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Results tagged with universal-algebra
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user 22131
The study of algebraic structures and properties applying to large classes of such structures. For example, ideas from group theory and ring theory are extended and considered for structures with other signatures (systems of basic or fundamental operations).
8
votes
1
answer
278
views
About the characterization of categories of model of algebraic theories
So, in his Handbook of categorical algebra Vol 2, Borceux states a theorem (the 3.9.1, page 158) that says that:
Given a category $C$ with a functor $U:C \to Set$, $C$ is the category of models of a ( …
- 42.4k
15
votes
Accepted
What is the smallest variety of algebras containing all fields?
If I'm not mistaken, your answer is 'yes' : Let $M(A)$ be the set of maximal ideal of your commutative inverse ring $A$. Then you have a map :
$$A \rightarrow \prod_{\rho \in M(A) } A / \rho $$.
Eac …
- 42.4k
6
votes
Accepted
Equivalence relations in arbitrary categories
Short answer :Yes, assuming $\overline{Q}$ exists and $C$ has kernel pairs (for example if it has finite limits).
For more details: The relation $\overline{Q}$ do not always exists, you need some ass …
- 42.4k
22
votes
Accepted
Why is the theory of small categories not algebraic?
This follows from two Facts:
1) A category monadic over Set/S is always an exact category. That is it has quotient by equivalence relation that are effective and universal. It is in particular a regu …
- 42.4k
21
votes
Accepted
Complete Boolean algebra not isomorphic to a $\sigma$-algebra
Let $\Sigma_0$ be the $\sigma$-algebra of Lebesgue measurable sets on the real interval $[0,1]$. Define $\Sigma$ to be $\Sigma_0$ quotiented by the relation $U \simeq V$ if $U$ and $V$ differ by a Leb …
- 42.4k
1
vote
About a construction of Borel $\sigma$-algebra associated to a lattice
I'm still leaving this answer, but I wrote it a long time thinking it was about distributive lattices. I would be very surprise if anything of this sort can be done in the non distributive case.
Le …
- 42.4k
20
votes
Accepted
What is a module over a Boolean ring?
Theorem: Given $A$ a boolean ring/boolean algebra then there is an equivalence of categories between the category of $A$-modules and the category of sheaves of $\mathbb{F}_2$-vector spaces on Spec $A$ …
- 42.4k