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).

learn more… | top users | synonyms

1
vote
0answers
32 views

Do you lose any more equational identities when you go past sedenions? [duplicate]

Every Cayley-Dickson algebra can be viewed as a $(+,-,*,0,1)$ algebra. The reals and the complexes share the same equational identities. The quaternions have a subset of the equational identities, ...
1
vote
0answers
17 views

Citations for the proof of universality of graph classes

In Automorphisms of graphs, Peter J. Cameron mentioned following classes of graphs which are universal structures. graphs of valency k for any fixed k > 2; bipartite graphs; strongly regular graphs; ...
1
vote
0answers
18 views

$C^*$-algebra desription of the complex Clifford algebra

I read somewhere a discription of the complex Clifford algebra as a $C^*$-algebra, but I don't know where... Is the complex Clifford algebra the universal $C^*$-algebra generated by elements $1$ and $...
3
votes
2answers
67 views

What is the smallest variety containing all monoids and all semigroups with a one-sided zero?

What is the smallest variety (in the universal algebra sense) containing all monoids, all semigroups with a left zero, all semigroups with a right zero, and as few other models as possible? So far, ...
0
votes
1answer
40 views

Can we do representation theory for algebras with >2 operations?

Suppose I define an algebra with 3 or more operations, perhaps using universal algebra. Would it be meaningful to talk about the representation theory of this algebra? In particular, I am interested ...
5
votes
2answers
75 views

What are permutable equivalence relations intuitively?

What are permutable equivalence relations, and what are they used for? What is the idea behind them? Could someone give me an example and a counterexample for finite sets? I have encountered the ...
1
vote
2answers
39 views

Finite generating sets and finitely-generated modules

(All my rings are commutative with $1$.) Reworded a little, a question in a previous Commutative Algebra exam goes like this: Let $A$ denote a ring, $X$ denote an $A$-module $F$ ...
3
votes
1answer
45 views

“One cannot hope to find any further essentially new lattice properties…”

I found the following passages in “A Course in Universal Algebra” by Burris and Sankappanavar. One cannot hope to find any further essentially new lattice properties which hold for the class of ...
0
votes
0answers
31 views

Does this algebra whose signature is (1,1) have a name?

Let there be an algebra $(S,f,t)$ with the laws: $$ f(t(x)) = t(x) \\ t(f(x)) = t(x) $$ or, put another way, $$ f \circ t = t \\ t \circ f = t. $$ Does that particular algebra have a name? Does a ...
0
votes
1answer
15 views

Not congruence-permutable lattice

What is an example of a lattice having congruences that don't permute. Equivalently, a lattice $L$ such that there exist $\theta, \sigma \in Con L$ for which $\theta \vee \sigma = \theta \circ \sigma$...
1
vote
1answer
35 views

Cokernel in universal algebra

Let $(S,f_1,\ldots,f_n)$ be an algebra of some variety and $(T,g_1,\ldots,g_n)$ be another algebra of the same variety. Next let $\varphi:S\to T$ be a homomorphism. I understand well that $\ker\varphi=...
1
vote
0answers
48 views

Slick Definition of the Category of Cartesian Closed Categories

I can produce elementary definitions by just inspecting the definition on nlab, but is there a readily available abstract definition? I vaguely remember seeing that they could be defined as algebras ...
3
votes
1answer
128 views

Is there some kind of deep relationship between substitution and recursion?

Define $\mathbb{N}$ as the initial object in the following category: Objects. Sets $X$ equipped with a function $S : X \rightarrow X$ and an element $0:X$. Morphisms. Functions that preserves ...
6
votes
1answer
56 views

If finitely many algebraic identities do not imply some identity, is there always a finite counterexample?

This question just popped up while experimenting with Prover9 and Mace4. Say we have a finite signature and some finite set of identities $E_i$ in the sense of universal algebra, like the axioms for ...
1
vote
2answers
157 views

A categorical first isomorphism theorem

It is known, that for a morphism of universal algebras $f : A \to B$, if $R$ is the congruence relation given by $xRy \Leftrightarrow fx=fy$, then $\operatorname{im} f \cong A/R $. Here is an idea ...
3
votes
1answer
48 views

Does equivalence of algebraic categories imply bi-interpratibility of their theories?

By an algebraic theory $\mathcal{T}$ I mean any category with finite products such that the objects are given by all finite powers of some object $X$. Let $Alg\mathcal{T}$ be the concrete category of ...
4
votes
1answer
47 views

Kernel cokernel correspondence?

On page 367 of Categorical Foundations: Special Topics in Order, Topology, Algebra, and Sheaf Theory there is mentioned a kernel co-kernel correpsondence, which says there's an equivalence between ...
0
votes
3answers
53 views

Question about universal quantifier

when I was reading a paper about the universal quantifier, I met this equation, says we can do conversions like the following: A -> B ≡ ¬A v B can anyone help ...
3
votes
1answer
39 views

Sufficient conditions for the category of group objects to have coproducts

For a category $\mathbf{C}$ with finite products, denote by $\mathbf{C}_{\text{Grp}}$ the category of group objects in $\mathbf{C}$. Using the fact that $G\in \operatorname{Obj}(\mathbf{C})$ is a ...
1
vote
0answers
27 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 ...
1
vote
1answer
33 views

finite presentations of algebraic structures

I've read an algebraic variety $A$ is finitely presented if there's a coequalizer diagram $$F(m)\rightrightarrows F(n)\twoheadrightarrow A$$ where $F(n)$ is the free model on $n$ generators. I'm ...
2
votes
2answers
53 views

How, intuitively, does commuting with filtered colimits capture “smallness”?

Definition. A compact object is an object representing a copresheaf which commutes with filtered colimits. In algebraic categories, the compact objects are the finitely presented ones, so commuting ...
7
votes
3answers
123 views

What is the importance of “variety of algebras” in Universal Algebra?

Given an algebraic category, Birkhoff's Variety Theorem gives a categorical characterization of the full subcategories whose object-class forms a variety (i.e. can be defined by equations in the sense ...
5
votes
3answers
49 views

Constants in a signature

This is my first post so I hope it works! Taking the axioms for a group as an example, the literature defines a group in (at least) two different ways: Method 1 A signature of $(G,\circ,\,^{-1})$ ...
1
vote
1answer
29 views

Finite algebraic structure where there is no finite generating set of equations

Let $A$ be an algebra whose carrier set is finite. Must it be the case that there is a finite set of equations which generate all the universally valid equations in that structure? If not, can anyone ...
1
vote
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 ...
6
votes
0answers
112 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 ...
3
votes
1answer
75 views

Equational identities of real multiplication augmented by a real number

Consider the structure $(\mathbb R, *, r)$, where $r$ is a real number that is neither $0$, $1$, or $-1$. Are the commutative and associative identities already sufficient to derive all the ...
4
votes
1answer
62 views

Is groups with binary operation alone a variety?

In the signature (+, 0, -), the class of groups are a variety, because they can be defined by a set of universal equations. But is it already a variety in the signature (+), by itself? The more ...
1
vote
1answer
38 views

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 ...
5
votes
1answer
44 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 ...
3
votes
1answer
125 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 \text{...
0
votes
0answers
27 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?
1
vote
1answer
60 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
votes
0answers
76 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 \mathsf{...
0
votes
1answer
64 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 $d_1$, $d_2 \in D$ ...
1
vote
0answers
32 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
votes
2answers
143 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
votes
1answer
34 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 $...
1
vote
1answer
55 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 ...
3
votes
0answers
56 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
votes
1answer
83 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
votes
1answer
87 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
votes
1answer
60 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 $a$. ...
1
vote
0answers
42 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 ...
8
votes
2answers
133 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
75 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 ...
1
vote
0answers
81 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
35 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 ...
1
vote
0answers
31 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 ...