Boolean Ring - Notations

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 = x ∧ y 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 (given just above) 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) (Also note that, when a Boolean ring has an identity, then a complement operation becomes definable on it, and a key characteristic of the modern definitions of both Boolean algebra and sigma-algebra is that they have complement operations.)