|
In mathematics, a Boolean ring ''R'' is a ring for which ''x''2 = ''x'' for all ''x'' in ''R'', such as the ring of integers modulo 2. That is, ''R'' consists only of idempotent elements. 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 ∨). ==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 +. 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 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.) 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「boolean ring」の詳細全文を読む スポンサード リンク
|