Khác biệt giữa bản sửa đổi của “Đại số Boole”

== Định nghĩa ==

 

* Given any [[linearly ordered]] set ''L'' with a least element, the interval algebra is the smallest algebra of subsets of ''L'' containing all of the half-open intervals [''a'', ''b'') such that ''a'' is in ''L'' and ''b'' is either in ''L'' or equal to ∞. Interval algebras are useful in the study of [[Lindenbaum-Tarski algebra]]s; every countable Boolean algebra is isomorphic to an interval algebra.

 

* For any [[natural number]] ''n'', the set of all positive [[divisor]]s of ''n'', defining ''a''≤''b'' if ''a'' [[divides]] ''b'', forms a [[distributive lattice]]. This lattice is a Boolean algebra if and only if ''n'' is [[square-free integer|square-free]]. The bottom and the top element of this Boolean algebra is the natural number 1 and ''n'', respectively. The complement of ''a'' is given by ''n''/''a''. The meet and the join of ''a'' and ''b'' is given by the [[greatest common divisor]] (gcd) and the [[least common multiple]] (lcm) of ''a'' and ''b'', respectively. The ring addition ''a''+''b'' is given by lcm(''a'',''b'')/gcd(''a'',''b''). The picture shows an example for ''n'' = 30. As a counter-example, considering the non-square-free ''n''=60, the greatest common divisor of 30 and its complement 2 would be 2, while it should be the bottom element 1.

* Other examples of Boolean algebras arise from [[topology|topological spaces]]: if ''X'' is a topological space, then the collection of all subsets of ''X'' which are [[Clopen set|both open and closed]] forms a Boolean algebra with the operations ∨:= ∪ (union) and ∧:= ∩ (intersection).