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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 ...
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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 $$ ...
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1answer
44 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
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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 ...
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1answer
80 views

What are the properties of this Poisson algebra?

I have the following (real) quantities (which are from a Classical Mechanics problem): $$A_1=\frac 1 4(x^2 +p_x^2-y^2-p_y^2 ) \quad A_2=\frac 1 2(x y +p_x p_y)$$ $$A_3=\frac 1 2(x p_y - y p_x )$$ ...
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0answers
17 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.
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0answers
142 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 ...
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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
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1answer
79 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 ...
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2answers
157 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
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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 ...
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4answers
214 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 ...
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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 ...
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1answer
49 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 ...
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1answer
58 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
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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
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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 ...
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2answers
58 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
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1answer
83 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
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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
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0answers
62 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
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1answer
27 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
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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
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1answer
57 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 ...
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1answer
113 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 ...
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1answer
55 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 $x$.)
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2answers
49 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
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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
73 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
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1answer
55 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 ...
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1answer
21 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
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2answers
66 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 ...
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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
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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 ...
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2answers
86 views

For what categories do category algebras exist?

The monoid algebra $R[M]$, for a commutative ring $R$ and a monoid $M$, can be described as the free $R$-algebra on $M$. We think of $R[M]$ as the set of finite formal sums of elements of $M$ with ...
3
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2answers
124 views

Should axioms be viewed as part of the signature?

I included category theory in the tags in order to get feedback from the categorial logic community. It goes without saying that this isn't really category theory. A semigroup can be defined as a ...
0
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1answer
51 views

Basic question about the definition of a variety (in universal algebra)

According to wikipedia, A variety is a class of algebraic structures of the same signature that is closed under the taking of homomorphic images, subalgebras and (direct) products. Isn't ...
4
votes
1answer
47 views

Which sentences survive the passage from $X$ to the set of all functions $I \rightarrow X$?

Suppose $X$ is a mathematical structure with a single underlying set which we will also denote $X$, equipped with some functions and relations. Letting $I$ denote an arbitrary non-empty set, we see ...
0
votes
1answer
56 views

Relation between quotients and subalgebras

If I have two algebras $A,B$, and one is the quotient of the other, i.e. there exists a surjective morphism $\phi : A \to B$. Then is $B$ isomorphic to some subalgebra of $A$? I think so, because I ...
3
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1answer
82 views

Completing a Partially Defined Associative Binary Operation

This is more like a question about terminology. I would like to hear some recommendations of books that discuss algebraic structures with one partially defined associative binary operation, and the ...
3
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0answers
86 views

What big families of theories/structures are there?

To gain a big picture of (pre-categorical) mathematics, is it correct to divide mathematical theories resp. structures in two big families? universal algebra: classes of objects with arbitrary ...
1
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0answers
70 views

Congruence lattice of a partial algebra is algebraic

A partial operation on a nonempty set $A$ is a map $f:\mathrm{dom}(f,A)\to A$ where $\mathrm{dom}(f,A)\subseteq A^n$ for some $n\in\mathbb{N}$. A partial algebra is an ordered pair $(A,P)$ where $A$ ...
2
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2answers
169 views

Isomorphism of algebras $\mathbb{Q}(\cdot , +, 1)$ and $\mathbb{Z}(\cdot , +, 1)$

I have these two algebras and I need to know if they are isomorphic: $\mathbb{Q}(\cdot , +, 1)$ and $\mathbb{Z}(\cdot , +, 1)$ Are there some general tricks how to deal with this type of tasks?
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2answers
55 views

Notation for “incommensurate” elements?

Say that $x\wedge y\not=x,y$, that is neither $x\leq y$ nor $y\leq x$. Is there a way to denote this? I've been saying $x<>y$ but that's completely made up.
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1answer
232 views

What is a subdirect product?

I'm having trouble understanding what a subdirect product is. Say $G$ is a subdirect product of $H=\prod H_i$ - this means that the homomorphisms $f_i:G\to H_i$ are surjective, which can be ...
2
votes
1answer
31 views

Local smallness of Lawvere theories

Reading this blog post, I'm trying to care about foundational matters. To summarize the first part of the article, living in a univers $\mathcal V$ of sets, one defines a Lawvere theory as follow : ...
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2answers
107 views

The kernel of the kernel.

From Wikipedia-Entry on Equivalence Relatin:Lattices The possible equivalence relations on any set X, when ordered by set inclusion, form a complete lattice, called Con X by convention. The ...
2
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1answer
85 views

Does axiomatizability in zeroth-order logic have important consequences?

If a theory is equationally axiomatizable, this has important consequences (that are studied e.g. in universal algebra). However, many theories fail to be equationally axiomatizable - examples ...
2
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0answers
61 views

A subalgebra of $P(M, \Delta)$ is a Peano Algebra

Let $M$ be any set and let $\Delta=(n_i)_{i \in I}$ be an algebraic type. Let $P=P(M, \Delta)$ be the algebra such that the generalized Peano Axioms hold: (P1) $f_{i}(a_0,\dots,a_{n-1}) \notin M$ ...