All Questions
Tagged with ra.rings-and-algebras universal-algebra
69 questions
2
votes
1
answer
197
views
Varieties of rational languages and (pseudo-)varieties of finite monoids, question regarding closure property
Let $\mathcal Rat(A)$ denote the class of rational (or regular) languages over the alphabet $A$, a subset $\mathcal V(A) \subseteq \mathcal Rat(A)$ is called a variety of (rational) languages iff
...
- 798
9
votes
1
answer
712
views
Generalizing detropicalization
Given an identity in max,plus arithmetic, are there ways to turn it into an ordinary algebraic identity it other than by replacing addition by multiplication and replacing max by series-plus or by ...
- 19.7k
4
votes
1
answer
1k
views
Commutative associative rational binary operations
What are all the nondegenerate rational binary operations that are commutative and associative? (Examples: $(x,y) \mapsto x+y$, $xy+x+y$, $xy/(x+y)$.)
Feel free to re-tag if you can think of ...
- 19.7k
5
votes
2
answers
974
views
Shape of axioms in algebraic structures
When defining algebraic structures (like monoids, groups, etc...), are there some constraints on the shape of the axioms, for the structure to have good properties that we implicitly use in many ...
- 1,341
13
votes
3
answers
678
views
IBN for algebraic theories
Let us say that a finitary algebraic theory $\tau$ has IBN (invariant basis number) if the free functor $F : \mathsf{Set} \to \mathsf{Mod}(\tau)$ reflects the isomorphism relation: If $S,T$ are sets ...
- 63.1k
4
votes
4
answers
388
views
What is an ideal-supporting algebra?
I'm sorry if this question is too elementary, but I asked it at MathStackExchange and it received no responses.
On the Wikipedia page for congruence relation it mentions how for groups and rings, ...
- 357
7
votes
1
answer
485
views
Two questions about commutative theories
Let $\mathcal{T}$ be a commutative algebraic theory (for example sets, abelian groups, commutative monoids, but not groups etc.). References include the nlab and Borceux' Handbook of Categorical ...
- 63.1k
2
votes
1
answer
195
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Cyclic Distribution on the reals?
Do there exist binary operators *, **, and *** on the real numbers, such that ...
- 21
22
votes
3
answers
6k
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Why are ring actions much harder to find than group actions?
I admit freely that the following question is a bit of a fishing expedition inspired by this lovely "definition" of a module as found on Wikipedia:
A module is a ring action on an abelian group.
...
- 15.5k
11
votes
3
answers
939
views
What is the smallest variety of algebras containing all fields?
A field is a ring whose nonzero elements form a commutative group under multiplication. A field is also a commutative inverse semigroup with respect to multiplication. The unique multiplicative ...
- 2,464
5
votes
1
answer
298
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Algebras with supremum-founded subalgebra lattice
I am interested in algebras whose subalgebra lattice is supremum-founded. Let us call those algebras small.
A complete lattice $(L, \leq)$ is called supremum-founded, if for any two elements $x < ...
- 1,498
9
votes
3
answers
670
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Algebraic axiomatization for AB+BA^T operation on matrices
Let us consider a matrix algebra $\operatorname{Mat}_{n\times n}(K)$, where $K$ is a field, $\operatorname{char} K \neq 2$.
It is well-known that the axiomatization of commutator operation $[A,B]=AB-...
- 413
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
9
votes
3
answers
1k
views
Does "finitely presented" mean "always finitely presented", considered in general
I'm wondering about the question
"If we have a finitely presented __, is it necessarily finitely presented with respect to any finite generating set for it?"
I know this is true for groups and ...
- 2,585
5
votes
2
answers
1k
views
Is there a notion of congruence relation for essentially algebraic structures?
In universal algebra there is the notion of congruence relation: Consider a (1-sorted) algebraic structure, i.e. a set $A$ with a bunch of finitary operations $f_i$ satisfying equations.
A congruence ...
- 12.3k
15
votes
2
answers
1k
views
Free division rings?
Does it make sense to talk about, say, the free division ring on 2 generators? If so, does the free division ring on countably many generators embed into the free division ring on two generators?
- 255
3
votes
1
answer
224
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Equational definition of Residuated Lattices
The usual axiomatization of residuated lattices involve using ≤. I know I can expand away the use of ≤ using a definition such as (x ≤ y) := (x ∧ y = x), but I fear I will get a set of messy axioms.
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- 1,439
7
votes
1
answer
732
views
Does ⬦ generate all De Morgan algebras?
(Asked by Nathaniel Hellerstein on the Q&A board at JMM)
This question is about De Morgan algebras (see also Wikipedia), which are something like Boolean algebras, but with a different weaker ...
- 1,119
62
votes
5
answers
10k
views
Does "finitely presented" mean "always finitely presented"? (Answered: Yes!)
Precisely, if an R-module M has a finite presentation, and Rk → M is some unrelated surjection (k finite), is the kernel necessarily also finitely generated?
Basically I want to believe I can ...
- 11.2k