All Questions
Tagged with ra.rings-and-algebras universal-algebra
69 questions
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 ...
- 11.3k
22
votes
3
answers
6k
views
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
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 ...
- 20.5k
16
votes
1
answer
548
views
Does every commutative variety of algebras have a cogenerator?
By a commutative variety $\mathcal{V}$ I mean a classical variety of algebras for some $(\Sigma,E)$, such that each pair of operations in $\Sigma$ commutes.
Equivalently (i) every interpretation of ...
- 1,271
16
votes
0
answers
218
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 $f(xy)=f(x)f(y)$ ...
- 161
15
votes
3
answers
843
views
Is the Amitsur-Levitzki identity essentially unique?
Let us consider the matrix algebra. $Mat_n(\mathbb{C})$. The Amitsur-Levitzki identity states that for any matrices $X_1, X_2, ..., X_{2n} \in Mat_n(\mathbb{C})$ the sum $\Sigma_{\sigma \in S_{2n}} ...
- 1,842
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
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
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
11
votes
3
answers
942
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
10
votes
3
answers
1k
views
Natural associative law for a ternary "group"?
Suppose one were to define a group-like structure based on a set $G$
with a ternary (rather than binary) operator $g( a, b, c ) = \left< a, b, c \right>$.
One possible definition for the ...
- 151k
10
votes
2
answers
473
views
Varieties where every algebra is projective?
Is it possible to classify all varieties (in the sense of universal algebra) where every algebra is projective?
Several years ago I asked a similar question, with "free" in place of "...
- 63.9k
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, ...
- 2,023
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
9
votes
2
answers
1k
views
Ternary associative multiplication
In this answer Brian M. Scott describes the following generalization of a binary associative multiplication to a ternary one: it is a function $$[\cdot,\cdot,\cdot] : G\times G \times G \to G$$ such ...
- 4,780
9
votes
2
answers
661
views
Birkhoff's completeness theorem put into practice
Birkhoff's completeness theorem (see here, Theorem 14.19) states that an equation which is true in all models of an algebraic theory can be proven in equational logic.
Question. Does the proof of ...
- 63.1k
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
9
votes
3
answers
670
views
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
8
votes
2
answers
596
views
If a semigroup embeds into a group, then is it a subdirect product of groups?
The title has it all:
Q. If a semigroup $S$ embeds into a group, then is $S$ (isomorphic to) a subdirect product of groups?
If yes, then $S$ is a subdirect product of subdirectly irreducible groups,...
- 10.5k
8
votes
1
answer
485
views
Jordan algebra identities
A Jordan algebra is a vector space with a commutative bilinear operation $\circ$ obeying an identity that's often written as
$$ (x \circ y) \circ (x \circ x) = x \circ (y \circ (x \circ x)) . $$
...
- 22.3k
8
votes
1
answer
1k
views
First isomorphism theorem for sets?
Let $f\colon S\to T$ be any function. There is the obvious refinement of $f$, by replacing the codomain $T$ with the image. Thus, every function factors into a surjection followed by an injection (...
- 18.7k
7
votes
2
answers
578
views
Deriving consequences of identities
Suppose we are given a variety in the universal algebra sense.
For concreteness, suppose that we have two binary operations $+,\cdot$, three unary operations $-,\ast,'$, and two zeroary operations $0,...
- 18.7k
7
votes
2
answers
488
views
Is every cancellative semigroup a subdirect product of subdirectly irreducible cancellative semigroups?
By a classical result of Birkhoff (that is, Theorem 2 in [G. Birkhoff, Subdirect unions in universal algebra, Bull. AMS, 1944]) and the trivial fact that the class of semigroups is closed under the ...
- 10.5k
7
votes
1
answer
193
views
Free median algebras and maximal linked systems
$\DeclareMathOperator\MLS{MLS}$Recall that the median operation, on the power set $2^Y$ of subsets of a set $Y$, is the ternary law $m(A,B,C)$ mapping a triple of subsets to the set of elements ...
- 63.9k
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
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
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
7
votes
0
answers
378
views
Is there a theory of algebraic universal algebra?
An algebraic group is a group that is also an algebraic variety. There is also a theory of algebraic monoids. Is there are version of universal algebra that incorporates these examples, and other ...
- 6,870
6
votes
1
answer
299
views
Can a compact object be a nontrivial self-retract?
Let $\mathcal C$ be a locally finitely-presentable category, and let $X$ be a finitely-presentable object of $\mathcal C$.
Question: Can there exist a nontrivial idempotent on $X$ whose fixed points ...
- 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 ...
- 20.5k
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
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
5
votes
1
answer
172
views
An elementary proof of the equivalence of the Bol and Moufang identities
By a well-known result of Bol (1937) and Bruck (1946), for any loop the following two identities are equivalent:
B: $x(y(xz))=((xy)x)z$
M: $(xy)(zx)=(x(yz))x$.
A proof of the equivalence (B)$\...
- 41.9k
5
votes
1
answer
298
views
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
5
votes
1
answer
245
views
What are algebraic systems and algebraic closure as defined by Kenjiro Shoda? Which are his main results on them?
In On Utumi's ring of quotients, Canad. J. Math. 15(1963), 363-370, J. Lambek says:
As a matter of historical record, the minimal injective extension of a module is a special case of the "algebraic ...
- 2,992
4
votes
2
answers
366
views
Notion of prime congruences
We have the idea of a prime ideal in a commutative ring $R$ but in universal algebra, we generalize the notion of ideal to that of a congruence. I’ve thought over the question of what a prime ...
- 293
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
4
votes
1
answer
372
views
On the tree-ishness of magmas and the stringiness of groups
Let me start off by saying that I suspect the answer to my question might fall under the domain of universal algebra, which is why I'm giving it that tag. However, I know only the very basics of ...
- 309
4
votes
1
answer
434
views
Regarding a new algebraic structure
By "left semigroup-joined-semigroup" I mean an algebraic structures $(S,\cdot,*)$ such that both $\cdot,*$ are associative, and the following property holds (see this )
$$
x*(y\cdot z)=x*y*z\;\; ; \;...
- 995
4
votes
2
answers
507
views
Are algebraic structures uniquely identifed by their free objects?
It might be a naive question, as I am not a specialist in this field.
This is a follow-up to this question.
I want to study varieties of objects generalizing ordered monoids, in particular using an ...
- 1,341
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
4
votes
2
answers
339
views
Are gyrogroups useful for anything else other than the Einstein velocity addition rule?
Gyrogroups were discovered by Ungar in modelling the Einstein velocity addition rule in relativity. Have they been shown to be useful elsewhere in mathematics (or mathematical physics)?
- 2,369
4
votes
0
answers
319
views
Polynomial objects in any concrete category
EDIT: The original question had a trivial answer: it's just a coproduct. New question below
New Question: As shown below, in the category of commutative unital rings, the coproduct of a ring $R$ with $...
- 438
4
votes
0
answers
113
views
Closing Subsets Under Operations
My question is about closing sets under operations. First, I need a definition:
Definition: Let $A$ be a set and take a function $f : A^n \rightarrow A$ for $n \in \mathbb{N}_{\geq 0}$. For a set $S$,...
user30211
4
votes
0
answers
172
views
Poincaré-Birkhoff-Witt theorem for Leibniz algebras
Leibniz algebras can be seen as a non-skew-symmetric generalization of Lie algebras. I have already taken a look at some papers related to Leibniz algebras and extending main results of Lie algebras ...
- 367
3
votes
1
answer
328
views
What does it mean for the surreal numbers/partizan games to be "universally embedding"?
In "On numbers and games", Conway writes that the surreal Numbers form a universally embedding totally ordered Field. Later Jacob Lurie proved that (the equivalence classes of) the partizan ...
- 83
3
votes
2
answers
219
views
Polynomial identities of supercommutative-gradable algebras
All algebras below are associative, and not assumed unital, and, to fix ideas, over the complex numbers.
An algebra $A$ is supercommutative-gradable if it admits a grading $A=A_0\oplus A_1$ in $\...
- 63.9k
3
votes
1
answer
224
views
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.
...
- 1,439
3
votes
1
answer
360
views
Is the equational theory of commutative vN regular rings decidable?
I wanted to check whether $A(x,y):=\frac{xy}{x+y}$ is an associative operation in every commutative vN regular ring. Now $A(-1,A(1,1))=A(-1,\frac{1}{2})=1\neq 0 =A(0,1)=A(A(-1,1),1)$. On the other ...
- 2,464
3
votes
1
answer
191
views
Is every $n$-ary semigroup a subalgebra of an algebra derived from a binary semigroup?
Let $(A,f)$ be an $n$-ary semigroup ($n \ge 2$). Then there exists a ($2$-ary) semigroup $(\overline A,*)$ with an inclusion homomorphism $A \hookrightarrow \overline A$ such that that the restriction ...
- 190