1
vote
0answers
21 views

Question about definition of some classes of bimodules.

Suppose that we have a ring $R$ and a $R$-$R$ bimodule $M$ such that: For every $r\in R$ and $m\in M$ there exists $r'\in R$ such that $m\cdot r=r'\bullet m.$ Examples of this bimodules can be seen ...
3
votes
0answers
41 views

The automorphism group of the real line with standard topology

How much is known about the automorphism group of the real line with the standard topology? Can it be described by a set of generators? I've been told that $\mathbb R$ has many weird and unexpected ...
1
vote
1answer
27 views

What are the roots of unity quadratic integers?

The article of Roots of unity in wikipedia implies that the following roots of unity are quadratic numbers: $$ \{\pm 1\}, \{\pm 1,\pm i\}, \{\pm 1,\pm \zeta,\pm \zeta^2\}. $$ where $\zeta=\exp(2\pi ...
1
vote
1answer
37 views

Primitive elements for $K=\Bbb{Q}(\sqrt{2},\sqrt{3})$

The key lemma for proving the primitive Element Theorem (for finite extension of a field $F$ with characteristic $0$) in Artin's Algebra (2nd edition) is the following: Suppose $char F=0$ and ...
2
votes
1answer
50 views

What sort of algebra is this?

Let us say that I have a set of symbols, $S$. The symbols can be operated on by a set of $n$-ary operators, $O$. Importantly, some of these operators are in the set of symbols, i.e. $S \cap O \neq ...
1
vote
1answer
25 views

Arrow's Impossibility Theorem Using Boolean Algebra

I am currently working on a research project which involves using Boolean matrices for the proof of Arrow's Impossibility Theorem and various other lemmas and results related to quasi ordered sets. In ...
11
votes
2answers
204 views

Putting down axioms for some symbols. Playing with their consequences qualitatively and symbolically. Building theories. The book?

I am interested in the design and building of theories. By building theories, I mean putting down axioms of various kinds, over various fields, exploring their perhaps interesting, or probably boring, ...
4
votes
0answers
137 views

Conjugation Quandles and… “Quandle-Groups”? From quandles to Groups.

A quandle $(Q,*,/ )$ is a idempotent right-distributive and right invertible structure. 1) $a*a=a$ 2) $(a*b)*c=(a*c)*(b*c)$ 3) $(a*b) /b=(a/b)*b=a$ If we have a group $(G, \cdot, ...
1
vote
0answers
40 views

coalgebra/algebra of the identity endfunctor

Let $\mathbb{C}$ be a small/locally small category and let $T:\mathbb{C} \to \mathbb{C}$ be an endofunctor. One can then have $T$-algebras and $T$-coalgebras in the usual way: for $X,Y \in ...
0
votes
0answers
29 views

Graph algebra papers

The following graph multiplication appears to be quite natural: Let $g_1=(V_1,E_1)$ and $g_2=(V_2,E_2)$ be two graphs ($V_i$ are sets of vertexes and $E_i$, sets of edges). Intuitively, the product I ...
6
votes
3answers
174 views

Categorical introduction to Algebra and Topology

I am self-studying Mathematics in my free time. At the moment I am reading books on Algebra and on Category theory. More exactly, I started working through the book $\textit{Algebra}$ by Serge Lang. I ...
0
votes
0answers
64 views

Relational algebraic structures

Recently I came across the notion of relational $\beta$-algebra, defined as a set $S$ and a binary relation $\xi:\beta S-S$, where $\beta S$ denotes the set of ultrafilters on $S$ (and $\beta$ is the ...
1
vote
0answers
27 views

integral basis for an arbitrary cubic Galois field

I wonder where I can find some information (possible a book) about finding an integral basis for cubic Galois fields? I know that for pure cubic fields there exists a simple criterion according to ...
0
votes
0answers
17 views

reference that any two elements of $S_n$ are conjugate iff they have the same cycle structure [duplicate]

How can I show/find a reference that any two elements of $S_n$ are conjugate iff they have the same cycle structure.
2
votes
0answers
48 views

The composition of functors and algebraic structures

An algebraic structure can be viewed as a finite-product-preserving functor $X : \mathbf{L} \rightarrow \mathbf{C},$ where $\mathbf{L}$ is a Lawvere theory and $\mathbf{C}$ is a category. Thus given ...
4
votes
1answer
43 views

Reference Request: Characters of Finite General Linear Groups

I've been looking at J.A. Green's article The Characters of Finite General Linear Groups and it seems that Green in this article comes up with a way of calculating all irreducible characters of a ...
1
vote
0answers
31 views

Book for partitions and equivalence classes

I couldn't find a similar question on the site; if it exists I will delete this Our teacher is covering a lot of theory and theorems about equivalence classes, partitions and basics of those. Now he ...
2
votes
0answers
76 views

Calculating sylow subgroups of some concrete groups

Question is to : exhibit all sylow $3$ - subgroups of $S_4$ What i have done so far is : Number of elements in symmetric group is $|S_4|=4.3.2=2^3.3$ number of elements of order $3$ in $S_4$ ...
5
votes
1answer
68 views

The infinite Direct Sum in the category Ring

If you don't have strong personal feelings about it already, most of you have at least witnessed the opposing factions on how we should define a ring and, by extension, how we should define a ring ...
7
votes
0answers
202 views

Modern research into Grassman's “theory of forms”?

I quote from Petsche's Hermann Graßmann: Biography (emphasis mine): The mathematical part of the book begins with the conception of the “General Theory of Forms”. Starting with a perspective on ...
2
votes
0answers
42 views

Connection between class number and the theory of Ideals/Quadratic Fields

I've been studying the classic results in integer binary quadratic forms, mainly the equivalence and reduction of quadratic forms and the class number $H(d)$ (the definition I got for $H(d)$ is the ...
5
votes
0answers
38 views

Is there a name for the algebra of substructures?

Let $X$ denote an entropic algebra (see here), which just means that all the operations of $X$ are homomorphisms $X^n \rightarrow X.$ Abelian groups are the classic example. Then for any operation of ...
3
votes
1answer
94 views

Counterexample for $A[[x, y]] = A[[x]][[y]]$

Maybe this is an idiot question, but I've heard that $A[[x, y]] = A[[x]][[y]]$ does not hold for $A$ an arbitrary commutative ring with identity, so I would like to know a counterexample, since the ...
4
votes
2answers
98 views

Have action/predicate systems (or similar) been considered in the literature?

Question. Has the following concept, or anything similar, been considered in the literature? Definition. An action/predicate system consists of sets $A$ (the actions) and $X$ (the predicates) such ...
1
vote
0answers
36 views

Resources for self-learning “relational” abstract algebra? [please see body of post for details]

I have been studying Grassman and Clifford algebras a bit, and it is fascinating to see how, for example, the rules defining the inner product operator are enough to the capture something of the ...
2
votes
1answer
55 views

Orbits that 'coalesce'

Let $R$ be a commutative ring, $G$ a group scheme over $\mathrm{Spec}\;R$, and $X$ a scheme over $\mathrm{Spec}\;R$ on which $G$ acts $R$-morphically via $G\times X\to X$. Suppose $S$ is another ...
4
votes
1answer
73 views

Weakened associativity axiom: $(x * y) * z \leq x*(y*z).$

Call a partially ordered magma $(X,*)$ sub-associative iff it satisfies the following axiom. $$(x*y)*z \leq x*(y*z).$$ Basically, this is saying that we may shuffle brackets right to get a larger ...
7
votes
3answers
314 views

Theorems implied by Yoneda's lemma?

Ok, so I was reading the Wikipedia article on Yonedas lemma. And I've heard before that when you prove things in category theory you automatically get a lot of results by proving it in abstract ...
1
vote
1answer
50 views

Models for Lie algebra E8 and octonions

I've heard that one can construct the exceptional Lie algebra $E_8$ as the Lie algebra of the group of isometries of projective plane over octonions, or something of this form. Unfortunately, I do not ...
1
vote
1answer
79 views

What are the conjugacy classes in $\mathrm{Aut}(G)$?

Let $G$ be an arbitrary group, and let $\mathrm{Aut}(G)$ be the group of automorphisms of $G$ (with composition of morphisms as multiplication). I'd like to learn more about the problem of ...
3
votes
0answers
45 views

Endomorphism rings of MCM Modules

Let $k$ be a field (algebraically closed of characteristic not equal to two, if you like) and let $R = k[[t^2, t^{2n+1}]]$. It is well known $R$ has finite type and the MCM (maximal Cohen-Macaulay) ...
2
votes
0answers
66 views

Do subhomomorphisms / subfunctors have a standard name, and where can I learn more?

Sorry for all the mistakes in the original! I think they're mostly fixed now. Thank you for your patience. Part 1. If $A$ and $B$ are models in $\mathrm{Pos}$ of an algebraic signature $\sigma$, ...
1
vote
1answer
122 views

How should I understand $R[x]/(f)$ for a ring $R$?

The following is a proposition in Artin's Algebra: Proposition 11.5.5 Let $R$ be a ring, and let $f(x)$ be a monic polynomial of positive degree $n$ with coeeficients in $R$. Let $R[\alpha]$ ...
2
votes
1answer
51 views

Algebraic signatures as quivers; is there somewhere I can learn more about these definitions?

In my opinion, a cool definition of "algebraic signature" is as follows: An algebraic signature on the sort symbols $\mathcal{X} = \{X_0,...,X_{n-1}\}$ is precisely a quiver whose underlying set ...
1
vote
1answer
35 views

Do we need really need associativity in Wedderburn's little theorem?

Wedderburn's little theorem states that a finite division ring $R$ is necessarily commutative. At first sight this sounds plausible due to the following very informal reasoning: the division property ...
2
votes
1answer
29 views

Embedding distance constraints in the plane

Let me first state my question somewhat vaguely. I am interested for which sets of "distance constraints" between $n$ points can be realised in the Euclidean plane. More precisely suppose you're ...
0
votes
0answers
50 views

When is a vector space (over field $K$) also a ring (with subring $K$)?

(Apologies in advance for the very naive question. I'm just learning about all this. Also, for the sake of expedience, below I use the word "ring" when it would more correct for me to use ...
6
votes
0answers
140 views

Point set topology from an algebraic perspective?

I got this idea of viewing a topology as an operation on a ring of sets. Let $\mathcal R = (\mathcal P(X), \cap, \triangle)$ be a ring of sets. ($\triangle$ is the symmetric difference operation and ...
3
votes
1answer
193 views

Ideals in ring of continuous functions $\mathcal{C}[0,1]$ … NBHM- Algebra

I would like to compile all questions I have encountered with Ideals in the ring $\mathcal{C}[0,1]$ of all continuous real valued functions and ask if there are any gaps. Question is to see if : ...
6
votes
0answers
47 views

What can we learn about a magma by studying these monoids?

Given a magma $(X,*)$, we get three monoids in the following way. First, define a pair of functions $L,R : X \rightarrow (X \rightarrow X).$ $$(Lx)(y) = x*y,\quad (Rx)(y) = y*x$$ Then each of the ...
8
votes
1answer
190 views

Advice: Algebra and category theory for geometry?

I'm interested in learning a bit of geometry. To start I'm (slowly) working my way towards differential geometry via Lee's Introduction to Smooth Manifolds. But, later on, I'd also like to study some ...
4
votes
0answers
47 views

List of Primes in UFD

Are there websites/databases containing lists ordered by norm of prime/irreducible elements in domains like $\mathbb{Z}[\sqrt{-2}]$ and $\mathbb{Z}\left[\frac{-1+\sqrt{-3}}{2}\right]$ for easy ...
2
votes
3answers
54 views

Examples of loops which have two-sided inverses.

Are there any neat examples of non-associative loops such that for each element a in the loop there exists $a^{-1}$ so that $a*a^{-1}=1=a^{-1}*a$. Even cooler would be a commutative loop. Also: are ...
1
vote
0answers
22 views

Two ordered rings are isomorphic iff their positive semirings are isomorphic

I am looking for (a reference to) a proof that Two ordered rings are isomorphic iff their positive ordered semirings are isomorphic. The positive semiring of an ordered ring $R$ is here the ...
2
votes
0answers
44 views

Taking the (pseudo)inverse of a monoid operation.

Let $M$ be a monoid with binary operation $f : M \times M \to M$. I'm interested in functions $g : M \to M\times M$ that obey the property: $$ f(g(m)) = m $$ I want to understand what all of the ...
0
votes
1answer
83 views

Power series ring over a ring of integers

Let $K/\mathbb {Q}_p$ be a finite extension, $\mathcal{O} := \mathcal{O}_K$ the ring of integers of $K,$ $\frak p$ the maximal ideal of $\mathcal{O}$, and $\pi$ a uniformizer, i.e., $\frak{p} = ...
2
votes
0answers
52 views

What is the background needed for Von Neumann Algebra?

I've found a book on Von Neumann algebras and I thought it could be interesting, but I have no idea on what is needed for studying it and the book also doesn't mention it.
-1
votes
1answer
133 views

A Book for Linear Algebra [duplicate]

I want to start learning Linear Algebra, I have no background about this subject except high school mathematics that doesn't includes complex number and matrices. I found the following books: ...
1
vote
3answers
57 views

Examples of root, parabolic, and borel subgroups corresponding to roots

I'm interested in seeing a few examples of root, parabolic, and Borel subgroups given a specific reductive group $G$. Here is what I know. Let $G$ be a reductive algebraic group over an ...
2
votes
2answers
55 views

When the homsets of a category are structured.

If the objects of a category are algebraic structures in their own right, this often places additional structure on the homsets. Is there somewhere I can learn more about this general idea? Example ...