Boolean algebra

:For a basic intro to sets, Boolean operations, Venn diagrams, truth tables, and Boolean applications, see Boolean logic.

Boolean rings, ideals and filters

Every Boolean algebra (A, land, lor) gives rise to a ring (A, +, *) by defining a + b = (a land ¬b) lor (b land ¬a) (this operation is called "symmetric difference" in the case of sets and XOR in the case of logic) and a * b = a land b. The zero element of this ring coincides with the 0 of the Boolean algebra; the multiplicative identity element of the ring is the 1 of the Boolean algebra. This ring has the property that a * a = a for all a in A; rings with this property are called Boolean rings.

Related Topics:
Ring - XOR - Boolean ring

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Conversely, if a Boolean ring A is given, we can turn it into a Boolean algebra by defining x lor y = x + y − xy and x land y = xy. Since these two operations are inverses of each other, we can say that every Boolean ring arises from a Boolean algebra, and vice versa. Furthermore, a map f : A → B is a homomorphism of Boolean algebras if and only if it is a homomorphism of Boolean rings. The categories of Boolean rings and Boolean algebras are equivalent.

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An ideal of the Boolean algebra A is a subset I such that for all x, y in I we have x lor y in I and for all a in A we have a land x in I. This notion of ideal coincides with the notion of ring ideal in the Boolean ring A. An ideal I of A is called prime if I ≠ A and if a land b in I always implies a in I or b in I. An ideal I of A is called maximal if I ≠ A and if the only ideal properly containing I is A itself. These notions coincide with ring theoretic ones of prime ideal and maximal ideal in the Boolean ring A.

Related Topics:
Ring ideal - Prime ideal - Maximal ideal

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The dual of an ideal is a filter. A filter of the Boolean algebra A is a subset p such that for all x, y in p we have x land y in p and for all a in A if a lor x = a then a in p.

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