Mathematics Stack Exchange
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Questions tagged [universal-algebra]

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).

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 the guys in my office, and despite a very shady…
Asaf Karagila
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Can Erdős-Turán $\frac{5}{8}$ theorem be generalised that way?

Suppose for an arbitrary group word $w$ ower the alphabet of $n$ symbols $\mathfrak{U_w}$ is a variety of all groups $G$, that satisfy an identity $\forall a_1, … , a_n \in G$ $w(a_1, … , a_n) = e$. Is it true, that for any group word $w$ there…
Chain Markov
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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 mathematically. Can anyone explain what a monad is using as…
Casebash
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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 theory and universal algebra?
Nobody
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Subgroups defined by negative formulas

I start with a simple problem that I was able to solve: Let $G$ be a group. Let $a\in G$. Assume that $H := \{g \in G : g^2 \neq a\}$ is a subgroup of $G$. The question: Can we define $H$ with a "positive" formula, not involving the symbol $\neq$?…
Ali Nesin
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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 structures with interesting properties, but in the…
Just Some Old Man
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Can a free complete lattice on three generators exist in $\mathsf{NFU}$?

Also asked at MO. It's a fun exercise to show in $\mathsf{ZF}$ that "the free complete lattice on $3$ generators" doesn't actually exist. The punchline, unsurprisingly, is size: a putative free complete lattice on $3$ generators would surject onto…
Noah Schweber
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Expressing associativity with only two variables

I'm wondering if it is possible to axiomatize associativity using a set of equations in only two variables. Suppose we have a signature consisting of one binary operation $\cdot$. Is it possible to find a set $\Sigma$ of equations containing only…
Tom
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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 identity and closure under multiplication. Is there a more…
sdcvvc
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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 one of those things a lot of people know about (and…
Shaun
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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 I explain my attempt for a definition of this,…
Corsin Pfister
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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 structures. For instance, rather than take particular…
James Fair
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$x^n = x$ implies commutativity, a universal algebraic proof?

I read in an answer on MO that Nathan Jacobson had given a universal algebraic proof that a ring satisfying the equation $x^n=x$ is commutative. The sketch given in the answer is very clear : wlog one may assume that $R$ is subdirectly irreducible,…
Maxime Ramzi
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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 modules $A$, $B$. Or is it even true that the…
k.stm
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What is the most general algebraic structure that a finite set has?

For an object $X$ in a category with finite products, define its endomorphism Lawvere theory to be the Lawvere theory generated by $X$: its $n$-ary operations are given by $\text{Hom}(X^n, X)$, and so accordingly it describes the most general…
Qiaochu Yuan
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