All Questions
Tagged with ra.rings-and-algebras universal-algebra
69 questions
4
votes
1
answer
112
views
An elementary proof of the equivalence of the Bol and Moufang identities
By a well-known result of Bol (1937) and Bruck (1958), 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)$\...
- 41.8k
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 ...
CommunityBot
- 1
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 ...
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8
votes
2
answers
596
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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,...
- 38.6k
16
votes
0
answers
218
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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
1
vote
1
answer
215
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Reference about cancellation property for semigroups
Have the semigroups with the following cancellation property been studied?
Property: Let $S$ be a semigroup and $x,y\in S$ such that $xz=yz,$ for all $z\in S,$ then $x=y$.
- 468
2
votes
0
answers
81
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The involutive structure on a division ring
This question is motivated by foundations of geometry, namely, by studying scalars in affine spaces.
Let $F$ be a field (or better a division ring). It has the operations of addition and ...
- 41.8k
9
votes
2
answers
661
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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 ...
- 31
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 ...
- 255
0
votes
0
answers
105
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Definition of term functions, in universal algebra
According to the definitions in Sankappanavar's universal algebra :
Assume $p$ is a term, then $p(x_1,x_2,...,x_n)$ indicates that the variables occurring in $p$ are among $x_1,...,x_n$. But there is ...
- 63.9k
0
votes
0
answers
101
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Is a principal filter in a free Heyting algebra a projective Heyting algebra?
A Heyting algebra is a bounded distributive lattice $(L,\vee,\wedge,0,1)$ together with a binary operation $\rightarrow$ called implication or relative pseudocomplementation with the property that, ...
- 63.9k
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-...
- 33.8k
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
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 ...
- 8,688
2
votes
1
answer
111
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Algebras determined by their globals
If $A= (A, f_1, f_2, ...f_n)$ is an algebra, then its global (sometimes referred to as complex algebras) $\mathcal{U}(A)$ is defined on the power set $\wp(A)$ in the usual way.
It is known that $\...
- 63.9k
62
votes
5
answers
10k
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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 ...
- 63.9k
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 ...
- 12.9k
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 ...
- 12.9k
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 (...
- 14.6k
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 ...
- 14.6k
1
vote
0
answers
66
views
First isomorphism theorem for inverse semigroups together with v-prehomomorphisms?
In this old paper D. B. McAlister has introduced another class of morphisms for inverse semigroups, called v-prehomomorphisms. For such a morphism $\theta : S \to T,$ instead of preserving the ...
- 1,093
1
vote
1
answer
349
views
Lawvere theory of Lawvere theories
There is a coloured operad $sOp$ such that $sOp$-algebras are single-coloured operads. This operad has a simple description in terms of generators and relations, say, as an operad $F(X)/R$. There is a ...
- 103
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)?
- 188k
1
vote
2
answers
221
views
Example of idempotent left quasigroups which are right-distributive but not left-distributive
I am looking for examples of the following algebraic structure: a set (X,.) which satisfy the axioms
(idempotent) x.x = x
(left quasigroup) the equation a.x = b has a unique solution denoted by x = ...
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
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 ...
- 118k
2
votes
0
answers
286
views
Union star symbol in set theory
In the slides Provenance for Database Transformations, page 24, they provide a semiring for lineage, which include a $\cup^*$ symbol. However, I can not find any related materials about the meaning of ...
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
2
votes
1
answer
260
views
Universal constructions that factor through endomorphisms
If $\cal A$ is a variety of algebras (e.g., all groups) and $\cal B$ is a subvariety defined by some set of identities $X$ (e.g., abelian groups with $X = \{xy \simeq yx\}$), then there is a functor $...
- 879
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
$$ (x \circ y) \circ (x \circ x) = x \circ (y \circ (x \circ x)) . $$
...
- 32.5k
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 $\...
- 16.9k
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 ...
- 63.9k
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 ...
- 63.9k
9
votes
3
answers
1k
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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 ...
- 63.9k
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 ...
- 63.9k
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.
...
- 4,713
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 ...
- 63.1k
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
12
votes
5
answers
2k
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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 ...
- 63.9k
2
votes
0
answers
169
views
What is the difference between a monosemiring and a semigroup?
What is the difference between a monosemiring and a semigroup?
The following definitions are for clarity of my question.
A semigroup $S$ is a non empty set that satisfies closure and associativity ...
- 203
3
votes
0
answers
95
views
Lie structure over $R$-module
In Higgins' paper Baer invariants and the Birkhoff-Witt theorem (J. Algebra 11 (1969) 469–482, doi:10.1016/0021-8693(69)90086-6) the following definition is given:
A Lie structure over the $R$-module ...
- 427
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 < ...
- 14.6k
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
0
votes
1
answer
654
views
Book on algebraic structures
What is the most complete book on algebraic structures that deals with the complete taxonomy from magmas to Lie algebras and inner product spaces?
4
votes
0
answers
113
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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 $S$,...
- 4,713
15
votes
2
answers
1k
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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?
- 4,713
1
vote
0
answers
138
views
Is every monosemiring an idempotent semiring?
Is every monosemiring an idempotent semiring?
To make my question clear, let me give definitions as follows:
A semiring $(R, +, .)$ is said to be monosemiring if $x.y= x+y$ for all $x, y$ in $R$. And ...
- 203
3
votes
0
answers
79
views
Semigroups containing an ideal with a local identity
I'm looking for some classes of semigroups containing a (proper) ideal with a local identity (i.e., ideal submonoid). Can somebody give some examples or/and theorems for the followings cases:
(a) ...
- 4,713
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\;\; ; \;...
- 44.4k
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 ...
- 14.6k