# Tagged Questions

The study of algebraic structures and properties applying to large classes of such structures. For example, ideas from group theory and ring theory are extended and considered for structures with other signatures (systems of basic or fundamental operations).

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### In (relatively) simple words: What is an inverse limit?

I am a set theorist in my orientation, and while I did take a few courses that brushed upon categorical and algebraic constructions, one has always eluded me. The inverse limit. I tried to ask one of ...
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### Simple explanation of a monad

I have been learning some functional programming recently and I so I have come across monads. I understand what they are in programming terms, but I would like to understand what they are ...
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### A structural proof that $ax=xa$ forms a monoid

During the discussion on this problem I found the following simple observation: If $M$ is a monoid and $a \in M$ then $\{x: ax = xa\}$ is a submonoid. This is trivial to prove by checking ...
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### Why are particular combinations of algebraic properties “richer” than others?

Pedagogically, when students are exposed to algebraic structures it seems standard for the major emphasis, if not all the emphasis, to be on groups, rings, R-modules, and categories. These are rich ...
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### Category Theory vs. Universal Algebra - Any References?

After seeing the answer to the question Category theory, a branch of abstract algebra, I would like to ask Are there literature discussing the difference/indifference/comparison between category ...
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### Embeddings $A → B → A$, but $A \not\cong B$?

Are there any nice examples of structures (groups, modules, rings, fields) $A$ and $B$ such that there are embeddings $A → B → A$ while $A \not\cong B$? I would especially like to see an example for ...
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### Is there a concept of a “free Hilbert space on a set”?

I am looking for a "good" definition of a Hilbert space with a distinct orthonormal basis (in the Hilbert space sense) such that each basis element corresponds to an element of a given set $X$. Before ...
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### Difference between abstract algebra and universal algebra

Wikipedia give this answer "Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic ...
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### Different ways of constructing the free group over a set.

This could be too broad if we're not careful. I'm sorry if it ends up that way. Let's put together a list of different constructions of the free group $F_X$ over a given set $X$. It seems to be ...
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### An exercise in infinite combinatorics from Burris and Sankappanavar

Exercise 6.7 in chapter IV of Burris and Sankappanavar's A Course in Universal Algebra starts as follows: Show that for $I$ countably infinite there is a subset $S$ of the set of functions from $I$...
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### What equational properties of a group only need to be checked on a generating set?

Let $G$ be a group and $S\subset G$ a generating set. Let $P$ (short for $P(x_1,\dots,x_n) = 1$) be an equational property that may or may not be satisfied by all $n$-tuples of elements of $G$. My ...
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### In a slice category C/A of a category C over a given object A, What is the role of the identity morphism of A in C with respect to C/A

In a slice category $C/A$ of a category $C$ over a given object $A$, what is the role of the $C$ identity morphism, $A\to A$ ($1_A$), in $C/A$, particularly with respect to composition? I ...
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### What is the importance of “variety of algebras” in Universal Algebra?

Given an algebraic category, Birkhoff's Variety Theorem gives a categorical characterization of the full subcategories whose object-class forms a variety (i.e. can be defined by equations in the sense ...
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### Distributor? Distributive analog of commutator and associator?

Motivation: "the commutator gives an indication of the extent to which a certain binary operation fails to be commutative" (http://en.wikipedia.org/wiki/Commutator). For example (courtesy of wikipedia)...
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### This is just the Eilenberg-Moore category, right?

Let $F$ denote a monad on $\mathbf{Set}$. Write $\mathbf{Set}_F$ for the corresponding Kleisli category and $\mathbf{Set}^F$ for the Eilenberg-Moore category. On the train home today, it occured to me ...
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### $\mathbf{N}_5$ as a congruence lattice

A finite lattice is said to be representable if there exists a finite algebra whose congruence lattice is isomorphic to that lattice. As I was reading a paper, I came across the line: "The reader can ...
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### Software for some universal algebra issues

I am looking for some mathematical software that can help me with a very common task in the realm of universal algebra (as far as I know programs like prover9/mace4 and uacalc do not help with this ...
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### Modus Ponens: implication versus entailment

Would it be inconsistent to write Modus Ponens using only implication, not entailment? $(p \wedge (p \to q)) \to q$ The way I understand is that implication ($\to$) is an operator that yields a new ...
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### If finitely many algebraic identities do not imply some identity, is there always a finite counterexample?

This question just popped up while experimenting with Prover9 and Mace4. Say we have a finite signature and some finite set of identities $E_i$ in the sense of universal algebra, like the axioms for ...
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### Obtaining a binary operation on $X \rightarrow Y$ from a binary operation on $Y$. What, if anything, to make of this observation?

Let $X$ and $Y$ denote sets. Then if $+$ is a binary operation on $Y$, then we can obtain a new binary operation $+'$ on $Y^X$ in a canonical way as follows. $$(f+' g)(x) = f(x)+g(x)$$ Question. The ...
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### Are pseudoheaps and heaps the same thing?

An exercise in a category textbook asked me to show that the category of pointed heaps and the category of groups are isomorphic. But my proof somehow didn't use the most unintuitive of the defining ...
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### Do torsionfree abelian groups form a (possibly many-sorted) algebraic category?

By "(possibly many-sorted) algebraic category", I mean "category of models of a (possibly many sorted) Lawvere theory. I have arguments for both affirmative and negative answers, and I can't find a ...
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### Functorial approach to Ideals and Quotients, Multiplicative Sets and Localizations

I have been playing with substructures of commutative rings today and noticed that there is a strong analogy between the formation of quotients and kernels with the formation of localizations with ...
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### What is a simple axiomatisation of groups using division?

I recall from an old exercise I did as an undergrad that groups can be axiomatised using division rather than multiplication: A group is a non-empty set equipped with a binary division operator / ...
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### How does this definition capture the intuitive notion of an algebra?

On page 15 of this document, the author writes: Definition 1.1.1. Let $\mathcal{E}$ be any category. Given an endofunctor $\Gamma : \mathcal{E} \rightarrow \mathcal{E}$, a $\Gamma$-algebra ...
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### Constants in a signature

This is my first post so I hope it works! Taking the axioms for a group as an example, the literature defines a group in (at least) two different ways: Method 1 A signature of $(G,\circ,\,^{-1})$ ...
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### Which sentences survive the passage from $X$ to the set of all functions $I \rightarrow X$?

Suppose $X$ is a mathematical structure with a single underlying set which we will also denote $X$, equipped with some functions and relations. Letting $I$ denote an arbitrary non-empty set, we see ...