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
232 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 ...
1
vote
3answers
278 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
741 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 ...
10
votes
3answers
632 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 ...
5
votes
1answer
268 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 ...
1
vote
1answer
175 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 $Id(V)...
3
votes
0answers
93 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 xhytxy=...
5
votes
3answers
326 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 ...
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, ...
4
votes
1answer
130 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
117 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
146 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 ...
3
votes
1answer
106 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
109 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
178 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 $J(...
5
votes
3answers
727 views

Presenting Lawvere theories?

(Re)Reading Lawvere theories versus classical universal algebra, I was reminded of a question I have had for quite some time: What is the best way to present Lawvere theories? By present, I mean ...
10
votes
3answers
526 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 ...
7
votes
2answers
291 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 ...
-3
votes
1answer
111 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 ...
-1
votes
1answer
96 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
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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 ...
6
votes
3answers
246 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 ...
7
votes
3answers
469 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 ...
5
votes
1answer
188 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
83 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
votes
4answers
622 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
243 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?
25
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 ...
3
votes
2answers
335 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 ...
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 ...
1
vote
1answer
206 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 ...
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 ...
1
vote
1answer
180 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
217 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
335 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 ...
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
126 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
461 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 ...
9
votes
1answer
425 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 ...
13
votes
2answers
420 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+...
2
votes
0answers
116 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 \...
3
votes
1answer
814 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 ...
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
268 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
192 views

Cyclic Distribution on the reals?

Do there exist binary operators *, **, and *** on the real numbers, such that ...