Boolean ring

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In mathematics, a Boolean ring R is a ring (with identity) for which x2 = x for all x in R; that is, R consists only of idempotent elements.

A Boolean ring is essentially the same thing as a Boolean algebra, with ring multiplication corresponding to conjunction or meet ∧, and ring addition to exclusive disjunction or symmetric difference (not disjunction ∨).

Contents

Notations

There are at least four different and incompatible systems of notation for Boolean rings and algebras.

  • In commutative algebra the standard notation is to use x + y = (x ∧ ¬ y) ∨ (¬ x ∧ y) for the ring sum of x and y, and use xy for their product.
  • In logic, a common notation is to use x ∧ y for the meet (same as the ring product) and use x ∨ y for the join, given in terms of ring notation by x + y + xy.
  • In set theory and logic it is also common to use x · y for the meet, and x + y for the join x ∨ y. This use of + is different from the use in ring theory.
  • A rare convention is to use xy for the product and x ⊕ y for the ring sum, in an effort to avoid the ambiguity of +.

The old terminology was to use "Boolean ring" to mean a "Boolean ring possibly without an identity", and "Boolean algebra" to mean a Boolean ring with an identity. (This is the same as the old use of the terms "ring" and "algebra" in measure theory).

Examples

The simplest Boolean ring is the two-element Boolean algebra, with set the Boolean domain, conventionally written B = {0, 1}.

One example of a Boolean ring is the power set of any set X, where the addition in the ring is symmetric difference, and the multiplication is intersection. As another example, we can also consider the set of all finite or cofinite subsets of X, again with symmetric difference and intersection as operations. More generally with these operations any field of sets is a Boolean ring. By Stone's representation theorem every Boolean ring is isomorphic to a field of sets (treated as a ring with these operations).

Relation to Boolean algebras

Since the join operation ∨ in a Boolean algebra is often written additively, it makes sense in this context to denote ring addition by ⊕ (which is the same as subtraction in any Boolean algebra), a symbol that is often used to denote exclusive or.

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