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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4
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1answer
669 views

How do finite door spaces work?

Recall that a door space is a topological space where every set is either open or closed (or both). A topological space is finite if it has finitely many points. I'm interested in learning about ...
4
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1answer
347 views

Reversible varieties

We say that a variety $V$ is reversible if for each $n>0$ and $n$-ary fundamental operation $f$, there is some $m\geq n$ and $r$ along with terms $T_{2},\dots,T_{r},S_{1},\dots,S_{m}$ such that $G,...
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1answer
308 views

How many sublattices are contained in the powerset lattice of a finite set?

How many sublattices does the powerset lattice $2^n$ contain for $n$ finite? (up to equality, not isomorphism) I thought for sure this would be easy to find on OEIS, but so far I am coming up empty. I ...
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0answers
85 views

Minimization of second-order unifiers

We know that first-order unification is decidable. More generally, if there exists a unifier for a first-order unification problem, then there exists a most general unifier. I'm interested in the ...
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300 views

The free complete lattice on three generators, beyond ZF

This was originally asked at MSE; although it is still under bounty it seems unlikely to be answered there. $\mathsf{ZF}$ proves that there is no free complete lattice on three generators since any ...
6
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1answer
426 views

First isomorphism theorem for sets?

Let $f\colon S\to T$ be any function. There is the obvious refinement of $f$, by replacing the codomain $T$ with the image. Thus, every function factors into a surjection followed by an injection (...
5
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3answers
289 views

Universal property of the cocomplete category of models of a limit sketch

Let $\mathscr{S}$ be a limit sketch in a small category $\mathcal{E}$, i.e. just a collection of cones in $\mathcal{E}$. Then its category $\mathbf{Mod}(\mathscr{S})$ of models (i.e. functors $\...
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129 views

Axiomatizability of image of functor

Let $\sigma$ and $\tau$ be first-order languages and let $S$ resp. $T$ be theories (sets of sentences) for $\sigma$ resp. for $\tau$ ($S$ and $T$ possibly empty). Let $\mathcal C$ resp. $\mathcal D$ ...
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3answers
278 views

Algebras with finite essential arity

We are talking about algebras in the universal algebraic sense, that is, a set that $A$ is equipped with a set $F$ of finitary operations on $A$. Definition: An algebra $(A,F)$ is said to have ...
3
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1answer
77 views

Do almost-point-transitive algebras generate almost-point-transitive varieties?

Say that an algebra $\mathfrak{A}$ (in the sense of universal algebra) is point-transitive iff for every $a,b\in\mathfrak{A}$ there is a $\pi\in Aut(\mathfrak{A})$ with $\pi(a)=b$. While genuinely ...
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2answers
939 views

Ternary associative multiplication

In this answer Brian M. Scott describes the following generalization of a binary associative multiplication to a ternary one: it is a function $$[\cdot,\cdot,\cdot] : G\times G \times G \to G$$ such ...
6
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1answer
522 views

Does the likelihood of these tables exist?

Probably it does, and may be a number near $e^{-3/2}$ for 2-deficient tables. First some background. Early on in my studies of universal algebra, I encountered a result of Vadim Murskii, with the ...
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281 views

What is going on in the field of algebraic logic these days?

I'm doing my masters in Mathematics and took a class in universal algebra and there I learned that for example: Boolean algebras have direct connection with classical logic, Heyting algebras with ...
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46 views

First isomorphism theorem for inverse semigroups together with v-prehomomorphisms?

In this old paper D. B. McAlister has introduced another class of morphisms for inverse semigroups, called v-prehomomorphisms. For such a morphism $\theta : S \to T,$ instead of preserving the ...
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Function and algebraic system

Let $f$ be a surjective function from $X$ to $\{1,2,3\}$. Let $* :X^2 \to X$, such that $$f(x)\neq f(y) \implies f(x) \neq f(xy)\neq f(y), $$ and $$f(x)=f(y)\implies f(x)=f(xy).$$ Let's call the ...
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1answer
208 views

Lawvere theory of Lawvere theories

There is a coloured operad $sOp$ such that $sOp$-algebras are single-coloured operads. This operad has a simple description in terms of generators and relations, say, as an operad $F(X)/R$. There is a ...
6
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120 views

Birkhoff's theorem with language expansion?

Let $\mathcal V$ be a variety (in the sense of universal algebra). Recall that Birkhoff's theorem characterizes when a class $\mathcal W \subseteq \mathcal V$ of $\mathcal V$-algebras forms a ...
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164 views

Were algebraic theories and abstract clones defined independently?

Algebraic theories (by which I mean the formalism based on bijective-on-objects functors) and abstract clones both capture universal algebraic structure, and are well-known to be equivalent. Algebraic ...
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1answer
1k views

Does sine interact equationally with addition alone?

$\DeclareMathOperator\Eq{Eq}\DeclareMathOperator\Th{Th}$Originally asked at MSE without success: For a structure $\mathcal{A}$ whose signature only contains function and constant symbols, let $\Eq(\...
39
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1answer
1k views

Identities of commutators

Let $G$ be a group and set $[x,y]:= x^{-1}y^{-1}xy$ as usual and consider it as a binary operation. Question: Is there a description of the identities that the operation $[.,.]$ satisfies for all ...
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989 views

Uses of Zorn's Lemma when the thing is actually unique

Are there uses of the sledgehammer Zorn's Lemma that are embedded in arguments similar to the more delicate one below, showing that the thing being constructed is actually unique? Many uses of Zorn's ...
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1answer
82 views

Algebras for general transfors

Algebras for endofunctors bridge the gap between functors acting on a category and structures defined in it. An algebra for an endofunctor $F$ is instantiated by some morphism $Fa \to a$, and more ...
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2answers
159 views

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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2answers
284 views

Are gyrogroups useful for anything else other than the Einstein velocity addition rule?

Gyrogroups were discovered by Ungar in modelling the Einstein velocity addition rule in relativity. Have they been shown to be useful elsewhere in mathematics (or mathematical physics)?
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1answer
84 views

Ordering preserved by an inverse frame homomorphism

Recall that a frame homomorphism $h:L\to M$ is called ($L$ and $M$ are frames): Dense if, for any $x ∈ L$, $h(x) = 0$ implies $x = 0$. Codense if, for any $x ∈ L$, $h(x) = 1$ implies $x = 1$. ...
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2answers
179 views

Example of idempotent left quasigroups which are right-distributive but not left-distributive

I am looking for examples of the following algebraic structure: a set (X,.) which satisfy the axioms (idempotent) x.x = x (left quasigroup) the equation a.x = b has a unique solution denoted by x = ...
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3answers
131 views

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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1answer
97 views

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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3answers
393 views

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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173 views

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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2answers
385 views

Comparing "axiomatized function spaces"

This was previously asked and bountied at math.stackexchange with no response. Let $C(\mathbb{R}^2,\mathbb{R})$ be the space of all continuous functions $\mathbb{R}^2\rightarrow \mathbb{R}$ with the ...
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0answers
109 views

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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2answers
855 views

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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83 views

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, ...
9
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2answers
382 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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5answers
8k views

Does "finitely presented" mean "always finitely presented"? (Answered: Yes!)

Precisely, if an R-module M has a finite presentation, and Rk → M is some unrelated surjection (k finite), is the kernel necessarily also finitely generated? Basically I want to believe I can ...
3
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1answer
202 views

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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167 views

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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1answer
218 views

Generalized cancelation properties ensuring a monoid embeds into a group

Context: an obvious necessary condition for a monoid to embed into a group (as submonoid) is to satisfy the left and right cancelation rules: $$xy=xz \quad\Longrightarrow y=z;$$ $$yx=zx \quad\...
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65 views

Equational identities of sedenions and beyond in the signature of multiplication

This is similar to my previous question about Cayley–Dickson algebras. However, now I am considering only multiplication. Consider a Cayley–Dickson algebra $(X,+,-,*,0,1)$, that is, an algebra ...
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182 views

Cohomology without comonad?

TL;DR. Many cohomologies can be unified using comonads. Question: which cohomologies cannot be? For each algebraic theory, there is an adjunction, and therefore a (co)monad (or called a (co)triple. A ...
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1answer
212 views

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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1answer
114 views

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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53 views

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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1answer
695 views

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 ...
3
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1answer
82 views

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 ...
5
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1answer
188 views

"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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215 views

Is there a finite equational basis for the join of the commutative and associative equations?

I asked this on math stack exchange, but I was told to post it on mathoverflow. Consider the lattice of equational theories of a single binary operation $*$. The meet of the theory axiomatized by the ...
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0answers
180 views

Invariant theory in universal algebra

Let $\mathcal{L}$ be a finite first-order language with no relation symbols, and $\mathcal{K}:=\mathcal{V}(\Theta)$ a variety in this language definited by a set of identities $\Theta$. My questions ...
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252 views

Polynomial objects in any concrete category

EDIT: The original question had a trivial answer: it's just a coproduct. New question below New Question: As shown below, in the category of commutative unital rings, the coproduct of a ring $R$ with $...

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