# Tagged Questions

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### What is the “opposite” of a forgetful functor?

Consider a category $C$ and a monoid $M$. Consider a functor $F:C\to M$. It maps the objects of $C$ into the only object of $M$. But I don't want it to map every morphism of $C$ into the identity on ...
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### Is there a name for the generalization of the concept “Abelian group” where the axiom $-x+x = 0$ is weakened to the following?

Is there a name for the generalization of the concept "Abelian group" where the axiom $−x+x=0$ is replaced by the following list? $−0=0$ $−(x+y)=−x+−y$ $−(−x)=x$ $x+(-x)+x = x$ In multiplicative ...
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### What's the name of the mathematical structure with is an abstraction of things like linear Independence?

This is a terrible question, I know. I can't remember the details for some reason, but I think (hope?) that anyone who's familiar with this object will immediately know what I'm talking about.
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### 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 ...
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### 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 ...
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### 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 ...
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### Meaning of 'Isomorphism (with respect to inclusion)'

This is the first time that I see this phrase. I'm reading Commutative Algebra by N.Bourbaki. I'll extract 2 propositions that use this phrase. The first one is on page 68 of the book. ...
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### 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 $\sigma$, ...
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### The Empty Relation?

In elementary set theory, a relation on sets $A,B$ is usually defined as a subset of $A\times B$. We know that there are $2^{|A\times B|}$ subsets of $|A\times B|$. One of these subsets is the empty ...
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### What is the correct usage of the terms ‘under’ and ‘over’ in abstract algebra?

For example, the Wikipedia article on rings states that a ring is: an abelian group under addition a monoid under multiplication multiplication distributes over addition
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### Name for this axiom $(X \bullet R) \sqcup (Y \bullet S) \equiv (X \sqcup Y) \bullet (R \sqcup S)$

I am trying to give a name to this axiom in a definition: $(X \bullet R) \sqcup (Y \bullet S) \equiv (X \sqcup Y) \bullet (R \sqcup S)$ (for all $X, Y, R, S$) where $\sqcup$ is the join of a ...
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### Definition of $p$-torsion module

Let $p$ be a prime. What is the definition of a $p$-torsion module ? I have googled but was not convinced.
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### Is there a name for magmas in which $y*(a*b) = (y*a)*(y*b)$?

Let $X$ and $Y$ denote magmas, suppose $f$ is a homomorphism $X \rightarrow Y$, and let $y \in Y$ satisfy the following condition. $$\forall a,b \in Y : y*(a*b) = (y*a)*(y*b)$$ Then $y * f$ is a ...
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### 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 ...
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### Terminology: How should we call $\mathbb{Z}[\sqrt{5}]$?

I'm wondering, what shall we call the ring $\mathbb{Z}[\sqrt{5}]$? I know that $\mathbb{Z} \left[ \frac{1+\sqrt{5}}{2} \right]$ is called a quadratic integer ring. But do we have something similar for ...
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### Is there a name for magmas with $[x+y]+[x'+y'] \equiv [x+x']+[y+y']$?

Is there a name for magmas (written additively) satsisfying the following identity? The square brackets have no particular signifance, but will hopefully promote readability in what follows. ...
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### Why are group theory and ring theory a part of abstract algebra?

I have followed the courses Algebra 1, which was about group theory and Algebra 2, which was about ring theory. I don't think I really understand why those subjects are part of abstract algebra. What ...
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### Terminology regarding elements of monoids

In what follows, the symbols $a,b$ and $n$ implicitly range over $\mathbb{N} = \{0,1,2,\cdots\}.$ Are there names for the following properties that an element $x$ in a monoid may or may not possess? ...
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### A module as an external direct product of the kernel and image of a function

If $f:A\rightarrow A$ is an R-module homomorphism such that $ff=f$, show that $$A=Ker\,\,f\oplus Im\,\, f$$ Here is a part of what I made as a proof. Let $a\in A$. $f(a)\in Im\,\,f$, ...
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### First Homomorphism Theorem and terminology

I am not finding clear terminology in my abstract algebra book to be clear at least and my questions are simple. Consider the construction of a quotient group G': G/K = G' ...
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### Definition: finite type vs finitely generated

The mathematical term "finite type" appears more and more in the modern articles nowadays. But it is still hard to be found in the standard textbooks. I learned the definition of it from Stacks ...
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### What is meant by 'runs through'?

I'm independently studying abstract algebra for fun (not my forte...) and I'm reading Herstein. He has a question in the chapter on rings: Let $p$ be an odd prime and ...
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### Why are they called orbits?

When we study actions in group theory, we consider sets of the form $$\text{Orb}_G(x)=\{gx\mid g\in G\}$$ that are called orbits. Although, the only reason I find convincing for that name is that in ...
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### Looking for an algebraic structure

I'm looking for the name of algebraic structures (in which the elements are partially ordered) with the following properties: Top element defined, bottom optional; Join defined for all elements, ...
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### What is an indeterminate in a polynomial ring?

I am currently studying polynomial ring. I have a basic doubt. What is $x$ in a polynomial? A polynomial is an expression $\sum_{k = 0}^n a_k x^k$, where $n,k \in \mathbb{N}$ and $a_k \in R$ where ...
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### Origin of the terminology “connected algebra”

I was wondering what is the origin of the word "connected" for a connected algebra ? To be more precise, why is a graded $R$-algebra $A_{\ast}$ with an augmentation $A_{\ast} \to R$ that restricts to ...
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### Is the kernel of any ring homomorphism a subring, according to this definition?

This is an exercise taken verbatim from Birkhoff and MacLane, A Survey of Modern Algebra: Show that if $\phi: R \rightarrow R'$ is any homomorphism of rings, then the set $K$ of those elements in ...
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### Ideal:Kernel :: Filter:?

I understand that the notion of a filter is in some sense dual to the notion of an ideal, at least in the context of Boolean algebras1. Let $f:{\mathbf A} \to {\mathbf B}$ be a Boolean algebra ...
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### Is there a name for a function whose square is an involution?

An involution is a function $f:X\to X$ such that $f\circ f=\text{id}$. Is there a name for a function $g:X\to X$ such that $f\equiv g\circ g$ is an involution? An example is multiplication by $\pm i$ ...
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### Why the words “inner” and “outer” to designate products?

Does anyone know what's the rationale for using the adjectives inner and outer for certain algebraic products? Also, I've seen the term exterior algebra. Does the exterior here have anything to do ...
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### Is there a difference between abstract vector spaces and vector spaces?

I am following my Oxford syllabus and my next step is abstract vector spaces, in my linear algebra book I've found vector spaces. I've searched a little and made a superficial comparison between ...
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### Looking for standard and consistent notation/terminology for the finite sequences/heaps on a set

Question 1. Does anyone know of standard and consistent notation for the following? The set of non-empty finite sequences on a set. As above, but including the empty sequence. The set of finite ...
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### Help to understand the ring of polynomials terminology in $n$ indeterminates

In the Hungerford's book, page 150, the author defines a ring of polynomials in "n" indeterminates in the following manner: After the author defines the operations in this ring with a theorem: ...
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### Why the terms “unit” and “irreducible”?

I'm trying to understand why in a ring we choose the names unit to an invertible element and irreducible element in this definition Maybe historical reasons? For example, I suppose the second ...
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### Should every group be a monoid, or should no group be a monoid?

Question: What is more convenient/useful? Writing mathematics as if every group is a monoid, or as if these two classes are disjoint? Additional discussion. Define a monoid as follows. Defn 1. A ...
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### Confusing with the concept of normalizer $N_G(H)$

I'm Confusing with the concept of normalizer $N_G(H)$. It's a stupid question, sorry I'm new in this subject. Following the Hungerford's concept: If $H$ acts by conjugation on the set $S$ of all ...
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### $(g\circ f)^{-1}=f^{-1}\circ g^{-1}$

When $f$ and $g$ are invertible functions we have $$(g\circ f)^{-1}=f^{-1}\circ g^{-1}$$ I think we have this in every Group $(A,*,u,')$ $(a*b)=c$ $(a*b)*b'=a=c*b'$ $c'*a=c'*(c*b')=b'$ ...
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### Name of $a*b=c$ and $b*a=-c$

$A_+=(A,+,0,-)$ is a noncommutative group where inverse elements are $-a$ $A_*=(A,*)$ is not associative and is not commutative $\mathbf A=(A,+,*)$ is a structure where 1) if $a*b=c$ then $b*a=-c$ ...
In a paper of Kollar and Szabo there is a lemma (Lemma $1$) in which the following terminology is used: "Every representation $H\rightarrow GL(n,K)$ has an $H$-eigenvector" ($H$ is an algebraic ...