0

The set of measurable sets (of points) in a n-dimension space, equipped with certain operations (like union, intersection, set difference, which are often called “Boolean”), make up a generalized Boolean algebra (GBA). A GBA is a Boolean algebra with a top element; the top element in the GBA of measurable sets in an n-dimension space is missing. Can I express dimension in terms of Boolean operations? Is there a condition satisfied by GBAs isomorphic to GBAs of measurable subsets of real line (of plane, of 3D space) and only by them?

Ioachim Drugus
  • 656
  • 1
    Do you mean "A GBA is a Boolean algebra without a top element"? Otherwise it doesn't make sense, since all BAs have a top element, and then following sentence you say that "the top element in a GBA of measurable sets in an n-dimension space is missing". – amrsa Jan 07 '20 at 11:34
  • It's also not clear why you say the top element is missing. Maybe you are talking about the algebra of sets of finite Lebesgue measure? – Eric Wofsey Jan 07 '20 at 19:54
  • In any case, one possible version of what your question might mean is a duplicate of https://math.stackexchange.com/questions/2465128/are-the-measurable-spaces-mathbbrn-bor-mathbbrn-and-mathbbrm. – Eric Wofsey Jan 07 '20 at 19:55
  • My question is more precisely expressed in the title than in explanation below it. Now, I see I should have written in my explanation measurable sets of finite measure”. However, my question is not about measure – it is about dimension or better `dimensionality'. I prefer the word `dimensionality’, but the spell checker of this site treats this word as incorrectly written. What I intended to ask I will now put down like thisCan the notion of dimensionality of a figure (planary, spatial) be somehow expressed in terms of Boolean operations?''. – Ioachim Drugus Jan 08 '20 at 00:30

1 Answers1

1

You will want to look at Beran's book on orthomodular lattices and the appendix of Coxeter's book on regular polytopes.

Since the relationship between Boolean lattices and ortholattices does not depend upon generators in the sense of free Boolean lattices, you might also find Hu's book on threshold logic to be of interest.

Also, Tarski's work on cylindric algebra involves dimensionality.

mls
  • 31