N = a i 2 − b 2 = ( a i + b ) ( a i − b ) {\displaystyle N=a_{i}^{2}-b^{2}=(a_{i}+b)(a_{i}-b)} is a (potentially non-trivial ...
In a non-distributive lattice, there may be elements that are distributive, but not dual distributive (and vice versa). For example, in the depicted pentagon lattice N 5, the element x is distributive, [2] but not dual distributive, since x ∧ (y ∨ z) = x ∧ 1 = x ≠ z = 0 ∨ z = (x ∧ y) ∨ (x ∧ z).
A non-associative algebra [1] (or distributive algebra) is an algebra over a field where the binary multiplication operation is not assumed to be associative.That is, an algebraic structure A is a non-associative algebra over a field K if it is a vector space over K and is equipped with a K-bilinear binary multiplication operation A × A → A which may or may not be associative.
The generalized distributive law (GDL) is a generalization of the distributive property which gives rise to a general message passing algorithm. [1] It is a synthesis of the work of many authors in the information theory , digital communications , signal processing , statistics , and artificial intelligence communities.