All Questions
Tagged with boolean-algebras universal-algebra
11 questions
18
votes
4
answers
2k
views
Complete Boolean algebra not isomorphic to a $\sigma$-algebra
Does there exist a complete Boolean algebra that is not isomorphic to any $\sigma$-algebra? If so, what is an easy or canonical example or construction?
- 24.8k
16
votes
1
answer
1k
views
What is a module over a Boolean ring?
Recall that a (unital) Boolean ring is a (unital) commutative ring $A$ where every element is idempotent; it follows that $A$ is of characteristic 2. There is an equivalence of categories between ...
- 63.9k
13
votes
4
answers
843
views
What is a "general" relation algebra?
I'm trying to understand why (or if) the axioms of relation algebras are "the right ones." For example, we can back up the idea that the group axioms exactly capture the notion of "...
- 20.9k
13
votes
3
answers
1k
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About a construction of Borel $\sigma$-algebra associated to a lattice
Let $(\mathcal{A}, \cup, \cap)$ a lattice (with minimum and maximum elements $\bot$ and $\top$).
Let $X\subset \mathcal{A}$ a generator set (a set of minimal cardinality that generate $\mathcal{A}$ i....
- 4,165
12
votes
5
answers
2k
views
Jonsson Boolean algebras?
Let us say that a mathematical structure of cardinality $\omega_1$ is Jonsson whenever every one of its proper substructures is countable.
There are examples of Jonsson groups due to Shelah or ...
- 11.3k
6
votes
1
answer
934
views
The universal algebra of a $\sigma$-algebra
I am searching for the 'dual' algebraic structure of a $\sigma$-algebra. The notion of duality is like in the case of the Boolean algebra and set algebra.
If $X$ is a set, the complement and ...
- 191
6
votes
1
answer
367
views
If questions are formalized as ideals of a boolean algebra, what kind of algebra of questions appears from Stone representation theorem?
Affirmative propositions make up a Boolean algebra, and Boolean algebras became part of classical algebra for over one century ago - in this sense they are "simple". But I did not encounter in ...
- 889
4
votes
0
answers
207
views
What algebraic identities does the iteration of forcing operation satisfy?
Let $G$ be the set of all formulas $\phi(x)$ in the language of such that $ZFC\vdash\exists x\phi(x)$ exists, $ZFC\vdash\phi(x)\rightarrow``x\,\textrm{is a complete Boolean algebra}"$, $ZFC\vdash``\...
3
votes
1
answer
133
views
What is the name for Boolean algebra's version of $\models$ between sets of identities and identities?
On p62 in Schaum's Outline of Theory and Problems of Boolean Algebra and Switching Circuits by Elliott Mendelson (1970),
Part (b) of the corollary says that if an identity is satisfied by some ...
- 357
1
vote
0
answers
126
views
Minimizing all aspects of the definition of Boolean algebra
There are many equivalent ways to describe Boolean algebras. There are a number of different ways to "minimize" the description. We can:
Minimize the number of function symbols.
Minimize ...
- 18.7k
0
votes
1
answer
514
views
Representation of free Boolean sigma-algebras
By a theorem of Loomis and Sikorski, for every Boolean $\sigma$-algebra $\mathfrak{A}$ there exists a $\sigma$-field of sets $\mathcal{F}$ and a $\sigma$-ideal $\Delta$ such that $\mathfrak{A}$ is ...
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