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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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 ...
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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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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\...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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-...
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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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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 ...
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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 ...
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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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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). ...
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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 ...
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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 ...
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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)....
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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*...
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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)...
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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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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 ...
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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+...
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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 ...
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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?
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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+...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $G_\...
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Birkhoff's completeness theorem put into practice
Birkhoff's completeness theorem (see here, Theorem 14.19) states that an equation which is true in all models of an algebraic theory can be proven in equational logic.
Question. Does the proof of ...
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$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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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 ...
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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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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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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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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 $...
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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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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 ...