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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84 views

In which algebraic theories do 'free' and 'projective' coincide?

Free models of algebraic theories are always projective objects in the category of models, but the converse is not always true. For instance, some (actually, all) projective modules are direct ...
6
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107 views

Most general form of Cayley's theorem?

For many classes of algebraic structures, there exists a family of structures such that any member of the class can be embedded in some member of the family (groups and symmetric groups, unital rings ...
6
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74 views

cofree coalgebra: explicit description from its universal property

Let $k$ be a commutative ring. There is a forgetful functor $$U : \mathsf{Coalg}_k \to \mathsf{Mod}_k$$ from $k$-coalgebras to $k$-modules. This has a right adjoint $R : \mathsf{Mod}_k \to ...
6
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82 views

Do torsionfree abelian groups form a (possibly many-sorted) algebraic category?

By "(possibly many-sorted) algebraic category", I mean "category of models of a (possibly many sorted) Lawvere theory. I have arguments for both affirmative and negative answers, and I can't find a ...
5
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100 views

Functorial approach to Ideals and Quotients, Multiplicative Sets and Localizations

I have been playing with substructures of commutative rings today and noticed that there is a strong analogy between the formation of quotients and kernels with the formation of localizations with ...
5
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52 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 ...
3
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55 views

Is there a systematic way of “discovering” an algebra from observations of its universe?

I am faced with the following situation: I have a finite set of some $m$ positive integers $Q^m \in \mathbb{N}$ These integers go through a series of $N$ possible black boxes that transform them. ...
3
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65 views

When do (universal-algebraic) varieties have (enough) injectives?

Following Zhen Lin's suggestion in my previous question, I ask this question separately. Suppose $\mathcal{V}$ is a variety in the sense of universal algebra. considered as a category. When does ...
3
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94 views

Prove that $\Pi_{i \in I} \mathbf{A}_i \cong \Pi_{j \in J} (\Pi_{i \in I_j} \mathbf{A}_i)$ where $\{I_j \ | \ j \in J\}$ is a partition of I.

My problem is following: prove that $\Pi_{i \in I} \mathbf{A}_i \cong \Pi_{j \in J} (\Pi_{i \in I_j} \mathbf{A}_i)$, where $\langle \mathbf{A}_i \ | \ i \in I \rangle$ is an indexed set of similar ...
3
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0answers
51 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 ...
3
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90 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 ...
2
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30 views

Definition of finitely generated model

Let $M$ be a model of an algebraic theory $\mathcal T$. $M$ is said to be finitely generated if it is the quotient of a free model $F(n)$ over a finite set $n$. Here, quotient means there exists a ...
2
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32 views

Clone of operations acting bicentrally

Let $(A,F)$ be an algebra, let $F^{\star}$ be the centralizer of $F$ and $F^{\star\star}:=(F^{\star})^{\star}$ the bi-centralizer. Let $[F]$ denote the clone of operations generated by $F$. We say ...
2
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0answers
62 views

Name for classes of algebras closed under products and quotients

A class of algebras closed under products, quotetiens and subalgebras is a variety. Is there a name for a class of algebras closed under products and quotients? Could you refer me to any theorems ...
2
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262 views

Algebraic theory and definition of multiplication: tensor product v.s. Cartesian product

In a monoidal category, one can define multiplication on a object $M$ as a morphism \begin{equation} M\otimes M\longrightarrow M, \end{equation} or a morphism \begin{equation} M\times ...
2
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0answers
91 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 ...
2
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94 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) $f_{i}(a_0,\dots,a_{n-1}) \notin M$ ...
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26 views

Non-roots of unity auxillary constants in a group?

Let $A$ be a set, together with a set $F$ of n-ary operations on A, which may include constants of $A$ as 0-ary operations. A set $G$ of operations on $A$ is said to be auxillary with respect to the ...
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0answers
23 views

what can we say if we just know the global section has a given universal algebra structure?

Given an universal algebra A, how to reconstruct a topological space X with some universal (and natural) property (e.g initial,terminal...)in the category of topological spaces whose global sections ...
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31 views

Does $\mathbf{Tos}$ generate $\mathbf{DLat}$ as a variety?

Let $\mathbf{DLat}$ denote the variety of distributive lattices and let $\mathbf{Tos}$ denote the subclass of $\mathbf{DLat}$ consisting of the totally-ordered sets. Question. Does $\mathbf{Tos}$ ...
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0answers
41 views

Optimal object in category? Metric, objective function on category objects. Optimization over category?

Is there notion about optimal object in category (that can be found by some algorithms, or - more importantly - that can be constructed (if unknown) by some algorithms), about metric and objective ...
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63 views

Existential and Universal Equivalence Proof

Taking ¬∀x:X.r≡∃x:X.¬r Is there a way of actually formally proving this? Not implementing it but proving how to go from a negated universal quantifier to a an existential with a negated element... ...
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0answers
35 views

Are there any preservation theorems for quotients of subalgebras?

Let $X$ denote an algebraic structure. Then: Every subalgebra of $X$ satisfies each quasi-identity that is satisfied by $X$. In other words, taking subalgebras preserves quasi-identities. Every ...
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28 views

What is the standard term for the property of an order-embedding $h \colon P \to P$ such that for all $a \in P$, $a \leq h(a)$?

In an essay, I want to talk about order-embeddings $h \colon P \to P$ on a partially ordered set P such that for all $a \in P$, $a \leq h(a)$. Does somebody know the standard term for a function ...
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46 views

Question on free Boolean algebras

Every Boolean algebra $A$ is isomorphic to a field of set. In particular, if $A$ is finite, then $A$ is isomorphic to the power set of its atoms. Now, suppose that $A$ is free Boolean algebra with 2 ...
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25 views

Example of relatively free lattice

"A relatively free lattice with $n$ generators has exactly $n$ maximal sublattices, each obtained by removing one of the generators (which are doubly irreducible). Thus there exist arbitrarily large ...
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31 views

Question about “immediate” observation about finitely presentable objects

The following is an excerpt from volume II of Borceux: Here $F$ is the left adjoint to the forgetful functor $U:\mathsf{Mod}_\mathcal{T}\longrightarrow \mathsf{Set}$. I'm confused and can't manage ...
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0answers
58 views

Generators of free Boolean algebras

Suppose $\mathfrak{A}$ is a free Boolean algebra and $G$ a countable set of free generators of $\mathfrak{A}$. What is the cardinality of $\mathfrak{A}$ if $G$ is countably infinite, but we only ...
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0answers
58 views

Universal Algebra: partial algebras and homomorphisms

I'm looking for an accessible introductory text on universal algebra, one which discusses homomorphisms in the context of partial algebras. I have been recommended the Grätzer book, "Universal ...
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0answers
144 views

Definition of $\Omega$-algebra

I'm studying universal algebra. I have this definiton: given a signature $\Omega$, an $\Omega$-algebra is comprised of a "carrier" for the algebra and an "interpretation" for every operation symbol. ...
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128 views

Is this a general structure for constructs?

Here a construct is a category where the objects are sets and the morphisms are structure preserving functions. Common examples are groups, graphs and topological spaces. As far as I can see there is ...
1
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0answers
46 views

Is there accepted terminology for algebraic structures whose every subalgebra is free?

Is there accepted terminology for algebraic structures whose every subalgebra is free? Examples: Any free group Any vector space More generally, any free module over a PID. In fact, this ...
1
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0answers
79 views

Congruence lattice of $N_5$

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
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0answers
32 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 ...
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0answers
163 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
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0answers
57 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
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0answers
81 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 $A$ ...
1
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0answers
73 views

Which “conditions” generate subalgebras?

While looking at this question I suddenly wondered about a more general question. Think of your favorite class of algebra (groups or rings, say), and forgive me for vocabulary mistakes while I try to ...
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0answers
81 views

Another way of saying that algebraic objects are isomorphic

From a universal algebraic perspective, let's say we have two isomorphic groups. Then can I speak of their isomorphic nature by saying the binary operations of multiplication of the two groups are ...
1
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0answers
87 views

direct product of different algebras?

Is it possible to define a "direct product" of two algebras with different signatures? For example, boolean lattice $\otimes$ monoid? Perhaps we need to take some quotient to make sense of it (e.g Q ...
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0answers
210 views

amalgam of structures

Trying to refine my question here. This is a response to the questions here: Homomorphisms between structures My objective is to take a set of $S-$structures and form an amalgam object out of that ...
0
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0answers
24 views

It has at most one absorbing element

Let $G = (M, \circ )$ be a groupoid and let $2^G = (2^M, \circ_K)$ a groupoid ( $\circ_K$ is the Product of group subsets). How can show that $G$ has at most one absorbing element?
0
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0answers
64 views

Equivalence of two definitions of complete distributivity

I would like to know if the following alternative definitions of complete distributivity are equivalent. Let's begin with defining choice functions: For any set $S$ and $U\in ...
0
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0answers
38 views

Homomorphism and similarity

Does a homomorphism from one Boolean algebra $A$ to another Boolean algebra $B$ necessarily makes the algebra $B$ similar to $A$?
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30 views

Regarding absolutely free structures

Suppose that you have a signature $\mathfrak{F}$ containing at least one constant symbol $f$ of arity $0$. How does the absolutely free algebra (genrated by $X$) interpret this symbol? I know that the ...
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0answers
19 views

Definition of (minimal) domain?

Consider the following links: http://www.glottopedia.org/index.php/Domain_%28Syntax%29 http://www2.let.uu.nl/uil-ots/lexicon/zoek.pl?lemma=Minimal+domain&lemmacode=542 What kind of mathematical ...
0
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0answers
41 views

Where can I find Kan's paper “On c.s.s. categories” from 1957.

Does anyone know how I can find the following article by Daniel Kan: Kan, D. M. On c.s.s. categories Bull. Soc. Math. Mexicana (1957), 82-94. Quillen lists it as a reference in his paper Rational ...
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0answers
51 views

Algebraic systems

Are there any books about algebraic systems without having Mal'cev book? Are there books in general about the varieties and quasi-varieties?
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91 views

An example of a free algebra in SP(K)

The following theorem is found in the book "Universal Algebra, Fundamentals and Selected Topics" by Clifford Bergman (pp.98). Theorem 4.28. Let $U$ be free for $K$ over $X$. Then, $U/\lambda_k$ is ...
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103 views

A Course in Universal Algebra (Millennium edition), page 74

The line before Theorem 10.12 says that "In general $F_K(\overline{X})$ is not isomorphic to a member of K (for example, let K={L} where L is a two-element lattice, then $F_K(\bar{x}, \bar{y}) \notin ...