# 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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### A magma with the property (xy)(zt) = x(yz)t

Let $(M,\cdot)$ be a magma. Does the property $(x\cdot y)\cdot(z\cdot t) = x\cdot(y\cdot z)\cdot t$ have a special name? Thanks.

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### Name for generalization of property: $f^n(x) \ne x$ for all $n > 0$

I am curious about how to specify with standard terminology that a certain function is non-periodic, in the following sense:
In the simple case of a unary operation $f: X \to X$, this property would ...

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### Can a Shelah semigroup be commutative?

A semigroup $S$ is called
$\bullet$ $n$-Shelah for a positive integer $n$ if $S=A^n$ for any subset $A\subset S$ of cardinality $|A|=|S|$;
$\bullet$ Shelah if $S$ is $n$-Shelah for some $n\in\...

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### Standard terminology for morphisms of binary relations

Dealing with relations in a set theoretic context, i.e. as just sets of ordered pairs what would one call a function $f:\text{fld}(R)\to\text{fld}(L)$ for any relations $R$ and $L$ in each of these ...

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### Is there a theory of algebraic universal algebra?

An algebraic group is a group that is also an algebraic variety. There is also a theory of algebraic monoids. Is there are version of universal algebra that incorporates these examples, and other ...

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### Book on algebraic structures

What is the most complete book on algebraic structures that deals with the complete taxonomy from magmas to Lie algebras and inner product spaces?

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### Dual space of polynomial one-form

Recently I read a paper "Quasi-particles models for the representations of Lie algebras and geometry of flag manifold". In section 2, author gives a fact without proof. Now I rephrase this fact as ...

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### Quandle homomorphism does not always induces group homomorphism on inner automorphism groups of quandles

Let $X$ and $Y$ be two quandles and $f: X \rightarrow Y$ be a quandle homomorphism. Then we can define a map $\bar f: Inn(X) \rightarrow Inn(Y)$ as $\bar f(S_a)=S_{f(a)}$, where $a \in X$. Then $\bar ...

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### Relation between monads, operads and algebraic theories (Again)

This question (as the title obviously suggests) is similar to, or a continuation of, this question that was asked years ago on MO by a different user.
The present question, though, is different from ...

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### Closing Subsets Under Operations

My question is about closing sets under operations. First, I need a definition:
Definition: Let $A$ be a set and take a function $f : A^n \rightarrow A$ for $n \in \mathbb{N}_{\geq 0}$. For a set $S$,...

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### Amalgamated free-product of semigroups (definition)

I am self-studying some concepts including the title one. I reached the definition of an amalgamated free-product ${S_1}{*_U}S_2$ where $[S_1, S_2; U, w_1,w_2]$ is an amalgam of semigroups. Let $S_1=\...

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### Presentation of amalgamated sum as a quotient of the direct sum

I am currently reading Arthur Ogus' "Lectures on Logarithmic Algebraic Geometry" (https://math.berkeley.edu/~ogus/preprints/log_book/logbook.pdf).
I'm trying to understand why the amalgamated sum of ...

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### Generalization of results from specific algebraic theories to Universal Algebra

I'm relatively new to universal algebra, but it seems that lots of theorems from specific algebraic theories (groups, rings) can be stated in the context of universal algebra, perhaps I'm wrong.
...

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### Finite lattice representation problem checking

[Grätzer and Schmidt 1963] proves that every algebraic lattice is isomorphic to the congruence lattice of a universal algebra. A finite lattice is algebraic. The finite lattice representation problem ...

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### Some questions about homogroups

Every semigroup containing an ideal subgroup is called a homogroup. Let $(S,\cdot)$ be homomgroup, hence it contains an ideal $I$ that is also a subgroup. It is easy to see that $I$ is the least ideal,...

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### Generalisation of a loop concept

Suppose that $(M, \circ)$ is a set $M$ over which there is defined a binary operation $\circ$ so that we have:
1) For every $(a,b) \in M \times M$ we have $a \circ b \in M$
2) For every $a \in M$ ...

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### What kinds of free algebras can rank-into-rank embeddings produce?

Let $n$ be a natural number. Then let $\mathcal{V}_{n}$ be the variety consisting of all algebras $(X,*,T)$ where $*,T$ satisfy the identities
$$x*(y*z)=(x*y)*(x*z)$$ and
$$x*T(x_{1},\dots,x_{n})=T(x*...

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### Can finite binary self-distributive algebras fit into small $n$-ary self-distributive algebras

A binary operation $*$ is said to be self-distributive if it satisfies the identity $x*(y*z)=(x*y)*(x*z)$. An $n+1$-ary operation $t$ is said to be self-distributive if it satisfies the identity
$$t(...

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### Is the variety of ternary self-distributive algebras generated by its finite members?

A ternary self-distributive algebra is an algebra $(X,t)$ that satisfies the identity
$$t(u,v,t(x,y,z))=t(t(u,v,x),t(u,v,y),t(u,v,z)).$$
Is the variety of ternary self-distributive algebras generated ...

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### Is the equational theory of the variety of ternary self-distributive algebras decidable?

A ternary self-distributive algebra is an algebra $(X,t)$ that satisfies the identity $$t(u,v,t(x,y,z))=t(t(u,v,x),t(u,v,y),t(u,v,z)).$$
Is the equational theory of the variety of ternary self-...

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### How slowly can it takes for the Fibonacci terms in a partially permutative self-distributive algebra to stabilize?

A self-distributive algebra is a structure $(X,*)$ where $*$ is a binary operation that satisfies the identity $x*(y*z)=(x*y)*(x*z)$. Let
$L,T:X^{2}\rightarrow X^{2}$ be the operations where $L(x,y)=(...

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### Are there non-finitely generated algebras of elementary embeddings when one includes compatible $n$-ary operations?

Let $\mathcal{E}_{\lambda}$ be the set of all elementary embeddings $j:V_{\lambda}\rightarrow V_{\lambda}$. If $j\in\mathcal{E}_{\lambda}$, then let $j^{+}(A)=\bigcup_{\alpha<\lambda}j(V_{\alpha}\...

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### What do the algebras of elementary embeddings look like under V=HOD?

Let $\mathcal{E}_{\lambda}$ be the set of all elementary embeddings $j:V_{\lambda}\rightarrow V_{\lambda}$. If $A\subseteq V_{\lambda}$, then let $j^{+}(A)=\bigcup_{\alpha<\lambda}j(A\cap V_{\alpha}...

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### Let $X$ be the class of all classical Laver tables. Is $HS(X)=S(X)$?

Let $A_{n}=(\{1,\ldots,2^{n}\},*_{n})$ be the algebra defined by
$x*_{n}1=x+1\mod 2^{n}$ and $x*_{n}(y*_{n}z)=(x*_{n}y)*_{n}(x*_{n}z)$ for all $x,y,z\in\{1,\ldots,2^{n}\}$. Suppose that $X$ is a ...

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### What is the name of this substructure/embedding?

I am interested in the following property, be it on an abstract or concrete category:
$A$ is a substructure of $B$ such that every automorphism of $A$ extends uniquely to an automorphism of $B$. Or ...

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### Universal identities on cubic surfaces or hypersurfaces

This question is inspired by this previous one. Generally speaking, I ask what algebraic identities are universally valid for the composition law on cubic surfaces (or hypersurfaces); since the law ...

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### Name for this algebraic structure?

I've found myself looking at a structure $\mathbb{M}$ whose important properties are:
$\mathbb{M}$ is a discretely ordered additive monoid.
$\mathbb{M}$ has a least element, and this least element is ...

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### Word problem for finitely presented bounded lattices

There is a solution to the word problem for finitely presented (non-bounded) lattices, as well as a solution to the word problem for free bounded lattices. I am assuming that there is a solution to ...

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### What is the difference between a monosemiring and a semigroup?

What is the difference between a monosemiring and a semigroup?
The following definitions are for clarity of my question.
A semigroup $S$ is a non empty set that satisfies closure and associativity ...

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### Is every monosemiring an idempotent semiring?

Is every monosemiring an idempotent semiring?
To make my question clear, let me give definitions as follows:
A semiring $(R, +, .)$ is said to be monosemiring if $x.y= x+y$ for all $x, y$ in $R$. And ...

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### What is known about ideal and divisibility lattices of GCD domains and their generalizations?

The divisibility relation "$a$ divides $b$", or concisely, $a \vert b$ defined over a commutative integral domain $R$ with identity induces a partial order on the multiplicative semigroup $R/R^{\times}...

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### How many classes of compatible linear orders exist on the classical Laver tables?

Let $A_{n}$ be the unique algebra $(\{1,…,2^{n}\},*_{n})$ such that
$x*_{n}1=x+1\mod 2^{n}$ and
$x*_{n}(y*_{n}z)=(x*_{n}y)*_{n}(x*_{n}z)$ for all $x,y,z$. We say that a linear ordering $\preceq$ on $\{...

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### How slowly can the critical points of the Fibonacci terms grow?

Define the Fibonacci terms $t_{n}$ for all $n\geq 1$ by letting $t_{1}(x,y)=y,t_{2}(x,y)=x$, and $t_{n+2}(x,y)=t_{n+1}(x,y)*t_{n}(x,y)$ for all $n$. The Fibonacci terms are used in order to ...

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### Semigroups containing an ideal with a local identity

I'm looking for some classes of semigroups containing a (proper) ideal with a local identity (i.e., ideal submonoid). Can somebody give some examples or/and theorems for the followings cases:
(a) ...

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### What are some examples of inner endomorphisms?

Let $(X,\mathcal{F})$ be an algebraic structure. If $t$ is an $n+1$-ary term, then let $L_{t,a_{1},...,a_{n}}:X\rightarrow X$ be the mapping defined by
$L_{t,a_{1},...,a_{n}}(x)=t(a_{1},...,a_{n},x)$. ...

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### Representation of free Boolean sigma-algebras

By a theorem of Loomis and Sikorski, for every Boolean $\sigma$-algebra $\mathfrak{A}$ there exists a $\sigma$-field of sets $\mathcal{F}$ and a $\sigma$-ideal $\Delta$ such that $\mathfrak{A}$ is ...

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### Examples of finite algebras that resemble the algebras of elementary embeddings

Let $X$ be a set. Suppose that there exists an element $1\in X$ along with functions $L:X^{2}\rightarrow X,R:X^{2}\rightarrow X$ that satisfy the following identities:
$L(1,x)=x,R(1,x)=1,L(x,1)=1,R(x,...

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### What is the correct generalization of “sigma-free” to props?

This is a question about props, a generalization of operads (used to model operations with several inputs and several outputs).
By forgetting the composition structure of an operad one obtains a so ...

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### Relating three viewpoints on the semidirect product

It's known that giving a semidirect product $(X,m)\rtimes G$ of a $G$-group $(X,m)$ with $G$ (as defined in wiki) is the same as giving a split pair over $G$, i.e a pair of arrows $H\overset{s}{\...

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### Lawvere theory and the Maybe monad

The Maybe monad is based on the endofunctor $- + 1$ (coproduct with the singleton set). Its Lawvere theory $L$ is supposed to be generated by one nullary operation (...

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### How to count Isomorphism Types of arbitrary structures?

For all relational signatures $\sigma$ and nonnegative integers $n$, I want to count the number of isomorphism types of structures of order $n$ of the signature $\sigma$.
What I mean by structure is ...

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### Regarding a new algebraic structure

By "left semigroup-joined-semigroup" I mean an algebraic structures $(S,\cdot,*)$ such that both $\cdot,*$ are associative, and the following property holds (see this )
$$
x*(y\cdot z)=x*y*z\;\; ; \;...

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### Which semirings have enough injectives in their category of modules?

Let $R$ be a semiring and $Mod_R$ its category of modules. That is, $R$ is a monoid in the monoidal category of commutative monoids and $Mod_R$ is its category of modules in the usual sense.
Question ...

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### Can I recover a crossed module by its homomorphisms?

This is a follow up to this question.
Imagine there is a finitely presented crossed module $\mathcal{G} = (G,H, -\triangleright-\colon G \to \operatorname{Aut}(H), \delta\colon H \to G)$ which I don'...

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### Varieties of groups with certain properties

Is there an example of a periodic variety $\mathbf{V}$ of groups that satisfies all of the following properties?
$\mathbf{V}$ is finitely based
$\mathbf{V}$ contains finitely many subvarieties
$\...

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### Classification of finitely generated chain groups

An ordered pair $\ \mathbf X := (X\ d)\ $ is called a chain group $\ \Leftarrow:\Rightarrow\ X\ $ is an abelian group, $\ d:X\rightarrow X\ $ is an abelian group endomorphism, and $\ d\circ d= 0$.
A ...

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### Deciding equality in free models of a (generalized) Lawvere theory

Let $F : \mathcal{C} \rightarrow \mathcal{D}$ be functor of Lawvere theories $\mathcal{C}, \mathcal{D}$ (i.e. cartesian categories where every object is isomorphic to some power of a chosen object) ...

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### Topological universal algebra: what is a variety?

Very roughly, universal algebra is the study of those classes of algebraic structures which can be defined via a set of equations; such a class is called a variety. Of course there is far more to the ...

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### Why are there so few elements in the classical Laver tables with period 32?

Recall that the classical Laver table $A_{n}$ is the unique algebraic structure
$(\{1,\ldots,2^{n}\},*_{n})$ where
$x*_{n}(y*_{n}z)=(x*_{n}y)*_{n}(x*_{n}z)$, and
$x*_{n}1=x+1\mod n$ for all $x,y,z\in ...

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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 ...