ring of sets
Ring of Sets
for any , then ,
for any , then .
A ring of sets is a distributive lattice. The word “ring” in the name has nothing to do with the ordinary ring found in algebra. Rather, it is an abelian semigroup with respect to each of the binary set operations. If , then becomes an abelian monoid. Similarly, if , then is an abelian monoid. If both , then is a commutative semiring, since , and distributes over . Dualizing, we see that is also a commutative semiring. It is perhaps with this connection that the name “ring of sets” is so chosen.
Since is not required to be in , a ring of sets can in theory be the empty set. Even if may be non-empty, it may be a singleton. Both cases are not very interesting to study. To avoid such examples, some authors, particularly measure theorists, define a ring of sets to be a non-empty set with the first condition above replaced by
for any , then .
This is indeed a stronger condition, as . However, we shall stick with the more general definition here.
Field of Sets
An even stronger condition is to insist that not only is non-empty, but that . Such a ring of sets is called a field, or algebra of sets. Formally, given a set , a field of sets of satisfies the following criteria
is a ring of sets of ,
if , then the complement .
The three conditions above are equivalent to the following three conditions:
if , then , and
if , then .
A field of sets is also known as an algebra of sets.
It is easy to see that is a distributive complemented lattice, and hence a Boolean lattice. From the discussion earlier, we also see that (of ) is a commutative semiring, with acting as the multiplicative identity and both the additive identity and the multiplicative absorbing element.
Remark. Two remarkable theorems relating to of certain lattices as rings or fields of sets are the following:
|Title||ring of sets|
|Date of creation||2013-03-22 15:47:46|
|Last modified on||2013-03-22 15:47:46|
|Last modified by||CWoo (3771)|
|Synonym||lattice of sets|
|Synonym||algebra of sets|
|Defines||field of sets|