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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33 votes
7 answers
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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 ...
Qiaochu Yuan's user avatar
22 votes
0 answers
1k 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 ...
William DeMeo's user avatar
2 votes
1 answer
208 views

Terminology for a monoid $(H, \cdot)$ s.t. $ax=a$ or $xa =a$ only if $x$ is a unit

Let $(H, \cdot)$ be a (multiplicative) monoid. Is there any consolidated name for the following Property $\text{(P)}$, or for the class of monoids for which it is satisfied? $$\text{(P) If }\,xy = x\...
Salvo Tringali's user avatar
12 votes
3 answers
656 views

IBN for algebraic theories

Let us say that a finitary algebraic theory $\tau$ has IBN (invariant basis number) if the free functor $F : \mathsf{Set} \to \mathsf{Mod}(\tau)$ reflects the isomorphism relation: If $S,T$ are sets ...
Martin Brandenburg's user avatar
10 votes
3 answers
484 views

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 ...
James Propp's user avatar
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7 votes
0 answers
346 views

Adjunction algebra - is there anything similar to this in algebra?

I call adjunction algebra a universal algebra with one binary operation denoted as the punctuation sign (;) "semicolon" (but I will be using only one space after it, not on both sides - to avoid going ...
Ioachim Drugus's user avatar
5 votes
5 answers
2k views

Defining 'free monoid' without Nat?

Is there a definition of what is a 'free monoid' which does not pre-suppose that the natural numbers has already been defined? The definitions that I have been able to track down all use the natural ...
Jacques Carette's user avatar
4 votes
3 answers
401 views

Relatively free algebras in a variety generated by a single algebra

Suppose $A$ is an algebra of signature $\mathcal{L}$ and $V=Var(A)$ is the variety generated by $A$. I want to know is it possible to classify relatively free elements of $V$? As a special case, for a ...
Sh.M1972's user avatar
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3 votes
4 answers
898 views

Examples of algebras satisfying (a+b)(c+d)=ac+bd

Is there a known example of an algebra $(A, +, \cdot)$ with two binary commutative (see P.S below) and idempotent operations $+$ and $\cdot$ satisfying the identity $(a+b)(c+d)=ac+bd$? Actually I ...
Sh.M1972's user avatar
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29 votes
4 answers
1k views

Varieties where every algebra is free

I'd like to know more about varieties (in the sense of universal algebra) where every algebra is free. Another way to state the condition is that the comparison functor from the Kleisli category to ...
Tim Campion's user avatar
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19 votes
4 answers
1k views

Representation theorem for modular lattices?

Birkhoff's representation theorem implies that every distributive lattice embeds into the lattice of subsets of a set. Is there also some representation theorem for modular lattices? For example, I ...
Martin Brandenburg's user avatar
17 votes
1 answer
2k views

So, did Poincaré prove PBW or not?

This seems to be a question whose answer depends on whom you ask. Maybe we can come up with a final answer? It is known that Poincaré, at least, invented something that can be called Poincaré-...
darij grinberg's user avatar
9 votes
3 answers
549 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 ...
მამუკა ჯიბლაძე's user avatar
9 votes
1 answer
288 views

Two notions of generalized quotient/substructure

Given a language $\Sigma$ and a $\Sigma$-algebra (in the sense of universal algebra) $\mathcal{A}=(A;\dotsc)$ and a function $f:A\rightarrow A$, let $\mathcal{A}_f$ be the $\Sigma$-algebra whose ...
Noah Schweber's user avatar
9 votes
1 answer
290 views

Equational theory of the orthocenter

Previously asked at MSE: Briefly speaking, I'm looking for a description of the equational theory of the orthocenter function, $\mathsf{orth}$. By $\mathsf{orth}$ I mean the (partial) function sending ...
Noah Schweber's user avatar
7 votes
0 answers
175 views

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 ...
Gro-Tsen's user avatar
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6 votes
2 answers
590 views

Classification of plethories over $\mathbb{Q}$

Let $k$ be a commutative ring. For every cocommutative bialgebra $A$ over $k$ the symmetric algebra of the underlying $k$-module $S(A)$ carries the structure of a $k$-plethory (Borger, Wieland, 2.5). ...
Martin Brandenburg's user avatar
6 votes
2 answers
453 views

Heyting algebras originating from directed graphs

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 ...
Frank's user avatar
  • 457
5 votes
3 answers
526 views

Congruences that aren't "finite from above"

Let $\mathfrak{A}=(A;...)$ be an algebra in the sense of universal algebra. Say that a congruence $\sim$ on $\mathfrak{A}$ is parafinite iff there is an equivalence relation $E\subseteq A^2$ with ...
Noah Schweber's user avatar
4 votes
0 answers
143 views

Is the orthocenter "(roughly) equationally finitely-based"?

Let $T$ be the "almost everywhere" equational theory of the orthocenter function, "tweaked appropriately" to avoid partiality issues (see this earlier question of mine for details)....
Noah Schweber's user avatar
3 votes
0 answers
346 views

Is there any research of universal algebras axiomatized by non-Horn clauses?

A Horn clause in the language of a universal algebra is a disjunction of equations and of at most one inequality ("equation" and "inequality" are the terms used by A.Horn in his paper "On sencences ...
Ioachim Drugus's user avatar
2 votes
4 answers
539 views

relatively free groups in $Var(S_3)$

Suppose $S_3$ is the symmetric group of order 6. Which elements of the variety $Var(S_3)$ are relatively free? This question is related to my previous question Relatively free algebras in a variety ...
Sh.M1972's user avatar
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1 vote
1 answer
140 views

Congruences that aren't "finite from above," take 2: semigroups

This is a hopefully less trivial version of this question. Briefly, say that a congruence is parafinite if it is the largest congruence contained in some equivalence relation with finitely many ...
Noah Schweber's user avatar
1 vote
0 answers
121 views

Basic questions about varieties of uniformly partially permutative algebras

Define the Fibonacci terms $t_{n}(x,y)$ for all $n\geq 1$ by letting $t_{1}(x,y)=y,t_{2}(x,y)=x,t_{n+2}(x,y)=t_{n+1}(x,y)*t_{n}(x,y)$. We say that an algebra $(X,*)$ is $N$-uniformly partially ...
Joseph Van Name's user avatar
-1 votes
2 answers
657 views

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 ...
Wlod AA's user avatar
  • 4,686
62 votes
5 answers
9k views

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 ...
Andrew Critch's user avatar
37 votes
19 answers
5k views

Counterexamples in universal algebra

Universal algebra - roughly - is the study, construed broadly, of classes of algebraic structures (in a given language) defined by equations. Of course, it is really much more than that, but that's ...
35 votes
6 answers
9k views

Do convolution and multiplication satisfy any nontrivial algebraic identities?

For (suitable) real- or complex-valued functions $f$ and $g$ on a (suitable) abelian group $G$, we have two bilinear operations: multiplication - $$(f\cdot g)(x) = f(x)g(x),$$ and convolution - $$(f*...
Darsh Ranjan's user avatar
  • 5,912
33 votes
21 answers
4k views

Why are so few operations with arity bigger than 2?

In usual algebraic structures, like groups, rings, monoids, etc, or in algebras coming from logics like Boolean algebras, Heyting algebras and the like the operations are usually of arity 0 (constants)...
24 votes
2 answers
2k views

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 ...
Qfwfq's user avatar
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21 votes
3 answers
2k views

Relation between monads, operads and algebraic theories

I've begun to interest in algebraic theories and their categorical models: in particular monads, generalized multicategories and operads, lawvere theories and their generalization. Is there any ...
Giorgio Mossa's user avatar
19 votes
2 answers
729 views

Do all subtraction-free identities tropicalize?

If you take a subtraction-free rational identity like $(xxx+yyy)/(x+y)+xy=xx+yy$ and replace $\times$,$/$,$+$,$1$ by $+$,$-$,min,$0$, do you always get a valid min,plus,minus identity like min(min($x+...
James Propp's user avatar
  • 19.4k
19 votes
0 answers
554 views

What algebraic properties are preserved by $\mathbb{N}\leadsto\beta\mathbb{N}$?

Given a binary operation $\star$ on $\mathbb{N}$, we can naturally extend $\star$ to a semicontinuous operation $\widehat{\star}$ on the set $\beta\mathbb{N}$ of ultrafilters on $\mathbb{N}$ as ...
Noah Schweber's user avatar
18 votes
4 answers
2k views

Complete Boolean algebra not isomorphic to a $\sigma$-algebra

Does there exist a complete Boolean algebra that is not isomorphic to any $\sigma$-algebra? If so, what is an easy or canonical example or construction?
Bjørn Kjos-Hanssen's user avatar
18 votes
1 answer
1k views

Is there an identity between the commutative identity and the constant identity?

I asked this on Math Stack Exchange, but it didn't get a single answer. So, I am now asking it here. Let our signature be that of a single binary operation $+$. I define the constant identity to be $x+...
user107952's user avatar
  • 2,063
16 votes
2 answers
1k views

Comparing "axiomatized function spaces"

This was previously asked and bountied at math.stackexchange with no response. I've also tweaked the language for clarity; see the edit history for the broader context, and note that the existing ...
Noah Schweber's user avatar
15 votes
2 answers
2k views

What is the status of (universal) algebra in type theory?

With the recent interest in homotopy type theory as a foundation for mathematics, it seems natural to develop algebra within the framework of type theory. So far, I can't find much literature ...
Cory Knapp's user avatar
12 votes
5 answers
5k views

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 ...
beroal's user avatar
  • 520
11 votes
3 answers
906 views

What is the smallest variety of algebras containing all fields?

A field is a ring whose nonzero elements form a commutative group under multiplication. A field is also a commutative inverse semigroup with respect to multiplication. The unique multiplicative ...
Thomas Klimpel's user avatar
9 votes
1 answer
687 views

Generalizing detropicalization

Given an identity in max,plus arithmetic, are there ways to turn it into an ordinary algebraic identity it other than by replacing addition by multiplication and replacing max by series-plus or by ...
James Propp's user avatar
  • 19.4k
9 votes
1 answer
747 views

What classes of groups can arise as "symmetry groups of terms"?

Let $\mathfrak{A}$ be an algebra (in the sense of universal algebra). To each term $t(x_1,...,x_n)$ in the language of $\mathfrak{A}$ in which each variable actually appears we can assign a group $G_\...
Noah Schweber's user avatar
9 votes
2 answers
563 views

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 ...
Martin Brandenburg's user avatar
9 votes
1 answer
426 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 ...
darij grinberg's user avatar
8 votes
1 answer
333 views

Example of trickiness of finite lattice representation problem?

I'm trying to come up with a good explanation for my students of why the finite lattice representation problem is difficult. I've already shown that the "greedy approach" to representing the ...
Noah Schweber's user avatar
8 votes
0 answers
614 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 ...
Victor Miller's user avatar
7 votes
0 answers
269 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) ...
Will Sawin's user avatar
  • 135k
7 votes
0 answers
330 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 ...
Jacques Carette's user avatar
7 votes
1 answer
586 views

Reversible varieties

We say that a variety $V$ is reversible if for each $n>0$ and $n$-ary fundamental operation $f$, there is some $m\geq n$ and $r$ along with terms $T_{2},\dotsc,T_{r},S_{1},\dotsc,S_{m}$ such that $...
Joseph Van Name's user avatar
7 votes
0 answers
1k 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). ...
boumol's user avatar
  • 788
7 votes
0 answers
582 views

A new and subtle order-theoretic fixed point theorem

Sometimes a very simple argument appears out of the blue and overturns a subject. It is not based on pre-existing theory and heavy involvement in such a theory is actually a handicap in finding such ...
Paul Taylor's user avatar
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