The tag has no wiki summary.

learn more… | top users | synonyms

11
votes
2answers
279 views

Is the Amitsur-Levitzki identity essentially unique?

Let us consider the matrix algebra. $Mat_n(\mathbb{C})$. The Amitsur-Levitzki identity states that for any matrices $X_1, X_2, ..., X_{2n} \in Mat_n(\mathbb{C})$ the sum $\Sigma_{\sigma \in S_{2n}} ...
2
votes
0answers
112 views

Is the equational theory of commutative vN regular rings decidable?

I wanted to check whether $A(x,y):=\frac{xy}{x+y}$ is an associative operation in every commutative vN regular ring. Now $A(-1,A(1,1))=A(-1,\frac{1}{2})=1\neq 0 =A(0,1)=A(A(-1,1),1)$. On the other ...
3
votes
2answers
107 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 $\mathcal{A}$ ...
10
votes
3answers
420 views

Natural associative law for a ternary “group”?

Suppose one were to define a group-like structure based on a set $G$ with a ternary (rather than binary) operator $g( a, b, c ) = \left< a, b, c \right>$. One possible definition for the ...
2
votes
0answers
62 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 ...
1
vote
1answer
160 views

Two questions about axiomatic rank of groups

Let $G$ be a group and $V=Var(G)$ be the variety generated by $G$. Suppose the axiomatic rank of $V$ is $n$. Let $Id(V)$ be the set of all identities of $V$. 1- Can we say that every element of ...
4
votes
1answer
111 views

When are the congruence lattices nicer?

This is a purely idle question, but one I'm increasingly interested the more thought I put into it: For $\mathcal{A}$ a universal algebra (that is, nonempty set together with some named functions), a ...
2
votes
0answers
104 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
votes
1answer
140 views

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 ...
4
votes
1answer
191 views

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 ...
3
votes
1answer
90 views

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) ...
2
votes
1answer
103 views

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 ...
3
votes
2answers
149 views

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 ...
2
votes
0answers
103 views

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 ...
9
votes
3answers
490 views

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 ...
2
votes
0answers
29 views

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 ...
6
votes
2answers
245 views

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 ...
-1
votes
1answer
87 views

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.
-3
votes
1answer
104 views

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 ...
3
votes
1answer
67 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
0answers
65 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
vote
0answers
52 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
votes
3answers
204 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
175 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 ...
3
votes
0answers
144 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
161 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
62 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
140 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 ...
7
votes
4answers
343 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
165 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
40 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?
13
votes
3answers
455 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 ...
10
votes
0answers
188 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 ...
1
vote
1answer
144 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
302 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
0answers
149 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 ...
24
votes
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
181 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 ...
0
votes
0answers
51 views

Semigroups admitting generating sets which induce a “weight” on elements of the semigroup

Neither universal algebra nor semigroup theory is something I really know much about, so this question might not be appropriate for MO; if so, I'll move it to MSE. Recently, I've been playing ...
3
votes
1answer
80 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 ...
1
vote
1answer
169 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
2answers
98 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
511 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
163 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
138 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 ...
4
votes
3answers
252 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 ...
29
votes
15answers
2k 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 ...
0
votes
1answer
151 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 ...
3
votes
3answers
187 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
234 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 ...