All Questions
Tagged with ra.rings-and-algebras lo.logic
54 questions
13
votes
2
answers
1k
views
What's the deal with De Morgan algebras and Kleene algebras?
The notion of Boolean algebras, and the corresponding classical propositional logic, is very standard, and it is easy to find information about them (for example, among many other such works, there is ...
- 32.4k
15
votes
0
answers
217
views
If a map between unital rings preserves multiplication and successor, does it preserve addition?
Welcome to my first MathOverflow posting!
This is a question about rings, all of them assumed to be both unital and associative.
Let $f\colon R\to S$ be a map between rings such that $f(xy)=f(x)f(y)$ ...
- 151
5
votes
1
answer
168
views
Countably compact Boolean algebras versus distributivity
Let us say that a complete Boolean algebra $B$ is:
countably distributive when for any sequence $(I_n)_{n\in\mathbb{N}}$ of sets and any elements $(u_{n,i})_{n\in\mathbb{N},i\in I_n}$ of $B$ we have
$...
- 32.4k
11
votes
2
answers
558
views
Whether an isotone bijection from a power set lattice to another sends singletons to singletons
By the work of Paul Cohen (on the continuum hypothesis), one can neither prove nor disprove from the axioms of ZFC that a bijection $f$ from the power set $\mathcal{P}(S)$ of a set $S$ to the power ...
- 10.5k
21
votes
1
answer
1k
views
Are the real numbers isomorphic to a nontrivial ultraproduct of fields?
Let $K_1, K_2, \dots$ be a countable sequence of fields, and let $\prod_{\mathcal F} K_i$ be the ultraproduct with respect to some nonprincipal ultrafilter $\mathcal F$.
Question: Can there be a field ...
- 63.9k
5
votes
3
answers
542
views
Congruences that aren't "finite from above"
Let $\mathfrak{A}=(A;...)$ be an algebra in the sense of universal algebra. Say that a congruence $\sim$ on $\mathfrak{A}$ is parafinite iff there is an equivalence relation $E\subseteq A^2$ with ...
- 21.1k
6
votes
0
answers
179
views
Elementary equivalence for rings
Let $\mathcal{L}$ be a first-order language, and $M$ and $N$ be two $\mathcal{L}$-structures. We say that $M$ and $N$ are elementarily equivalent (write $M \approx N$) if they satisfy the same first-...
- 3,318
12
votes
0
answers
542
views
Does Wedderburn's Little Theorem hold constructively?
Wedderburn's Little Theorem states that every finite division ring is commutative. Perhaps even more surprising, this implies that every finite reduced ring is commutative.
The proofs that I am aware ...
- 63.1k
7
votes
1
answer
494
views
Normal form for terms in language with two ring structures
Suppose I have two different ring structures on the same domain $\langle R,+,\cdot,0,1\rangle$, $\langle R,\oplus,\otimes,\bar 0,\bar 1\rangle$ and I throw the structures together into a single common ...
- 236k
5
votes
0
answers
302
views
Reference for countable and uncountable algebraic closures of $\mathbb{Q}$ in ZF
The following facts seem to be part of the folklore (where $\mathsf{ZF}$ means Zermelo-Fraenkel set theory with no axiom of choice):
it is consistent with $\mathsf{ZF}$ that there exists an ...
- 32.4k
13
votes
0
answers
571
views
Why is it so hard to give examples of differentially closed fields?
The theory of algebraically closed field, say in characteristic zero, and of differentially closed fields (of characteristic zero) have much in common: quantifier elimination and (hence) decidability; ...
- 32.4k
4
votes
0
answers
197
views
Are there any 1-decidable algebraic extensions of $\mathbb{Q}$ which are not decidable?
A model $M$ is decidable if the set of all first-order formulas which are true in $M$ is a recursive set. And a model is $1$-decidable if the set of all existential formulas which are true in $M$ is ...
- 4,587
4
votes
1
answer
287
views
What is the lowest complexity definition of $\mathbb{Z}$ in an infinite algebraic extension of $\mathbb{Q}$?
In 2009, Jochen Koenigsmann showed that $\mathbb{Z}$ is universally definable in the field $\mathbb{Q}$. And in 2012, Jennifer Park proved a result which implies that $\mathbb{Z}$ is $\exists\forall$-...
- 4,587
6
votes
1
answer
518
views
What is the Galois group of one ultrapower over another ultrapower?
Let $F$ be a field, let $E$ be a field extension of $F$, and let $U$ be an ultrafilter. Then my question is, what is the relationship between the Galois groups $Gal(\Pi_U E/\Pi_U F)$ and $Gal(E/F)$?
...
- 4,587
13
votes
1
answer
492
views
Applications of Robinson's consistency theorem in algebra?
This is crossposted from MSE. It's also my first time asking on MO, so please let me know if there's anything you need from me!
There are a family of results which, in many model theory books, are ...
- 291
6
votes
0
answers
294
views
Independence results on pure algebra
I think that the most celebrated result in this direction is Shelah's famous work on Whitehead's Problem:
Is every abelian group $A$ such that $Ext^1(A, \mathbb{Z})=0$ free?
This is known to be ...
- 3,318
13
votes
1
answer
811
views
Is there a ring for which the reducibility of a polynomial is undecidable?
Let $R$ be a ring such that all of its elements have a finite number of divisors, ie $\forall r\in R\, |\{x\in R: x|r\}|<\infty$.
Then we can decide whether a polynomial in $R[t]$ is reducible ...
- 377
1
vote
2
answers
157
views
Link btw. exponential and derivatives from an algebraic perspective [closed]
I have been attempting to understand my math education (as a bachelor in electrical engineering) from a more algebraic perspective recently. I would like to understand more about the link between ...
10
votes
0
answers
416
views
Equational theory in the signature (+,*,0,1) of sedenions and beyond
Consider a Cayley-Dickson algebra $(X,+,∗,0,1)$, that is an algebra generated from the reals by the Cayley-Dickson construction. From complexes to quaternions, we lose commutativity of multiplication, ...
- 2,013
4
votes
0
answers
127
views
Cyclic relation algebra
A relation algebra $\mathbf{R}$ is a structure $\langle |\mathbf{R}|, \vee, \neg, \circ, I, (-)^{op} \rangle$ such that:
$\langle |\mathbf{R}|, \vee, \neg \rangle$ is a Boolean algebra,
$\langle |\...
- 4,096
11
votes
4
answers
2k
views
When is it okay to intersect infinite families of proper classes?
For experts who work in ZFC, it is common knowledge that one cannot in general define a countable intersection/union of proper classes. However, in my work as a ring theorist I intersect infinite ...
- 18.7k
-1
votes
1
answer
416
views
Conversion of logic formula into algebraic formula
We know formula of boolean algebra in canonical disjunctive normal form has or may be converted to Zhegalkin polynomial.
Is there any approach to convert first order formula into algebraic function ...
- 3,084
19
votes
1
answer
977
views
Topological universal algebra: what is a variety?
Very roughly, universal algebra is the study of those classes of algebraic structures which can be defined via a set of equations; such a class is called a variety. Of course there is far more to the ...
- 21.1k
15
votes
1
answer
524
views
Non standard extension of real numbers via nonprincipal ultra filters
Assume That $U,V$ are two filters on the natural number $\mathbb{N}$.
We say that $U$ is equivalent to $V$ if there is a bijection $\phi: \mathbb{N} \to \mathbb{N}$ such that $\tilde{\phi}(U)=...
- 356
5
votes
2
answers
408
views
Lefschetz Principle for semisimplicity
I think I can prove the following using the compactness of first order logic and I am wondering what a purely algebraic proof would look like.
Let $R$ be a unital ring (not necessarily ...
- 38.6k
7
votes
1
answer
555
views
Fuzzy logic of Godel
In Gödel logic, is conjunction definable from implication, negation , and disjunction?
We know that conjunction in that logic is not definable from negation and implication.
- 137
3
votes
0
answers
115
views
Cardinality based results in Topological Vector Spaces?
Given a topological vector space $V$, let its density be the smallest cardinal $A$ such that a set of cardinality $A$ is dense in $V$. Naively, it seems one of two things happen:
TVS's $V$ of larger ...
- 437
8
votes
1
answer
385
views
For a ring $k$ and a set $X$, what are the $k$-algebra homomorphisms $k^X \to k$?
Let $k$ be a commutative ring. Feel free to assume it's a field.
Let $X$ be a set. This question is only interesting when $X$ is infinite.
Write $k^X$ for the $k$-algebra of functions $X \to k$, ...
- 27.7k
20
votes
3
answers
2k
views
How do I apply the Boolean Prime Ideal Theorem?
I have become aware of an amazing phenomenon from a myriad of questions and answers here on MathOverflow: many of the results that I would typically prove using the Axiom of Choice can actually be ...
- 5,407
24
votes
2
answers
2k
views
What do you do if you believe a problem is undecidable?
While the title of this question is subjective, I hope to make what I'm looking for quite concrete. The first, and main question is this: If you believe that a problem you are working on is formally ...
- 18.7k
8
votes
3
answers
825
views
Does a left basis imply a right basis, without AC?
If $_DV_D$ is a $D$-$D$-bimodule, and we have a $D$-basis for $V_D$, do we still need AC to get a $D$-basis for $_DV$?
(The original question appears below. But this shorter question gets at the ...
- 18.7k
1
vote
1
answer
272
views
Self-similarity for simple algebraic structures [closed]
I'm doing this thread because I have some ideas about how to define self-similarity in algebra, but I don't know if this is known at all. Any critics, comments and references are more than welcomed. ...
- 438
15
votes
1
answer
614
views
Computability of Brauer groups
A friend of mine and I were talking about computable algebra, and this question came up. The answer may already be known, but I couldn't find it with Google:
Suppose I have a countable field, $k$. ...
- 21.1k
2
votes
2
answers
377
views
Is there an intuitionistic generalized boolean algebra (of Stone)?
A "boolean algebra without the greatest element" was called by Stone "generalized boolean algebra" and he axiomatized it. Is there any publication about "preudo-boolean algebras without the greatest ...
- 889
2
votes
0
answers
300
views
on the Axiom of Choice and the Spectrum of Rings
consider the following theorem, when $R$ is a commutative ring with a non-zero identity:
A ring $R$ is zero-dimensional if and only if $\mbox{Spec(R)}$ is Hausdorff.
The proof uses the Axiom of ...
user30230
1
vote
1
answer
193
views
Is a variety of algebras a set?
Let $K$ be a field and $K\{X\}$ be the free non-associative algebra, freely generated by the countably infinite set $X$. We consider elements of $K\{X\}$ as (non-associative) polynomials in the ...
- 398
22
votes
3
answers
2k
views
Nice algebraic statements independent from ZF + V=L (constructibility)
Background and motivation
I've always been fascinated about algebraic statements independent from ZFC set theory. One such fascinating example comes from considering $\rm{Ext}^1_\mathbb{Z}(A,\mathbb{Z}...
user1437
16
votes
3
answers
2k
views
Continuum Hypothesis
I am new here, so forgive me if this question does not satisfy the protocols of the site.
I know there are so many equivalents to the AC (axiom of choice) and there are books that lists this ...
user30300
14
votes
1
answer
2k
views
Finite dimensional real division algebras
A celebrated theorem of Milnor and Kervaire asserts that any finite dimensional (not necessarily associative, unital) division algebra over the real numbers has dimension 1,2,4 or 8. This result is ...
- 2,550
15
votes
1
answer
1k
views
Is the class of additive groups of rings axiomatizable?
I know that it is impossible to axiomatize the multiplicative structures of rings, called $R$-semigroups. Is anything known about the first-order axiomatizability of the class of abelian groups which ...
- 1,445
2
votes
1
answer
604
views
Complete De Morgan algebra
Recall that an algebra $(A,\sim)$ is a De Morgan algebra if $A$ is a bounded distributive lattice and $\sim$ is a unary operation which satisfies:
${\sim} (x\vee y)={\sim} x\wedge {\sim} y$ and ${\sim\...
- 21
8
votes
2
answers
897
views
Is there something like a Heyting Ring?
I would like to know whether a Heyting algebra gives rise to ring in a similar way that a Boolean algebra gives rise to a Boolean ring. In a Boolean algebra $(B,\lor,\land,\lnot,0,1)$ I can define ...
Countably Infinite
15
votes
1
answer
1k
views
Are wild problems related to undecidable ones?
In representation theory, there is a well-known notion of a wild classification problem (such problems have been discussed often on this forum, for example, here). In logic, there is a notion of an ...
- 5,654
9
votes
2
answers
1k
views
Non-Standard Prime
Hello,
My question is about the non-standard models of the integers. If we add to the Peano's axioms $P$ of arithmetic the following axioms for a fixed constant $c$:
$c \neq 0$, $c \neq 1$, $c \neq 1+...
- 663
76
votes
9
answers
6k
views
Can we unify addition and multiplication into one binary operation? To what extent can we find universal binary operations?
The question is the extent to which we can unify addition
and multiplication, realizing them as terms in a single
underlying binary operation. I have a number of questions.
Is there a binary ...
- 236k
7
votes
3
answers
903
views
Construction of a maximal ideal
Hello,
Let R denote the ring of continuous functions defined on the real line, let I in R be the ideal consisting of functions with compact support. Obviously, I is not maximal, and by Zorn's Lemma ...
- 71
5
votes
1
answer
1k
views
Can the Knaster-Tarski theorem be proved using the Schroeder-Bernstein theorem?
The reverse can be done easily and the proof is well known I am wondering if the exact same argument can be used to prove reverse as well.
- 51
18
votes
1
answer
1k
views
Is Dependent Choice equivalent to the statement that every PID is factorial?
In this question, it was asked if AC is needed in the proof of the well-known fact that every principal ideal domain is factorial. As KConrad and Joel David Hamkins have pointed out, only DC, the ...
- 63.1k
7
votes
3
answers
915
views
Decidability of matrix algebra
Take multi-sorted first-order logic with equality, complex scalars, 1xn vectors, nx1 vectors, nxn matrices, addition and multiplication for each pair of sorts they make sense for, and hermitian ...
user5810
114
votes
2
answers
12k
views
How would you solve this tantalizing Halmos problem?
$1-ab$ invertible $\implies$ $1-ba$ invertible has a slick power series "proof" as below, where Halmos asks for an explanation of why this tantalizing derivation succeeds. Do you know one?
Geometric ...
- 4,736