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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Algebras with a self-dual congruence lattice

The well known (Mal'tsev) conditions that characterize certain properties of the congruence lattice of an algebra. The existence of a 3-ary term $p$ together with familiar identities $p(x,y,y) \sim x \...
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Mal'cev condition for variety of rings generated by finite fields to be arithmetical

This is an exercise of Burris & Sankappanavar (Universal Algebra), Chapter II, section 12. It asks to prove that, if $V$ is a variety of rings generated by finitely many finite fields, then $V$ is ...
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Examples of Arithmetical Varieties, and Varieties that admit a majority term.

I am looking for examples of varieties of Universal Algebras that admit a ternary majority term $p$. For example, boolean-algebras have such a term: $p(x,y,z) = (x \wedge y)\vee (x \wedge z) \vee (...
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Are Free Groups the “Smallest Group” Containing their Generators

I apologize if this is a duplicate; I was not sure how to search for this. When I say "the smallest group" I mean unique up to isomorphism of course. Specifically, is "the smallest group containing ...
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Is there a finite list of identites in the language of $(\mathbb{N},0,1,+,\times,\mathrm{gcd},\mathrm{lcm})$ that generates all the others?

Let $\Phi$ denote the set of all identities satisfied by $(\mathbb{N},0,1,+,\times,\mathrm{gcd},\mathrm{lcm}).$ Question. Is $\Phi$ finitely axiomatizable? If so, I'd like to see a list of ...
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What can we say about generating subsets if all free subsets are finite

Definitions An (universal) algebra is a pair $\mathcal A=(A, (f_1,\dots, f_n))$ where $A$ is a non-empty set and $(f_1, \dots, f_n)$ is a family of finitary operations on $A$. The notation $o(f_i)$ ...
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41 views

Prove or disprove the law of double pseudo-complements hold for subgraphs?

As for the categories of subgraphs of a given graph $X$ with their inclusions, we have a co-Heyting algebra (and Heyting algebra). In the co-Heyting algebra, we define $\sim$$A$ as the pseudo-...
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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, ...
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42 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; ...
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$C^*$-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 $...
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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, ...
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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 ...
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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 ...
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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 $F$ ...
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“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 ...
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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 ...
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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 $\theta \vee \sigma = \theta \circ \sigma$...
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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=...
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50 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 ...
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129 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 ...
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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 ...
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159 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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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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34 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 ...
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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 ...
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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 ...
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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 $(G,\circ,\,^{-1})$ ...
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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 ...
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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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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 ...
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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 ...
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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 ...
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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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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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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{...
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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?
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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 ...
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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{...
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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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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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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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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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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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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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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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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 ...