# 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).

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### Finitary endofunctors: “Support” of elements of $TM$ where $T:\mathrm{Set}\to\mathrm{Set}$ is finitary

Consider the word a.k.a. list monad $W:\mathrm{Set}\to\mathrm{Set}$ assigning to a given set $M$ the set of words over $M$. Now, given a set $M$, we can assign to every word $w\in WM$ a subset of $M$: ...
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### Finitely presentable grids and co-presentable topological spaces

In the 90s there has been some interest for the category $\mathsf{Top}^\circ,$ the dual of the category of spaces. Most of the relevant papers on the topic are co-authored by Pedicchio. Barr, ...
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### value (element of an algebra), constant, variable, ground and non-ground terms, free algebras : there is a need for clarification

I have been developing an algorithm to compute the congruence defined by a finite set of "generators" and a finite set of equations (in the sense of equational theories). The algorithm ...
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### Finitely presented algebra [closed]

I found in the book "Universal Algebra for Computer Scientists", by W. Wechler, the following statement : "In general, even for finite presentations, the word problem is unsolvable"...
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### Comparing the existing formulations of universal algebra and their levels of generality

I am a newcomer to universal algebra and I just read this (very good, IMO) book on the topic: Adámek, J., Rosický, J., & Vitale, E. M. (2010). Algebraic theories: a categorical introduction to ...
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### Birkhoff's HSP theorem in categories other than $\mathbf{Set}$

Fix a category $C$ with finite products and a set $L$ of function symbols (each equipped with an arity in $\mathbb N$). An $L$-algebra in $C$, $\mathbf A=(A,(f^\mathbf{A})_{f\in L})$, is given by some ...
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### Minimizing all aspects of the definition of Boolean algebra

There are many equivalent ways to describe Boolean algebras. There are a number of different ways to "minimize" the description. We can: Minimize the number of function symbols. Minimize ...
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### Why does abelianization preserve finite products, really?

The abelianization functor $(-)^{ab} : \mathrm{Grp} \to \mathrm{Ab}$ is left adjoint to the inclusion of abelian groups into groups. As such, it preserves all colimits, but it doesn't generally ...
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### Proving interpolation results through amalgamation

Notice: this is a cross-posting, I have asked essentially the same question on MSE (https://math.stackexchange.com/questions/4012960) but received no answers, and as this problem, although very basic, ...
356 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 "...
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### 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 ...
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### 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 ...
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### What is the name for Boolean algebra's version of $\models$ between sets of identities and identities?

On p62 in Schaum's Outline of Theory and Problems of Boolean Algebra and Switching Circuits by Elliott Mendelson (1970), Part (b) of the corollary says that if an identity is satisfied by some ...
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### Is equational logic in universal algebra a proof system not a logic system?

As far as I know a logic system defines its own semantics (e.g. $\models$), but not a proof calculus/system on its language. See p261 in Ebbinghaus et al's Mathematical Logic: In universal algebra, ...
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### Varieties where any subalgebra is a regular sub-object

Following [LPAC] Chap.3, p.132 let $S$ a set of sorts and $\Sigma$ a $S$-sorted signature. From the [LPAC] treatment we have the category $Alg(\Sigma)$ with a $(\mathit{regular.Epi,Mono})$-...
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### Are there axioms satisfied in commutative rings and distributive lattices but not satisfied in commutative semirings?

Consider the language of rigs (also called semirings): it has constants $0$ and $1$ and binary operations $+$ and $\times$. The theory of commutative rigs is generated by the usual axioms: $+$ is ...
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### Characterisation of presentations for varietal large equational theories

Let $T : \mathbf{Set}^\mathrm{op} \to \mathscr T$ be a large equational theory (i.e. a bijective-on-objects product-preserving functor). Following Linton in Some Aspects of Equational Categories, we ...
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### “Tietze-like transformations” for defining interesting bijections between algebraic structures

Consider the following two definitions of the natural numbers: The natural numbers are the algebraic structure $\mathbb{N}_1$ generated by one constant, $0$ and one unary function, $S$ (and no ...
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### Indecomposable weirdos (cnt.)

This post is a continuation of Weirdos but algebraic. Logically, the quoted post could follow the present one rather than precede it. Question Does there exist an indecomposable weirdo which is ...
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### Non-abelian variety of groups in which finite groups are abelian

Is there a non-abelian variety of groups $V$ such that any finite group from $V$ is abelian? This was posed in a paper by Hanna Neumann (1967), but I cannot find the solution.
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### 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, ...
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### 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 ...
Let’s define the set of outer-commutator group words $OC \subset F_\infty = F[x_0, x_1, …, x_n, …]$ using the following recurrence: $$\forall i \in \mathbb{N} \text{ } x_i \in OC$$ \forall u, v \...
Suppose $\mathfrak{U}$ is a group variety. Now let’s define $L(\mathfrak{U})$ as the class of all such groups $G$, such that $\forall g \in G$ $\langle \langle g \rangle \rangle \in \mathfrak{U}$ (...