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

Uppercase E notation for sets?

In Jónsson and Tarski's (1951) paper Boolean Algebras with Operators, Part I from the American Journal of Mathematics, they write formulae such as $L_i = \underset{u}{\mathbf{E}} \, [u \in At^m ...
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1answer
51 views

Finding the congruences of a lattice

Part of the excercise I am currently doing is finding the congruences of the following lattice: The problem I struggle with the most is what happends when $1 \sim d$ - how to find what is the ...
6
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0answers
71 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 ...
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1answer
53 views

Algebraic lattices and distributivity over joins of upward directed sets

I am reading Burris & Sankappanavar, Chapter 1 on lattices, and I am doing Exercise 6 in Section §4: If $L$ is an algebraic lattice and $D$ a subset of $L$ such that for each $d_1$, $d_2 \in D$ ...
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0answers
30 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 $\mathbf{Tos}$ ...
4
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1answer
113 views

How can you actually do universal algebra with monads?

Instead of digging deep into "classical" universal algebra, it seems more interesting or fruitful to look at universal algebra categorically. This should be doable with monads, since every category of ...
3
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1answer
28 views

Prime ideal theorem for modular lattices?

There's a well-known theorem for distributive lattices commonly referred to as the "prime ideal theorem:" Let $L$ be a distributive lattice, $I$ an ideal of $L$, and $F$ a filter of $L$ such that ...
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1answer
50 views

Vector spaces as free algebras

Exercise 4.6 of An algebraic introduction to mathematical logic asks: $K$ is a field. Show that vector spaces over $K$ form a variety $V$ of algebras, and that every space over $K$ is a free algebra ...
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53 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. ...
5
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1answer
80 views

Does a power-complete finite pasture exist?

Suppose we define a pasture to be an algebraic structure $\langle M, 0, +, \times, \wedge \rangle$ where $\langle M, 0, +, \times \rangle$ is a ring (not necessarily commutative or unital) $\wedge$ ...
12
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1answer
75 views

Aside from the obvious stuff, do the partial functions that solve the quadratic equation have any interesting properties?

Let us define partial functions $$f_+,f_- : \mathbb{R} \leftarrow \mathbb{R} \times \mathbb{R} \times \mathbb{R}$$ so as to return the zeros of the quadratic $ax^2+bx+c$ whenever they exist, such ...
2
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1answer
51 views

A generalized Boolean algebra gives rise to an implication algebra

A generalized Boolean algebra $G$ is relatively complemented distributive lattice with largest element 1. That is, an element $a\in G$ has a complement in any interval $[x\,,\,1]$ that contains ...
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40 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 ...
5
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2answers
110 views

Classification of Finite Topologies

Does there exist a classification of finite topologies? I define a finite topology as a finite Set $T$ of Sets which respects the following properties: $\forall a,b \in T: a \cap b \in T$, ...
4
votes
1answer
73 views

What does it mean if a free algebra has an unsolvable word problem?

I wonder how hard identity testing (similar to polynomial identity testing) can be for a free algebra. I thought that in a certain sense, the problem should always be semi-decidable, because the free ...
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0answers
42 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... ...
0
votes
1answer
29 views

Free commutative ring functor

The free commutative ring on a set $X$ is the polynomial ring with variables the elements of $X$. This polynomial ring is the free (additive) abelian group on the free (multiplicative) abelian monoid ...
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0answers
29 views

Coproduct in the category of $\Omega$-algebras [duplicate]

Let $\Omega$ be a type (or signature, depending on your terminology), and let $\mathbf {\Omega Alg}$ be the category of $\Omega$-algebras. What is the coproduct in this category? G. Bergman's "An ...
2
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1answer
39 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 ...
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3answers
52 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
30 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
33 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
61 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 ...
4
votes
1answer
44 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 ...
6
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1answer
58 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 ...
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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
43 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
votes
1answer
34 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
117 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 ...
9
votes
0answers
77 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
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 ...
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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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0answers
35 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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27 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
84 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
55 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 : ...
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0answers
42 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
27 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 $a \lor a^\perp=1$ ...
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0answers
24 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
24 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
votes
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
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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1answer
56 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 ...
2
votes
1answer
96 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 ...
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54 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
169 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
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1answer
83 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
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1answer
53 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? ...
4
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1answer
108 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
77 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 ...