Concept
Boolean ring
Overview
In mathematics, a Boolean ring 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 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 ∨, 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.
Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).
Possible passages
Found by search and not yet checked by a person; they may only mention the idea in passing.
The exercise of logical reason is always concerned with these absolutely general conditions. In its broadest sense, the discovery of mathematics is the discovery that the totality of these general abstract conditions, which are concurrently applicable to the relationships among the entities of any one concrete occasion, are themselves inter-connected in the manner of a pattern with a key to it. This pattern of…
Alfred North Whitehead, Science and the modern world
States what Boolean ring is.
Thus there arises a classification of modes of formation, or in the language commonly used by mathematicians, a classification of Functions. Any number, considered as formed from any other number, is called a function of it; and there are as many kinds of functions as there are modes of formation. The simple functions are by no means numerous, most functions being formed by the combination of several of the…
John Stuart Mill, A System of Logic, Ratiocinative and Inductive
States what Boolean ring is.
Division is defined as the operation inverse to multiplication; but begin by an example taken from the familiar notion of partition and show on this example that multiplication reproduces the dividend. There still remain the operations on fractions. The only difficulty is for multiplication.
Henri Poincaré, The Foundations of Science
States what Boolean ring is.
Philosophers who discussed Boolean ring
Related ideas
- Exclusive or
- Finite set
- If and only if
- Logical conjunction
- Mathematics
- Mathematical logic
- Intersection (set theory)
- Power set
- Set theory
- George Boole
- Boolean algebra (structure)
- Cardinality
- Decidability (logic)
Place in the tradition
Part of Outline of logic.