Boolean ring in the context of George Boole

Boolean ring in the context of George Boole

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⭐ Core Definition: Boolean ring

In mathematics, a Boolean ring $R$ is a ring for which $x = x$ for all $x$ in $R$ , that is, a ring that consists of only idempotent elements. 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 $\land$ , and ring addition to exclusive disjunction or symmetric difference (not disjunction $\lor$ , 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.

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⭐ Core Definition: Boolean ring
Boolean ring in the context of Boolean algebra (structure)

Boolean ring in the context of Boolean algebra (structure)

In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. This type of algebraic structure captures essential properties of both set operations and logic operations. A Boolean algebra can be seen as a generalization of a power set algebra or a field of sets, or its elements can be viewed as generalized truth values. It is also a special case of a De Morgan algebra and a Kleene algebra (with involution).

Every Boolean algebra gives rise to a Boolean ring, and vice versa, with ring multiplication corresponding to conjunction or meet ∧, and ring addition to exclusive disjunction or symmetric difference (not disjunction ∨). However, the theory of Boolean rings has an inherent asymmetry between the two operators, while the axioms and theorems of Boolean algebra express the symmetry of the theory described by the duality principle.

View the full Wikipedia page for Boolean algebra (structure)

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