# 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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### 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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### 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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### 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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### 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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### 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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### 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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### Universal constructions that factor through endomorphisms

If $\cal A$ is a variety of algebras (e.g., all groups) and $\cal B$ is a subvariety defined by some set of identities $X$ (e.g., abelian groups with $X = \{xy \simeq yx\}$), then there is a functor $...

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### 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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### Is the class of power-associative binars finitely axiomatizable?

A binar is simply a set $S$ equipped with a single binary operation $*$. A power-associative binar is a binar where the subalgebra generated by a single element is associative. Equivalently, they can ...

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### Jordan algebra identities

A Jordan algebra is a vector space with a commutative bilinear operation $\circ$ obeying an identity that's often written as
$$ (x \circ y) \circ (x \circ x) = x \circ (y \circ (x \circ x)) . $$
...

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### Morphism of sketches inducing a (co)reflective subcategory

Certain morphisms of theories gives rise to (co)reflective subcategories of their models. For the sake of simplicity, I'm going to fix my definition of sketch to be a category $\mathbb{C}$ with a ...

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### 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 ...

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### Polynomial identities of supercommutative-gradable algebras

All algebras below are associative, and not assumed unital, and, to fix ideas, over the complex numbers.
An algebra $A$ is supercommutative-gradable if it admits a grading $A=A_0\oplus A_1$ in $\...

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### Weirdos but algebraic

Weirdos generalize Abelian groups as well as an algebra of arithmetic mean of reals (or geometric mean of positive reals). But first, I'll define eccentrics. (I will not ask about eccentrics here ...

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### Topological connected eccentrics, not homeomorphic to commutative Lie groups

An eccentric is a universal algebra $\ (X\ \sigma\ \lambda\ \rho)\ $ such that operations
$\ \sigma\ \lambda\ \rho\,:\,X\times X\to X\ $ satisfy:
$\quad \forall_{x\ y\,\in X}\quad \lambda(\sigma(x\ y)...

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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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### Generalizations of Birkhoff's HSP Theorem

Let $\mathbf{C}$ be the class of algebraic structures of some fixed type satisfying some sentence $\phi$. Birkhoff's HSP theorem says that $\mathbf{C}$ is closed under homomorphisms, subalgebras and ...

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### What are algebraic systems and algebraic closure as defined by Kenjiro Shoda? Which are his main results on them?

In On Utumi's ring of quotients, Canad. J. Math. 15(1963), 363-370, J. Lambek says:
As a matter of historical record, the minimal injective extension of a module is a special case of the "algebraic ...

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### Universal anti-Horn classes?

Is there published work about universal anti-Horn classes?
Anti-Horn formulas are also sometimes known as dual Horn.
See also related question Is there any research of universal algebras axiomatized ...

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### The question about elementary equivalence of free products

Let $A,B,C,D$ be algebraic systems and $A$ and $B$ be elementary equivalent as well as $C$ and $D$. Are free products of $A,C$ and $B,D$ elementary equivalent if
$A,B,C,D$ are groups, or
$A,B,C,D$ ...

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### Shape of axioms in algebraic structures

When defining algebraic structures (like monoids, groups, etc...), are there some constraints on the shape of the axioms, for the structure to have good properties that we implicitly use in many ...

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### What is a module over a Boolean ring?

Recall that a (unital) Boolean ring is a (unital) commutative ring $A$ where every element is idempotent; it follows that $A$ is of characteristic 2. There is an equivalence of categories between ...

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### Algebraic/relational structures produced using evolutionary/machine learning algorithms?

Are there examples of algebraic structures which have been constructed using evolutionary algorithms and possibly machine learning algorithms? I am looking for algebraic structures like lattices ...

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### $fgf = f$, $gfg = g$, $fg$ not necessarily identity, what is this called?

A very simple question, I just totally forgot how it was called, and Google is not helping.
There's a pair of functions $f:X\to Y$, $g:Y\to X$.
$fgf = f$, $gfg = g$, but $fg$ and $gf$ don't need to ...

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### 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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### Short proof a monoid is a group iff every splitting is right homogeneous

In the paper "Schreier split epimorphisms between monoids" by Bourn, Nelson, Martins-Ferreira, Montoli and Sobral, Semigroup Forum
June 2014, the authors prove a characterization of groups among ...

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### Primitive recursive arithmetic via universal algebra

From the Wikipedia article on Primitive recursive arithmetic:
"Primitive recursive arithmetic, or PRA, is a quantifier-free formalization of the natural numbers. It was first proposed by Skolem[1] ...

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### Axiomatizing orientation in the complex plane

Lately I've begun to suspect that a certain ternary relation might play a role in $\bf{C}$ analogous to the role played by the binary relation $>$ in $\bf{R}$, namely, the relation that the ...

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### 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 ...

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### Motivation of filtered colimits

I am trying to move in categorical algebra beyond the basics. A Lawvere theory L is a small category with finite products. (I know that there also is a functor $(skeleton(FinSet))^{op}\to L$, which ...

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### If questions are formalized as ideals of a boolean algebra, what kind of algebra of questions appears from Stone representation theorem?

Affirmative propositions make up a Boolean algebra, and Boolean algebras became part of classical algebra for over one century ago - in this sense they are "simple". But I did not encounter in ...

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### Explicit lifting characterization of complete lattices among posets?

It's well-known that the complete lattices are characterized among all posets as the regular-injectives. That is, a poset $L$ is a complete lattice if and only if $L$ has the right lifting property ...

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### Does “finitely presented” mean “always finitely presented”, considered in general

I'm wondering about the question
"If we have a finitely presented __, is it necessarily finitely presented with respect to any finite generating set for it?"
I know this is true for groups and ...

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### What's the cokernel of a monoid homomorphism?

Let $f:A\to B$ be a monoid homomorphism. Where can I find an explicit description of the its cokernel? Are there any books on this topic?
By the cokernel of $f$, I mean the universal arrow which ...

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### For a new operation on a finite group of odd order giving a loop structure, when does this also gives a group

For finite groups $G$ of odd order, as $x \mapsto x^2$ is bijection (but no automorphism in general) then, we can define for each $g \in G$ the element $x^{1/2}$ by requiring $(x^{1/2})^2 = x$. Then ...

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### 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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### Why is the theory of small categories not algebraic?

In "Partial Horn logic and cartesian categories", E. Palmgren and S. J. Vickers state without proof that "The theory of categories is not algebraic." Is there a reference, or an elementary argument, ...

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### When will a monad satisfy Moggi’s “equalizer” property?

I’m interested in finding when ($V$)-monads will satisfy the universal property that $\eta$ equalizes $\eta_T, T\eta$. I’m particularly interested in the case for monads on presheaf categories or that ...

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### Varieties of non-distributive lattices with De Morgan negations and their ‘canonical’ members

After some revisions and checks, I have decided to repost the question.
Recall that the variety $\mathcal{DM}$ of (distributive) De Morgan lattices has the lattice $\mathbf{4}$ as its canonical ...

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### Does every commutative variety of algebras have a cogenerator?

By a commutative variety $\mathcal{V}$ I mean a classical variety of algebras for some $(\Sigma,E)$, such that each pair of operations in $\Sigma$ commutes.
Equivalently (i) every interpretation of ...

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### Formally undecidable problems on finitely presented quandles

In the literature, one sometimes sees the claim that finitely presented quandles (in particular, knot quandles) are "hard to deal with". Hence, a great deal of effort has gone into studying finite ...

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### Why are ring actions much harder to find than group actions?

I admit freely that the following question is a bit of a fishing expedition inspired by this lovely "definition" of a module as found on Wikipedia:
A module is a ring action on an abelian group.
...

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### Birkhoff's completeness theorem put into practice

Birkhoff's completeness theorem (see here, Theorem 14.19) states that an equation which is true in all models of an algebraic theory can be proven in equational logic.
Question. Does the proof of ...

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### Do non-associative objects have a natural notion of representation?

A magma is a set $M$ equipped with a binary operation $* : M \times M \to M$. In abstract algebra we typically begin by studying a special type of magma: groups. Groups satisfy certain additional ...

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### Braided lobsters

If $(X,m)$ is a median algebra, then for each $x\in X$, define an operation $\wedge_{x}$ by letting $y\wedge_{x}z=m(x,y,z)$. Then $(X,\wedge_{x})$ is a meet-semilattice with least element $x$. Define ...

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### Direct limit of lattice-ordered groups

In general, any abelian group can be expressed as a direct limit of its f.g. subgroups. For the case of $\ell$-group (lattice-ordered group) is that true or not? As an abelian group we do not have ...

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### Is finite verbal subgroup equivalent to finite index of marginal subgroup?

There is a well known fact:
If $G$ is a finitely generated group. Then $|G’| < \infty$ iff $[G:Z(G)]<\infty$.
Suppose $\mathfrak{U}$ is a group variety. Let’s denote the corresponding verbal ...