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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230 views

Contexts and notations for composing asymmetric simplices

Imagine the elements of a group-like structure as puzzle pieces with essential two sides, an IN-side and an OUT-side. You can compose two such pieces in two obvious ways: Now consider triangular ...
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752 views

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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678 views

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 ...
11
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361 views

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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299 views

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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188 views

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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446 views

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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540 views

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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541 views

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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138 views

“Generalized theory of polynomials” for a given commutative Lawvere Theory

I am trying to understand Nikolai Durov's "New Approach to Arakelov Geometry" right now and it got me thinking about a particular thing. Let $R$ be a commutative, associative ring with unit. We can ...
5
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183 views

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 ...
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905 views

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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214 views

$U\left(\mathfrak a\right) \otimes_{U\left(\mathfrak a\cap\mathfrak b\right)} U\left(\mathfrak b\right) \cong U\left(\mathfrak a + \mathfrak b\right)$ 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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256 views

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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263 views

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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182 views

What algebraic identities does the iteration of forcing operation satisfy?

Let $G$ be the set of all formulas $\phi(x)$ in the language of such that $ZFC\vdash\exists x\phi(x)$ exists, $ZFC\vdash\phi(x)\rightarrow``x\,\textrm{is a complete Boolean algebra}"$, ...
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88 views

Pseudovarieties of monoids

All (pseudo)varieties considered here are (pseudo)varieties of monoids. It is known that any (finite or infinite) monoid that satisfies the identities \begin{equation} xhxyty = xhyxty, \quad ...
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157 views

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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69 views

Can such categorical notion of action be formalized?

I'm wondering if it's possible to find an universal construction for a general concept of action for (single-sorted?) finite product sketches, such that one of those is "acting" on the second in the ...
2
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0answers
94 views

Generalizing disjointness

The following definition generalizes set-theoretic disjointess: Definition 0. (Autonomy). Given a Lawvere theory $\mathsf{T}$, a $\mathsf{T}$-algebra $X$, and an indexed family $S$ of subalgebras ...
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0answers
114 views

How distributive are the bad Laver tables?

Suppose that $n\in\omega\setminus\{0\}$. Then define $(S_{n},*)$ to be the algebra where $S_{n}=\{1,...,n\}$ and $*$ is the unique operation on $S_{n}$ where $n*x=x$ $x*1=x+1\,\text{mod}\, n$ and if ...
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69 views

Ultraproducts and subobjects of projectives

Usually the question whether the class of projective algebras in a given variety is closed under taking subalgebras seems to be quite hard. In varieties with well understood dual geometry (e. g. ...
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0answers
115 views

Non-finitely based varieties and pseudovarieties

The variety of semigroups defined by $B=\Big\{(x^py^p)^2=(y^px^p)^2:p \text{ is prime}\Big\}$ is non-finitely based (Isbell, 1970). Is the pseudovariety defined by $B$ also non-finitely based? More ...
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71 views

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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113 views

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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233 views

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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51 views

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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61 views

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 ...