Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

During my studies of real analysis I've come across the definition of a field (or algebra) of sets. My question is: What is the connection between this structure and the structure of a field or algebra known from abstract algebra?

share|improve this question
1  
The two use the same word. –  Mariano Suárez-Alvarez Nov 9 '10 at 16:10

1 Answer 1

The term "field" in "field / algebra of sets over A" has no relation to the term "field" as used in ring theory. Rather it denotes a Boolean subalgebra of the power set of A, i.e. a collection of subsets of A closed under union, intersection and complement. The term is well-known due to its frequent use in the famous Stone representation theorem: $\ $every Boolean algebra is isomorphic to a field of sets. For further information see this Wikipedia page and this sci.math thread.

Note: Boolean algebras can be considered as Boolean rings, i.e. rings where every element is idempotent, i.e. $\: x^2 = x\:$. However, except for the field of two elements, such rings are never fields: $\ 0 = x^2 - x = x\ (x-1)\ \Rightarrow\ x = 0\ \ {\rm or}\ \ x = 1\ $ in a field, i.e. fields have only trivial idempotents.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.