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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Understanding Quotienting by Relations vs Quotienting by Generators

I understand the idea of a quotient algebra $A / I$ where $A$ is a $K$-algebra and $I$ is a two-sided ideal, i.e. I understand the projection map as an algebra morphism. However, I'm unsure about how ...
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1answer
31 views

No simplifying identities for any single nonzero number under addition.

Consider the structure $(\mathbb{R}, +, r)$, where r is a nonzero real number. Are the commutative and associative identities already sufficient to derive all universally valid equations in that ...
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85 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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167 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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209 views

$M_3$ is a simple lattice

I'd like to prove (exercise 9.5 in Roman's Lattices and Ordered Sets, p.203) that the lattice $M_3$ is simple, meaning that the only congruences on $M_3$ are the trivial ones (the 'equality' ...
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52 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 ...
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333 views

Lattices are congruence-distributive

$\newcommand{\r}[1]{\mathrel{#1}}$ First, a few definitions. Given a lattice $L$, a congruence on $L$ is an equivalence relation $\theta$, compatible with the lattice operations, i.e. if ...
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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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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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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 $\mathbf{Tos}$ ...
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114 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 ...
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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 ...
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29 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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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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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) $\wedge$ ...
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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. ...
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52 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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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 ...
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111 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: $\forall a,b \in T: a \cap b \in T$, ...
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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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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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76 views

A conjecture in equational logic

In an algebra with a single binary operation g, is there a single equational identity that generates the same set of identities as the set {g(x,y)=g(y,x) , g(g(x,y),z)=g(x,g(y,z))}? My conjecture is ...
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43 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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30 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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41 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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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 ...
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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
35 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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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 ...
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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 ...
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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 ...
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59 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 $\ cong(R)‎\cong‎ Id(R)$?
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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 ...
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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 ...
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0answers
78 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 ...
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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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1answer
299 views

Combinatorics of term algebras

My question is about the number of terms of size $n$ in term algebras for an arbitrary (finite) signature. A signature is a map $\Sigma : S \rightarrow \mathbb{N}$ from a set $S$ of symbols. We ...
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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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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 ...
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2answers
278 views

Difference between abstract algebra and universal algebra

Wikipedia give this answer "Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic ...
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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 ...
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1answer
56 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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45 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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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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6answers
9k views

In (relatively) simple words: What is an inverse limit?

I am a set theorist in my orientation, and while I did take a few courses that brushed upon categorical and algebraic constructions, one has always eluded me. The inverse limit. I tried to ask one of ...
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2answers
195 views

Is there a concept of a “free Hilbert space on a set”?

I am looking for a "good" definition of a Hilbert space with a distinct orthonormal basis (in the Hilbert space sense) such that each basis element corresponds to an element of a given set $X$. Before ...