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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274 views

About a construction of Borel $\sigma$-algebra associated to a lattice

Let $(\mathcal{A}, \cup, \cap)$ a lattice (with minimum and maximum elements $\bot$ and $\top$). Let $X\subset \mathcal{A}$ a generator set (a set of minimal cardinality that generate $\mathcal{A}$ i....
20
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0answers
797 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 ...
13
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0answers
779 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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0answers
369 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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0answers
318 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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193 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) ...
7
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0answers
486 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 ...
7
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547 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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98 views

Finitely presented algebras with isomorphic semilattices of congruences

Let $\mathbb{T}$ be a finitary algebraic theory. For each $\mathbb{T}$-algebra $A$, let $Q (A)$ be the join semilattice of finitely generated congruences on $A$. There is an evident pushforward ...
6
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0answers
143 views

When are the categories of algebras over props (co)complete?

Suppose P is a (colored) prop in a closed symmetric monoidal locally presentable category C. Is the category Alg_P of algebras over P in C locally presentable? It seems that one can relatively easy ...
6
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546 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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144 views

Hemi-Semi Direct Product

In the category of racks (similarly quandles), instead of well-known semi direct product, we have hemi-semi direct product construction as seen on Wagemann & Crans. As far as I know, semi direct ...
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66 views

General approaches to extension theorems as Caratheodory

I would like to know if there are some general studies about extension-like theorem, in the sense which i'm going to describe. This paragraph is not rigorous; I just would like the idea to be clear. I ...
5
votes
0answers
164 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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0answers
206 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 axiomatized ...
5
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0answers
911 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). ...
4
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0answers
219 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 ...
4
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261 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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269 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 ...
3
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76 views

Poincaré-Birkhoff-Witt theorem for Leibniz algebras

Leibniz algebras can be seen as a non-skew-symmetric generalization of Lie algebras. I have already taken a look at some papers related to Leibniz algebras and extending main results of Lie algebras ...
3
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0answers
156 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 ...
3
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0answers
191 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, $ZFC\vdash\phi(x)\rightarrow``x\,\textrm{is a complete Boolean algebra}"$, $ZFC\vdash``\...
3
votes
0answers
93 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 xhytxy=...
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0answers
134 views

Is the class of Heyting algebras originating from directed graphs a variety?

The category RefGph of reflexive directed graphs is the functor category $\hat{∆}_1=\mbox{Fun}(∆^◦_1,$Set), where $∆_1$ is the simplex category truncated at level 1. Hence the poset Sub(X) of ...
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0answers
202 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
votes
0answers
98 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 ...
2
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0answers
81 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. ...
2
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0answers
116 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 ...
2
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0answers
81 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 ...
2
votes
0answers
115 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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0answers
236 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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53 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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0answers
67 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 ...