Editing Boolean ring (section)

== Notation ==
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 {{math|1=''x'' + ''y'' = (''x'' ∧ ¬ ''y'') ∨ (¬ ''x'' ∧ ''y'')}} for the ring sum of {{math|''x''}} and {{math|''y''}}, and use {{math|1=''xy'' = ''x'' ∧ ''y''}} for their product.
* In [[Mathematical logic|logic]], a common notation is to use {{math|1=''x'' ∧ ''y''}} for the meet (same as the ring product) and use {{math|1=''x'' ∨ ''y''}} for the join, given in terms of ring notation (given just above) by {{math|1=''x'' + ''y'' + ''xy''}}.
* In [[set theory]] and logic it is also common to use {{math|1=''x'' · ''y''}} for the meet, and {{math|''x'' + ''y''}} for the join {{math|1=''x'' ∨ ''y''}}.{{sfn|Koppelberg|1989|loc=Definition 1.1, p. 7|ps=none}}  This use of {{math|+}} is different from the use in ring theory.
* A rare convention is to use {{math|''xy''}} for the product and {{math|1=''x'' ⊕ ''y''}} for the ring sum, in an effort to avoid the ambiguity of {{math|+}}.

Historically, the term "Boolean ring" has been used to mean a "Boolean ring possibly without an identity", and "Boolean algebra" has been used to mean a Boolean ring with an identity.  The existence of the identity is necessary to consider the ring as an algebra over the [[GF(2)|field of two elements]]: otherwise there cannot be a (unital) ring homomorphism of the field of two elements into the Boolean ring. (This is the same as the old use of the terms "ring" and "algebra" in [[measure theory]].{{efn|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.}})