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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3
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1answer
70 views

H S class operator and its equality

$A \in S(K)$ iff $A$ is a subalgebra of some member of $K$ $A \in H(K)$ iff $A$ is a homomorphic image of some member of $K$ It is trivial to see the containment $SH \leq HS$. Taking a simple ...
1
vote
1answer
110 views

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 ...
1
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1answer
89 views

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 $F$?
6
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3answers
240 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 ...
5
votes
3answers
318 views

Locally finite varieties which are not finitely generated

Let $\Sigma$ be a signature consisting of operations with finite arity. Let $\mathcal{V}$ be a variety of algebras for this signature. Further suppose that $\mathcal{V}$ is locally finite i.e. every ...
4
votes
2answers
289 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 ...
5
votes
1answer
187 views

What do algebraic theories with strictly terminal trivial models look like?

By algebraic theory I mean one in the sense of Lawvere, i.e. a collection of finitary operations, including projections, together with a multi-composition satisfying the obvious axioms. (I believe ...
2
votes
0answers
80 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 ...
4
votes
1answer
150 views

Finite generation of vector identities

This question is partially motivated by Looking for a comprehensive referece for vector identities, although that question may not be appropriate for MO. Consider the set $\mathcal{E}$ of all valid ...
9
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4answers
608 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?
2
votes
1answer
238 views

The Universal Algebra of a sigma-Algebra

I am searching for the 'dual' algebraic structure of a Sigma Algebra. The notion of duallity is like on the case of the Boolean Algebra and Set Algera. If X is a set, the complement and intersection ...
0
votes
1answer
42 views

Product of two algebras with maximum condition

Suppose $A$ and $B$ are two algebras of the same signature, both having maximum condition on sub-algebras. Is it true that $A\times B$ has the same property?
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votes
4answers
800 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 ...
12
votes
1answer
281 views

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 ...
2
votes
1answer
192 views

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 ...
3
votes
2answers
334 views

Are algebraic structures uniquely identifed by their free objects?

It might be a naive question, as I am not a specialist in this field. This is a follow-up to this question. I want to study varieties of objects generalizing ordered monoids, in particular using an ...
2
votes
1answer
221 views

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 ...
25
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1answer
2k views

A preprint of Sela concerning the work of Kharlampovich-Miyasnikov

Yesterday, Z. Sela published a preprint in arXiv which claims that the solution of Olga Kharlampovich and Alexi Miyasnikov for the Tarski problem on decidablity of the first order theories of free ...
4
votes
1answer
188 views

Why the axiomatic rank of the variety of groups is equal to three?

I am thankful of Anton Klyachko who introduced axiomatic rank to me: the axiomatic rank of a variety is the minimum number of variables which we need to define that variety by identities. It seems ...
3
votes
1answer
89 views

Non finitely based varieties of groups defined by finitely many variables

A set $\Sigma$ of group identities is called bounded if there is $n\geq 1$ such that for any $(w\approx 1)\in \Sigma$, we have $w\in F(x_1, \ldots, x_n)$. A variety $\mathbf{V}$ is called bounded ...
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1answer
204 views

Varieties generated by a two element algebra

I have two questions regarding universal algebra, and also its ordered version. If a variety $\mathcal{V}$ is generated by a specific two element algebra $2 = \{0,1\}$, then is that the only ...
1
vote
3answers
275 views

Example of a non-finitely based variety with explicit set of defining identities

There are many examples of non-finitely based varieties. In a finite signature, is there an example of such variety with a known explicit set of identities?
2
votes
4answers
727 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 ...
3
votes
2answers
171 views

A variety of algebras satisfying some dual conditions

I would like prove that, under the conditions described below, no non-trivial variety exists. Let $\mathcal{V}$ be a variety of algebras e.g. rings, semigroups, semilattices. Further suppose ...
0
votes
3answers
147 views

Negated varieties and their relatively free algebras

During the past days, I asked some questions in order to gain a clear understanding of the notion of "free algebras". I suppose that the question below is the most clear image of the concept I have ...
5
votes
3answers
273 views

The existence of an algebra whose set of identities and first order theory are equivalent

Is there an algebra $A$ (for example a group) such that $Th(A)$ is logically equivalent to $id(A)$? In other words, is there an algebra $A$ such that $$ Mod(Th(A))=Var(A)? $$ Clearly finite algebras ...
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16answers
3k 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 ...
1
vote
1answer
177 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 ...
4
votes
3answers
214 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 ...
6
votes
1answer
327 views

Generalizations of Birkhoff's HSP Theorem

Let $\mathbf{C}$ be the class of algebraic structures of some fixed type satisfying some sentence $\phi$. Birkhoff's HSP theorem says that $\mathbf{C}$ is closed under homomorphisms, subalgebras and ...
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vote
0answers
52 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 ...
13
votes
2answers
1k 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 ...
2
votes
1answer
119 views

Varieties of rational languages and (pseudo-)varieties of finite monoids, question regarding closure property

Let $\mathcal Rat(A)$ denote the class of rational (or regular) languages over the alphabet $A$, a subset $\mathcal V(A) \subseteq \mathcal Rat(A)$ is called a variety of (rational) languages iff ...
5
votes
1answer
446 views

What are the essential properties of algebraic closure on an arbitrary structure?

Define the "model theoretic" notion of a closure function as follows: Definition (1): Let $D$ be a non-empty set. A function $cl:P(D)\longrightarrow P(D)$ called a closure function iff it has the ...
2
votes
0answers
114 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 ...
9
votes
1answer
423 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 ...
3
votes
1answer
808 views

Commutative associative rational binary operations

What are all the nondegenerate rational binary operations that are commutative and associative? (Examples: $(x,y) \mapsto x+y$, $xy+x+y$, $xy/(x+y)$.) Feel free to re-tag if you can think of ...
1
vote
1answer
207 views

Bodnarchuk, Kaluzhnin, Kotov, Romov’s Theorem on inclusion of Polymorphism ($Pol \rho \subseteq Pol \sigma$)

Bodnarchuk, Kaluzhnin, 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 (Bodnarchuk, ...
2
votes
1answer
158 views

How to prove that a binary relation is a strongly rigid relation? i.e. Polρ only contains projections

I first quote a definition from Clone theory in Universal Algebra: A binary relation $\rho$ on a set U is strongly rigid if every universal algebra on U such that $\rho$ is a subuniverse of its square ...
3
votes
2answers
265 views

Coequalizers in an Eilenberg-Moore category

Last month I proved that some category $\mathbf C$ that I happen to care about is isomorphic to the Eilenberg-Moore category for a monad on the category of bounded posets $\mathbf{BPos}$. I know from ...
2
votes
1answer
570 views
4
votes
2answers
570 views

Shape of axioms in abstract algebra

When defining abstract algebraic structures (like monoids, groups, etc...), are there some constraints on the shape of the axioms, for the structure to have good properties that we implicitly use in ...
3
votes
1answer
273 views

Iterating Monad-Comonads structures

Let $(T, \mu , \eta )$ a monad on the category $\mathscr{C}$ , with the usual EM (Eilenberg-Moore) adjunction $\langle F_T, U_T, \eta_Y, \epsilon_T \rangle: \mathscr{C}^T \to \mathscr{C}$ where ...
6
votes
1answer
387 views

Is there a general theory of “representation theorems”?

Let $V$ and $W$ be classes of algebraic structures, and suppose we have some canonical way of constructing objects of $W$ from objects of $V$. Let's call this construction $C$, so that for all $A\in ...
12
votes
1answer
447 views

Which categories are the categories of models of a Lawvere theory?

Background: a Lawvere theory $T$ is a category with finite products such that each object is a power of a fixed object $x$. Given a Lawvere theory $T$, the category $\text{Mod}_T$ of models of $T$ is ...
13
votes
0answers
762 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 ...
7
votes
0answers
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) ...
6
votes
3answers
220 views

Algebras with finite essential arity

We are talking about algebras in the universal algebraic sense, that is, a set that $A$ is equipped with a set $F$ of finitary operations on $A$. Definition: An algebra $(A,F)$ is said to have ...
11
votes
5answers
669 views

What are the invariant Pseudo-differential operators on a Lie group?

It is well-known that (left) $G$-invariant differential operators on a Lie group $G$, has an algebraic description, i.e. universal enveloping algebra of the Lie algebra of the group. On the other ...
13
votes
2answers
415 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 ...