**3**

votes

**2**answers

174 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 ...

**5**

votes

**3**answers

711 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

**3**answers

519 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

**2**answers

285 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

**1**answer

110 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

**1**answer

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**

votes

**1**answer

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

**3**answers

232 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

**3**answers

461 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

**1**answer

184 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

**0**answers

78 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

**1**answer

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

**4**answers

584 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

**1**answer

218 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

**1**answer

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?

**19**

votes

**6**answers

1k views

### 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 ...

**25**

votes

**1**answer

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

**2**answers

329 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

**1**answer

187 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

**1**answer

88 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

**1**answer

203 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

**2**answers

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

**3**answers

146 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

**3**answers

268 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

**1**answer

174 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

**3**answers

211 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

**1**answer

311 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 ...

**1**

vote

**0**answers

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

**2**answers

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

**1**answer

116 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

**1**answer

429 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

**1**answer

412 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

**2**answers

408 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 ...

**2**

votes

**0**answers

113 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

**1**answer

759 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

**1**answer

157 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

**2**answers

260 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

**1**answer

190 views

### Cyclic Distribution on the reals?

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

**2**

votes

**1**answer

541 views

### Suggestions on the best introductory Model Theory texts [closed]

Any recommendations on the best texts for introducing Model Theory?

**4**

votes

**2**answers

559 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

**1**answer

266 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 ...

**5**

votes

**1**answer

362 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 ...

**4**

votes

**2**answers

265 views

### Theorems (from clone theory) that can be stated only by using operations and their composition.

To state my question, I first have to explain what I mean by "using operations and their compositions only". In short, it means that I am looking for theorems from clone theory (or on operations and ...

**12**

votes

**1**answer

414 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

**0**answers

728 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

**0**answers

191 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) ...

**11**

votes

**5**answers

649 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 ...

**1**

vote

**1**answer

204 views

### What are sources for pathological and non-so-pathological Gabriel filters on commutative rings?

The heavy lifting in the theory of Gabriel filters is for noncommutative rings, and discussions I've been able to find all focus there.
I am trying to develop a theory of Gabriel-filter localization ...

**8**

votes

**0**answers

309 views

### When can I assure that the representation theory of a PROP is faithful?

Recall that a PROP is a symmetric monoidal category whose object-set is freely $\otimes$-generated by a single object $x\in P$. A representation or algebra for a prop $P$ in a symmetric monoidal ...

**2**

votes

**1**answer

214 views

### Methods to tell if a magma has idempotents

(Disclaimer: below, when I say "compact" I mean "compact Hausdorff.")
I asked a version of this question on math stackexchange ...