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).
440 questions
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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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Algebraic objects and lifts of their represented functors
I've seen the following theorem around in various forms:
To give an object $A \in \mathcal{C}$ the structure of a $\Omega$-algebra object in $\mathcal{C}$ is equivalent to giving a lift of the ...
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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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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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The word problem of the free left distributive algebra on one generator
A left distributive algebra is a set $A$ together with a binary operation, $\cdot$, satisfying $a\cdot(b\cdot c)=(a\cdot b)\cdot(a\cdot c)$.
One important example of left distributive algebras arises ...
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Is quasivariety generated by all perfect graphs finitely axiomatizable?
Fix logic $L$ with equality and a binary relation symbol $E$.
The class of graphs can be identified with the class of models of the universal first-order Horn $L$-sentences $\forall x,y\; E(x,y) \...
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Which algebraic theories have the property that $\mid$ is antisymmetric for all free algebras?
Let $T$ denote an algebraic theory.
Terminological Question. Let $X$ denote a $T$-algebra. Is there a name for the preorder $\mid$ defined on $X$ by asserting that $a \mid b$ iff there is a term ...
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What kind of algebra is the class of ordered pairs equipped with the binary operation which forms them?
There are many definitions of ordered pair in set theory, but all such definitions have the characteristic property of ordered pair:
$ \ \ \ \ \ \ (x, y) = (x', y') \leftrightarrow \ (x = x' \ and \ ...
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If questions are formalized as ideals of a boolean algebra, what kind of algebra of questions appears from Stone representation theorem?
Affirmative propositions make up a Boolean algebra, and Boolean algebras became part of classical algebra for over one century ago - in this sense they are "simple". But I did not encounter in ...
- 889
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Finite lattices whose number of join-irreducibles does not exceed its height
In a finite distributive lattice $L$ one has $height(L) = |J(L)|$ i.e. the size of the largest chain equals the number of join-irreducible elements.
Briefly, this follows by arranging the subposet $J(...
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Is the following a sufficient condition for being a primal algebra?
I have a question regarding universal algebra and, in particular, primal algebras:
Suppose that A is a finite simple algebra with no proper subalgebra, no automorphism except the identity map, with a ...
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Are norms intrinsically $\mathbb{R}$-valued?
Another way of phrasing this: are there any viable definitions of something which is norm-like but whose range is in a linearly ordered rig (for example) rather than $\mathbb{R}$?
I have searched a ...
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Dualities between varieties and quasivarieties at the finite level
Suppose one has two locally finite quasivarieties $\mathcal{V}$ and $\mathcal{W}$.
Further suppose that:
$\mathcal{V}$ is a variety.
The finite algebras $\mathcal{V}_f$ are dually equivalent to $\...
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Is HSP(A) = ISP(A) decidable?
Let $A$ be a finite algebra for some finitary signature.
Is it decidable whether $\mathbb{H}\mathbb{S}\mathbb{P}(A) = \mathbb{I}\mathbb{S}\mathbb{P}(A)$?
That is, whether the variety ...
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Variety of commutative semi group [closed]
V is a variety of commutative semi group satisfying the identity $x^2 = x^3$.
I need to prove that:
$|F_V(\{x_1\dots,x_n\})|$ = $3^n -1$.
Any hints on this ?
$F_V$ is V-free algebra.
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SHPS and SPHS inequality using monounary algebra
Let $A_n = \{(1,\ldots,n) , f \}$ where $f(i) = (i+1)$ if $i \neq n $ otherwise $f(n) = 1$.
This describes a mono unary algebra.
The proof for $HPS \neq SPHS$ I know uses metabelian groups and was ...
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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 ...
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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 ...
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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 $F$?
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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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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 ...
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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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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 ...
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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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Finite generation of vector identities
This question is partially motivated by https://mathoverflow.net/questions/158451/looking-for-a-comprehensive-referece-for-vector-identities, although that question may not be appropriate for MO.
...
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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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The universal algebra of a $\sigma$-algebra
I am searching for the 'dual' algebraic structure of a $\sigma$-algebra. The notion of duality is like in the case of the Boolean algebra and set algebra.
If $X$ is a set, the complement and ...
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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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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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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 ...
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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 $...
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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 ...
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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 ...
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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 ...
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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 ...
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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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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 ...
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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?
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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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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 that:
...
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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 ...
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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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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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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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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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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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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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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
...
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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 ...
user36136
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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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