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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48 views

'Finitely presented' implies 'always finite presented' for algebraic theories

In this MO question it is proven the answer is yes for modules. The proof given relies on the snake lemma, which does not generally make sense in the category of rings, groups, monoids, etc. It seems ...
1
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3answers
54 views

What does it mean that in a factor/quotient group certain elements get “glued” together

In these these notes on the generalized quaternion group it is written that: [...] $Q_{2^n}$ is made by taking a cyclic group of order $2^{n-1}$ and a cyclic group of order $4$ and "glueing" them ...
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1answer
32 views

Are projectives always the retracts of free objects in algebraic theories?

Is it always true that projective objects are retracts of free objects? I know that retracts of projective objects are always projective, so in particular, retracts of free objects are projective. To ...
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1answer
50 views

Are finitely generated projective models of an algebraic theory always finitely presented?

I know that for modules over rings, a finitely generated projective module is finitely presented. I was wondering whether this holds in full generality for algebraic theories, and if not, which parts ...
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0answers
67 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 \mathcal{P}(\mathcal{P}(...
4
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1answer
45 views

Every model of an algebraic theory is a quotient of a free model

I have stumbled upon the following proposition from Borceux: Proposition 3.8.9 Let $\mathcal{T}$ be an algebraic theory. Every $\mathcal{T}$-model $M$ is a quotient of a free model. More precisely,...
6
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1answer
81 views

Are pseudoheaps and heaps the same thing?

An exercise in a category textbook asked me to show that the category of pointed heaps and the category of groups are isomorphic. But my proof somehow didn't use the most unintuitive of the defining ...
1
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1answer
23 views

How can I prove that $\ cong(R)‎\cong‎ Id(R)$?

If $R$ is an arbitrary ring, $\ cong(R)$ is the set of all congruence of $R$ and $Id(R)$ is the set of all Ideals of $R$, How can I prove that $\ cong(R)‎\cong‎ Id(R)$?
2
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1answer
48 views

Nonexistence of infinite subdirectly irreducible algebras

I am trying to prove a theorem of Quackenbush (Theorem 3.8 in Chapter V of Burris & Sankappanavar): If $V$ is a locally finite variety with only finitely many finite subdirectly irreducible ...
2
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1answer
36 views

Image-preimage adjunction induced from regular epi respects regular monos?

Concretely, every set function $f:A\rightarrow B$ induces an adjunction $f_\ast \dashv f^\ast$ between image and preimage. For rings groups, the image and preimage along a surjective ring group ...
8
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1answer
119 views

What equational properties of a group only need to be checked on a generating set?

Let $G$ be a group and $S\subset G$ a generating set. Let $P$ (short for $P(x_1,\dots,x_n) = 1$) be an equational property that may or may not be satisfied by all $n$-tuples of elements of $G$. My ...
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0answers
90 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 ...
2
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0answers
31 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 ...
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0answers
36 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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0answers
29 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 ...
3
votes
1answer
88 views

What properties are shared by isomorphic universal algebras?

There is a consensus, that "isomorphy" (the "is isomorphic to"-relation) is the right kind of sameness between universal algebras (say groups, (single-sorted) vector spaces, lattices, ...), because ...
2
votes
1answer
68 views

Quotient structures and “Lagrange” like formulas as for groups and subgroups

Before I formulate my question, I give two examples to motivate it: i) Given a group homomorphism $\varphi : G \to H$ betwenn finite groups the kernel $N := \mbox{ker}(\varphi) = \{ g \in G : \varphi(...
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0answers
47 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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1answer
28 views

Nullary and unary operations defined on a uniquely complemented lattice?

A lattice is a set $L$ with two binary operations, $\lor$ "join" and $\land$ "meet". In a complemented lattice, for every element $a$ there exists an element $a^{\perp}$ such that $a \lor a^\perp=1$ ...
0
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0answers
32 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
27 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 ...
6
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1answer
87 views

Is it possible to delete undesired identifications in algebraic structures?

In algebraic topology, there is a notion of covering space, which essentially "de-identifies" points that look the same but which for certain purposes really shouldn't be considered the same. I was ...
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0answers
33 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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1answer
57 views

Proof forgetful functor $U:\mathsf{Mod}_\mathcal{T}\rightarrow \mathsf{Set}$ has a left adjoint

I'm confused by a proof in volume II of Borceux. First, two facts: Proposition 3.3.3 $\;\;\;$ Let $\mathsf{T}$ be an algebraic theory. Consider the functor $$U:\mathsf{Mod}_\mathcal{T}\...
2
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1answer
98 views

Why are algebras classified as being of a certain type?

In Grätzer's, Universal Algebra, page 33, Grätzer defines the concept of an algebra of type $\tau$, as follows: An algebra $\mathfrak{A} = \langle A ; F \rangle$ of type $\tau$ is a pair, where $A$...
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0answers
61 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 ...
5
votes
1answer
181 views

What can we actually do with congruence relations, specifically?

Let $T$ denote a Lawvere theory and $X$ denote a $T$-algebra. Under my preferred definitions: A subalgebra of $X$ consists of a $T$-algebra $Y$ together with an injective homomorphism $Y \...
7
votes
1answer
89 views

This is just the Eilenberg-Moore category, right?

Let $F$ denote a monad on $\mathbf{Set}$. Write $\mathbf{Set}_F$ for the corresponding Kleisli category and $\mathbf{Set}^F$ for the Eilenberg-Moore category. On the train home today, it occured to me ...
0
votes
1answer
55 views

Polynomial algebras

What is a polynomial algebra? I cannot find any definition of this concept. Is it a set of n-ary polynomial symbols defined over an algebra, as defined on p.29 of this document? http://www.illc.uva....
4
votes
1answer
117 views

Are there any free or fascist boolean algebras?

A boolean algebra is an algebra with the binary operations $\wedge$, $\vee$, an unary operation $\neg$, and constants $0$, and $1$, satisfying axioms. A heyting algebra is an algebra with the binary ...
2
votes
1answer
86 views

Free idempotent semigroup with 3 generators

On Finite free objects I asked examples of finite free objects. I got answer "The free idempotent semigroup satisfies $x^2=x$ for each element $x$. And I confirm it is finite if it is finitely ...
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3answers
79 views

Finite free objects [closed]

Free distributive lattice with any number of generators is finite. For example with 3 generator the lattice will have 20 elements. Is there other examples of free objects that are finite and have at ...
1
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0answers
59 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 ...
0
votes
1answer
87 views

Terminological conventions regarding group actions

Suppose: $G$ is a group $T$ is a Lawvere theory ("algebraic theory") and $X$ is a $T$-algebra What conventions surround the phrase: "action of $G$ on $X$"? In particular, does this mean: a ...
4
votes
1answer
95 views

Do the usual examples of “dimensive” Lawvere theories form an exhaustive list of such theories?

Call a Lawvere theory $T$ dimensive iff, letting $F_T : \mathbf{Set} \rightarrow \mathbf{Mod}(T)$ denote the free functor, we have the following. Every finitely generated $T$-algebra is free. From $...
3
votes
2answers
72 views

why only closed operations

Why does the carrier of an algebraic structure has to be closed under the operations of the algebraic structure? One could also consider $(\mathbb{N}^*, \div)$. But why isn't that an algebraic ...
4
votes
1answer
58 views

Can you find a plain aneloid?

I defined an "aneloid" to be a set endowed with two operations, adition and multiplication, with multiplication being distributive BOTH sides in relation to adition. I tried to find an example of "...
6
votes
1answer
67 views

Binomial rings closed under colimits?

A binomial ring is a ring (for the purposes of this question all rings are commutative and unital) which is torsion-free and has, for each $n$, a binomial function $\binom{x}{n}$ satisfying $n!\binom{...
4
votes
1answer
51 views

The identity elements of two multiplication satisfying interchange law.

Let $M$ be a set, and $\circ_1,\circ_2$ two binary operations defined on $M$ satisfying that both $(M,\circ_1),(M,\circ_2)$ are semigroups and $(a\circ_1 b)\circ_2(c\circ_1 d)=(a\circ_2 c)\circ_1(b\...
6
votes
0answers
86 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 ...
2
votes
1answer
63 views

An Exercise From Universal Algebra and Coalgebra by Denecke and Wismath

I'm going through this book right now, and I don't understand something about exercise 9 from chapter 1: Determine all elements of t he free algebra $F_V(X_2)$ for the following varieties: (i) ...
3
votes
1answer
31 views

Ternary algebra satisfying some identities is a join-semilattice

A join-semilattice with greatest element is an algebra $(S,\vee, 1)$ of type $(2,0)$ such that $\vee$ is idempotent, commutative, and associative, and $a\vee 1=1$ for all $a\in A$. Now, let $(A,m,1)$...
3
votes
1answer
63 views

A function symbol with more than one arity schema and type assigned to it in a signature

I am studding many-sorted algebra. In this paper (page 4), it is clearly said that in the signature $(S, \leq , \Sigma)$, $\Sigma$ is a family $\Sigma=\{ \Sigma_{w,s}\}_{(w,s)\in S^*\times S}$ of (...
4
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1answer
83 views

Is this generalization of Boolean algebras a variety?

I'm interested in the class of structures $\langle S,\top,\neg,\wedge\rangle$ defined by the axioms: $p \wedge q=q \wedge p$ $p \wedge (q \wedge r) = (p\wedge q)\wedge r$ $p = p \wedge p$ $p = \neg\...
2
votes
0answers
33 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 ...
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3answers
73 views

Constructing the groupification of a semigroup (Vakil 1.5.G)?

In the first chapter of Ravi Vakil's Algebraic Geometry notes, he suggests a construction of the groupification $H(S)$ of an abelian semigroup $S$ by considering $S\times S/\sim$ where $(a,b)\sim (c,d)...
7
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1answer
161 views

Distributor? Distributive analog of commutator and associator?

Motivation: "the commutator gives an indication of the extent to which a certain binary operation fails to be commutative" (http://en.wikipedia.org/wiki/Commutator). For example (courtesy of wikipedia)...
0
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1answer
36 views

why $(\theta _{1}\circ \theta _{2})^{\vee }\subset(\theta _{2}\circ \theta _{1})^{\vee } $

my questiones at this theorem: i coud not undrestand $a\Rightarrow b$ and why $(\theta _{1}\circ \theta _{2})^{\vee }\subset(\theta _{2}\circ \theta _{1})^{\vee } $ please guide me?
0
votes
1answer
48 views

how to find $\alpha(a\wedge b)=\alpha(a)\wedge\alpha(b)$

for $\alpha(a\wedge b)=\alpha(a)\wedge\alpha(b)$ how to do? answer: $\alpha $ and $ \alpha^{-1}$ are order-preserving and $a \leqslant b$ and $a=a\wedge b$ so $\alpha(a) = \alpha(a\wedge b)$ so $\...
0
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1answer
50 views

show that N(G) is a lattice?

if $G$ is a group let N(G) be the set of normal subgroups of $G$ define $\wedge$ and $\vee$ on$ N(G)$ by $N_{1}\wedge N_{2}=N_{1}\cap N_{2}$ and $N_{1}\vee N_{2}=N_{1} N_{2}=\{n_{1}n_{2}:n_{1}\in N_{...