In mathematics, a Boolean ringR is a ring for which x2 = x for all x in R, that is, a ring that consists of only idempotent elements.[1][2][3] An example is the ring of integers modulo 2.
Every Boolean ring gives rise to a Boolean algebra, with ring multiplication corresponding to conjunction or meet ∧, and ring addition to exclusive disjunction or symmetric difference (not disjunction ∨,[4] which would constitute a semiring). Conversely, every Boolean algebra gives rise to a Boolean ring. Boolean rings are named after the founder of Boolean algebra, George Boole.
mathematics, a Booleanring R is a ring for which x2 = x for all x in R, that is, a ring that consists of only idempotent elements. An example is the ring of integers...
of Boolean variables whose state is determined by other variables in the network Boolean processor, a 1-bit variable computing unit Booleanring, a mathematical...
any set becomes a Booleanring, with symmetric difference as the addition of the ring and intersection as the multiplication of the ring. The symmetric difference...
In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the...
Boolean algebras are models of the equational theory of two values; this definition is equivalent to the lattice and ring definitions. Boolean algebra...
In logic and computer science, the Boolean satisfiability problem (sometimes called propositional satisfiability problem and abbreviated SATISFIABILITY...
elements, so any module over a Booleanring is locally free, but there are some non-projective modules over Booleanrings. One example is R/I where R is...
distributive lattices. The smallest semiring that is not a ring is the two-element Boolean algebra, e.g. with logical disjunction ∨ {\displaystyle \lor...
algebras is isomorphic to the category of Booleanrings. Given a Boolean algebra B, we turn B into a Booleanring by using the symmetric difference as addition...
of Boolean variables whose state is determined by other variables in the network Boolean processor, a 1-bit variables computing unit Booleanring, a ring...
However, for all above routines to work, m must not exceed 63 bits. Booleanring Circular buffer Division (mathematics) Finite field Legendre symbol Modular...
every r, the ring is called Booleanring. More general conditions which guarantee commutativity of a ring are also known. A graded ring R = ⨁i∊Z Ri is...
such as complex numbers, polynomials, matrices, rings, and fields. It is also encountered in Boolean algebra and mathematical logic, where each of the...
Algorithmic information theory Booleanring commutativity of a booleanringBoolean satisfiability problem NP-completeness of the Boolean satisfiability problem...
over fields or rings in ring theory. Fields of sets play an essential role in the representation theory of Boolean algebras. Every Boolean algebra can be...
power set considered together with both of these operations forms a Booleanring. In set theory, XY is the notation representing the set of all functions...
Neumann regular rings. The ring of affiliated operators of a finite von Neumann algebra is von Neumann regular. A Booleanring is a ring in which every...
\alpha _{2}\in \mathbb {C} \}} . The prime spectrum of a Booleanring (e.g., a power set ring) is a compact totally disconnected Hausdorff space (that...
LCM are idempotent. In a Booleanring, multiplication is idempotent. In a Tropical semiring, addition is idempotent. In a ring of quadratic matrices, the...
In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied...
associativity. Jordan ring: a commutative nonassociative ring that respects the Jordan identity Booleanring: a commutative ring with idempotent multiplication...
In mathematics, the Boolean prime ideal theorem states that ideals in a Boolean algebra can be extended to prime ideals. A variation of this statement...
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