The universal-algebra tag has no wiki summary.

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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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### Bondarchuk, Kaluznin, Kotov, Romov’s Theorem on inclusion of Polymorphism ($Pol \rho \subseteq Pol \sigma$)

Bondarchuk, Kaluznin, Kotov, Romov’s paper [1] is well-known. Anne Fearnley [2] infered from it the following theroem and used it to prove the inclusion of polymorphisms.
Theorem (Bondarchuk, ...

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### Given a lattice L with n elements, are there finite groups H < G such that L $\cong$ the lattice of subgroups between H and G?

If there is no restriction on $n$, this is a famous open problem. I'm wondering if any recent work has been done for small $n>6$. I believe the question is answered (positively) for $n=6$ by ...

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### When can we tell if PROPs, Algebraic Theories, etc. are faithfully detected in a given category?

I am interested in understanding a certain phenomenon. I am hoping this sort of problem has been studied before, but I don't know the proper terminology and am having trouble finding answers. I am ...

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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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### Why did Bourbaki not use universal algebra?

I have seen a discussion about Bourbaki’s usage of categories before. So let me ask a different question: why did he not use universal algebra?
Well, universal algebra is not much older than category ...

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### When can I assure that the representation theory of a PROP is faithful?

Recall that a PROP is a symmetric monoidal category whose object-set is freely $\otimes$-generated by a single object $x\in P$. A representation or algebra for a prop $P$ in a symmetric monoidal ...

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### The name for a partial order

In a recent paper of mine, my co-authors found that a partial order that we were using was contained in a paper by Kundgen. In it, he called it "right-shifted partial order". I was curious, and found ...

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### Counting and understanging commuting functions.

Fix a positive integer $n$, and consider the functions from a set of size $n$
to itself. Let $cp(n)$ denote the number of ordered pairs $\langle f,g
\rangle$ of these functions which commute, i.e., ...

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### Binary Operation on a Cubic Surface

If $P$ and $Q$ are two sufficiently general points on a cubic surface, the line between them intersects the cubic surface at a unique third point, $f(P,Q)$. This gives a binary operation on (generic) ...

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### Invertible elements in generalized fields

Durov's theory of generalized rings also includes generalized fields (5.7.6), which are defined as generalized rings, which are not subtrivial and whose proper strict quotients are subtrivial. For ...

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### Undecidability degree of some elementary theories (two equivalence relations, …)

I have a question about some results in the paper
I. A. Lavrov. Effective inseparability of the sets of identically true formulae and finitely refutable formulae for certain theories (in Russian). ...

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### $U\left(\mathfrak a\right) \otimes_{U\left(\mathfrak a\cap\mathfrak b\right)} U\left(\mathfrak b\right) \cong U\left(\mathfrak a + \mathfrak b\right)$ over a ring containing $\mathbb{Q}$

While the Poincaré-Birkhoff-Witt theorem is usually proven (and sometimes even formulated) for free modules only, it is known (see also here) that it holds for arbitrary modules if the ground ring is ...

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### Embedding of relatively free groups of bigger rank into ones of smaller rank

This question is prompted by this one by Arturo Magidin: whether there exist varieties of groups in which the relatively free group of rank 2 is finite, and the relatively free group of rank 3 is ...

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### The graph of algebraic theories

Fix a logic $L$ and consider the category $\mathbf{AlgTh}_L$ with theories of $L$ as objects and theory interpretations as morphisms. For nice enough logics, this category has pushouts (which we will ...

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### Equational theories determined by “identities without variables”

How to characterize equational theories $T$ which have the following property: for any two terms $t(x_1,...,x_n)$ and $t'(x_1,...,x_n)$ in the signature of $T$, if for any closed terms (i. e. terms ...

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### Existence of a construction in Universal Algebra: infinite trees

Is anything known about the following construction? Fix a signature (function symbols with arities incl 0) Sigma and a Sigma-algebra A. Construct a new Sigma-algebra T(A) as follows: The carrier set ...

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### Algebras admitting quantifier elimination

I apologize if this question is meaningless or trivial:
What are examples of Algebras admitting quantifier elimination? Especially are there Groups admitting quantifier elimination?
I need to say ...

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### A fixed point problem in infinite dimensions for monotonic algebras

Let $A$ be an algebra over $[0,1]$, whose operations are all unary monotone (increasing or decreasing) bijections, except that $A$ also includes the infimum operation over finite or countably many ...

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### A generalization of quasi-identities

In universal algebra, a variety is axiomatized by identities $t \approx s$ between terms $t$ and $s$. More general are quasi-varieties that are axiomatized by quasi-identities of the form $$u_1 ...

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### Algebras with supremum-founded subalgebra lattice

I am interested in algebras whose subalgebra lattice is supremum-founded. Let us call those algebras small.
A complete lattice $(L, \leq)$ is called supremum-founded, if for any two elements $x < ...

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### When does a cogenerator determine a variety?

Two varieties of universal algebras are categorically equivalent iff their respective full subcategories of finitely generated free algebras are equivalent. Roughly speaking, this follows because they ...

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### Minimal generating sets of free algebras of varieties

Let $V$ be a variety and $F$ be a relatively free algebra in $V$. Suppose $X$ is a minimal generating set for $F$. Under what conditions we can deduce that $X$ is a free basis of $F$?

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### Looking for a uniform explanation of algebras with canonical generators.

Let $\mathcal{V}$ be a finitary variety i.e. the algebras for a signature whose operations have finite arity and for some arbitrary set of equations. Then any algebra $A \in \mathcal{V}$ has a ...

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### Semigroups admitting generating sets which induce a “weight” on elements of the semigroup

Neither universal algebra nor semigroup theory is something I really know much about, so this question might not be appropriate for MO; if so, I'll move it to MSE.
Recently, I've been playing ...