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4
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4answers
482 views

A doubt in Bergman's notes

On pg. 8 of these notes, Bergman says that a group $G$ contains an inverse operation $i:G\to G$, along with $\mu:G\times G\to G$ and a "neutral element" $e$. Hence, a group should be referred to as ...
12
votes
3answers
131 views

Embeddings $A → B → A$, but $A \not\cong B$?

Are there any nice examples of structures (groups, modules, rings, fields) $A$ and $B$ such that there are embeddings $A → B → A$ while $A \not\cong B$? I would especially like to see an example for ...
0
votes
1answer
18 views

Reducing a set of generators to a free base

Given an algebraic structure $ A $ which is free on some subset of its underlying set, does every generating subset of $ A $ contain a free generating set? For vector spaces, this is true, what about ...
3
votes
2answers
86 views

Examples of Stone algebras which are not Boolean algebras

Grätzer, in his Lattice Theory: Foundation, describes a Stone algebra as a distributive lattice with pseudocomplementation $L$ which satisfies the Stone identity: for every $a \in L$, $\neg a \vee ...
1
vote
2answers
57 views

When do DeMorgan's laws hold in a Heyting algebra

I'm working a bit with Heyting algebras (which are pseudocomplemented distributive lattives, right?) and I have a question about DeMorgan's laws. I know that, in general, it's not the case that $-(X ...
1
vote
0answers
34 views

Congruence lattice of N5

I calculated the the congruence lattice of $N_5$ using hit and trial and then verified it with Universal Algebra calculator. But I need to prove that it is the congruence lattice of $N_5$ How should I ...
1
vote
1answer
67 views

Textbook question on variety

Suppose a variety V is defined by an infinite minimal set of identities. Show that V is a subvariety of at least continuum many varieties.
1
vote
1answer
44 views

Variety satisfying an identity.

$V$ is a variety of commutative semigroup 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
109 views

Semilattices are congruence-semi-distributive

A semilattice $(S,\cdot)$ is a commutative idempotent semigroup. A congruence on a semilattice is an equivalence relation that preserves multiplication, i.e. $x_1\mathrel{\theta} y_1$ and ...
1
vote
1answer
53 views

Factor congruences of non trivial Lattices

A pair of congruences $\theta$ and $\theta^*$ are called factor congruences if $\theta \vee \theta^*$ = full congruence. $\nabla$ $\theta \wedge \theta^*$ = trivial congruence. $\triangle$ I need ...
0
votes
0answers
26 views

How does topological dense subgroup induces properties in the larger group?

Let G denote the group of orientation-preserving isometries of the plane; equivalently, the group of affine transformations of the complex field C of the form $z \rightarrow \alpha z + \beta$ ...
1
vote
0answers
27 views

Question on HSP and SHPS inquality.

In the screenshots attached above George Bergman outlines his way of proving $HSP \ne SHPS$ I understand the first definition as the group of affine transformations and each element of the group ...
3
votes
2answers
64 views

Why are the algebras of the associative operad unital?

According to the n-lab page: The associative operad Assoc is an operad which is generated by a binary operation $\Theta$ satisfying $$\Theta\circ(\Theta,1)=\Theta\circ(1,\Theta)$$ It then ...
0
votes
0answers
46 views

H P S class operators and their inequalities

First few definitions: $A \in I(K)$ iff $A$ is isomorphic to some member of $K$ $A \in S(K)$ iff $A$ is a subalgebra of some member of $K$ $A \in H(K)$ iff $A$ is a homomorphic image of some ...
2
votes
1answer
55 views

Inequality of Class operators H S and P

First few definitions: $A \in I(K)$ iff $A$ is isomorphic to some member of $K$ $A \in S(K)$ iff $A$ is a subalgebra of some member of $K$ $A \in H(K)$ iff $A$ is a homomorphic image of some ...
0
votes
0answers
60 views

What do we call functions that are definable by expressions?

Let $X$ denote a model of an algebraic theory $T$. What do we call the functions $f : X^n \rightarrow X$ that are definable by some expression in the language of $T$? e.g. If $S_3$ is the symmetric ...
0
votes
1answer
51 views

Clarification about the definition of term algebras

The following definition has been given in this article. A term algebra is an algebra $ \langle \mathcal{S}, \mathcal{G} \rangle $ where every time that $g_\alpha, g_\beta \in \mathcal{G}$ and $$ ...
0
votes
1answer
52 views

Operator generating subuniverse generated by X is algebraic closure operator

This is taken from Universal Algebra Text book by Stan Burris. I have a question regarding the last conclusion as to how does the author conclude that Sg is an algebraic closure operator. How do ...
3
votes
1answer
52 views

Can a quasi-identity express that a function $f$ is surjective? And if not, can this be explained by duality?

Consider a first-order theory having a unary function symbol $f$. Then the following quasi-identity expresses that $f$ is injective. $$\forall xy : f(x)=f(y) \rightarrow x=y$$ Alternatively, we can ...
0
votes
1answer
83 views

What are the properties of this Poisson algebra?

I have the following (real) quantities (which are from a Classical Mechanics problem): $$A_1=\frac 1 4(x^2 +p_x^2-y^2-p_y^2 ) \quad A_2=\frac 1 2(x y +p_x p_y)$$ $$A_3=\frac 1 2(x p_y - y p_x )$$ ...
0
votes
0answers
18 views

A filterbase generating filter F

Show that a non empty subset $X$ of a filter $F$ in $B$ is a base for $F$ iff $X$ generates $F$ and for all $x,y$ $\in$ $X$ $\exists$ $z $ $\in$ $X$ such that $z$ $\leqq$ x $\wedge$ y.
0
votes
0answers
143 views

Collections of Homomorphic (defined) structures via $f$

Long ago I read a text about a collection of algebraic sturctures all homomorphic (or isomorphic) via a unique homomorphism An Example similar to the construction I found was this: Lets take define ...
1
vote
1answer
55 views

More Information about Magmas

I first learnt of magmas on Wikipedia and have been trying to read more on them just out of my own interest. Whenever I try to search them on Google, though, the search results are overwhelmed by the ...
2
votes
1answer
83 views

Subdirect embedding of a quotient algebra

If $A$ is an algebra and $\theta_i$ $\in$ $Con(A)$, i $\in$ $I$, let $\theta$ = $\cap \theta_i$. Show that $A/\theta$ can be subdirectly embedded in $\prod$$A/\theta_i$. What intuitively I think of ...
10
votes
2answers
167 views

Suggestions for a learning roadmap for universal algebra?

I think a useful combination of resources for universal algebra would ideally, when taken together: Provide ample motivation behind the various developments in the field. Either provide powerful ...
5
votes
0answers
46 views

Is there a name for the algebra of substructures?

Let $X$ denote an entropic algebra (see here), which just means that all the operations of $X$ are homomorphisms $X^n \rightarrow X.$ Abelian groups are the classic example. Then for any operation of ...
15
votes
4answers
238 views

Why are particular combinations of algebraic properties “better” (richer and more pervasive) than others?

I am a second-year graduate student of pure mathematics. Like most graduate students, I have been exposed to many types of algebraic structures, and it seems standard for the major emphasis, if not ...
0
votes
2answers
37 views

How is modularity a weakened form of distributivity?

While reading an essay Lattice Theory- Its Birth and Life, the following line confused me: modularity is a weakened form of distributivity Just to be clear, here modularity and distributivity of ...
1
vote
1answer
50 views

Universal Algebras for Pseudovarities and their cardinality

A Birkhoff variety is a class of algebras closed under division and arbitrary products, a pseudovariety is a class of algebras closed under division and finite products. Now for each type of ...
1
vote
1answer
61 views

Construct an algebra from its finitely generated algebras

In the general sense of an algebra (a set with some operations, as in Universal Algebra courses), is it always possible to construct any full algebra (up to isomorphism) just from its finitely ...
1
vote
1answer
47 views

Why is the collections of all groups a variety

A variety is an equationally defined class of algebras. As I understand it equationally defined means defined by universally quantified equations, for example the variety of all semigroups could be ...
1
vote
0answers
53 views

What is the $K$-free algebra for the class of implication algebras, over a finite set

I suppose the title is pretty self explanatory. I have been struggling with the concepts of $K$-free algebras, where $K$ is some class of same-type algebras, over some set $X$. So, in trying to ...
1
vote
2answers
63 views

Do “equational theories” include sequents?

In equational logic, which of the following best describes the term "equational theory"? A collection of identities. A collection of quasi-identities, by which I mean sequents of the form ...
3
votes
1answer
85 views

$\mathrm{Pol}_m(\mathbb{A})$ viewed as a relation pp-definable from $\mathbb{A}$

First let me recall some (abbreviated, and possibly simplified to suit my situation) definitions: Let $A$ be a finite set and $\mathbb{A}$ some set of relations on $A$. Let $m, n$ be positive ...
3
votes
1answer
40 views

Confirmation needed of the fact that subcategory $\mathbf{Lat}$ is not full in $\mathbf{Pos}$

If you are familiar with this stuff then you probably don't need the information I have added. So let me start with the question: Can you prove that category $\mathbf{Lat}$ is not a full ...
2
votes
0answers
66 views

Which algebraic identities survive the journey to the powerset?

Given an algebraic structure $A$ (call its underlying set $U$) we can obtain a new algebraic structure $B$ with underlying set $V=\mathcal{P}(U)$ in the obvious way. In particular, if $f : U^n ...
3
votes
1answer
29 views

Signatures having precisely one constant symbol, and pointed categories.

Given an algebraic signature $\sigma$ having precisely one constant symbol, is it true that if $A$ is a set of quasi-identities in the language of $\sigma$, then the set-theoretic models of ...
3
votes
0answers
46 views

Seeking information about (category-theoretic) varieties, quasivarieties, and universal Horn classes.

I'm looking for a list of basic facts regarding (category-theoretic) varieties, quasivarieties, and universal Horn categories, as well as information about which forgetful functors preserve what. In ...
2
votes
1answer
58 views

Reference Request: Semi-Rings and Rings (System of Sets, not Algebraic Structures)

I studied Probability Theory (from a Measure Theory viewpoint) using only Sigma-Algebras. Recently, I got a book about measure theory that starts from Semi-Rings, but it's presentation is too ...
5
votes
1answer
117 views

$\mathbf{N}_5$ as a congruence lattice

A finite lattice is said to be representable if there exists a finite algebra whose congruence lattice is isomorphic to that lattice. As I was reading a paper, I came across the line: "The reader can ...
0
votes
1answer
58 views

Is there a standard name for a set equipped only with an idempotent binary operation?

Is there a name for an idempotent magma, or do they not arise often enough to warrant a special name? (By idempotent binary operation, I mean an operation $+$ such that $x + x = x$ for any $x$.)
2
votes
3answers
57 views

Class of algebras defined by quasi-identities is not closed under taking the quotients

Let $\mathbf{A}$ be an algebra (in the sense of universal algebra) of some signature $\Sigma$. By quasi-identity I mean the formula of the form $$(\forall x_1) (\forall x_2) \dots (\forall x_n) ...
6
votes
1answer
78 views

Obtaining a binary operation on $X \rightarrow Y$ from a binary operation on $Y$. What, if anything, to make of this observation?

Let $X$ and $Y$ denote sets. Then if $+$ is a binary operation on $Y$, then we can obtain a new binary operation $+'$ on $Y^X$ in a canonical way as follows. $$(f+' g)(x) = f(x)+g(x)$$ Question. The ...
4
votes
1answer
74 views

The function $f(x)=(x\vee a)\wedge b$ in a lattice.

Is there an algebraic modular lattice $(X,\vee,\wedge)$ and $a,b\in X$ with $a\le b$ such that the function $$f:X\to X$$ $$f(x)=(x\vee a)\wedge b$$ is not $\vee$-homomorphism?
0
votes
1answer
57 views

Who first described commutative algebraic theories explicitly?

Lately, I've been thinking that the concept of a commutative algebraic theory is really, really important. So I'm curious; who had the honor of first discovering this concept? In particular, I'd like ...
1
vote
1answer
22 views

Does the closure of the set of all irreducible elements always equal the whole set?

Let $X$ denote a set and $\mathrm{cl}$ denote a finitary closure operator on its powerset. Call $x \in X$ irreducible iff for all $A \subseteq X$ we have that if $x \in \mathrm{cl}(A)$, then $x \in ...
2
votes
2answers
68 views

Do formal polynomials make sense in arbitrary algebraic structures?

Let $R$ denote a commutative ring with unity and $X$ a set of formal indeterminates. Then intuitively, the set of all formal polynomials in $X$ with coefficients in $R$ can be defined as the free ...
4
votes
1answer
77 views

The “closed” subspaces of topological algebraic structures

Every set-theoretic model of an algebraic theory gives rise to notion of (algebraically) "closed subset" in a canonical fashion; namely, the closed subsets are those that cannot be escaped via the ...
1
vote
0answers
17 views

Does the notion of “commutative algebraic theory” generalize to many-sorted signatures?

I think that the notion of commutative algebraic theory only makes sense for unisorted signatures, and cannot sensibly be generalized to the case of more than one sort. Does anyone know of a ...
4
votes
1answer
97 views

How does this definition capture the intuitive notion of an algebra?

On page 15 of this document, the author writes: Definition 1.1.1. Let $\mathcal{E}$ be any category. Given an endofunctor $\Gamma : \mathcal{E} \rightarrow \mathcal{E}$, a $\Gamma$-algebra ...