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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A singleton set $\{g\}$ can be regarded as a unary relation in $G$. Why?

Theorem 1.1. A relation $R \subseteq M^n$ is definable if and only if every automorphism of every elementary extension of $M$ preserves $R$. For a proof, the reader can see [4]. Suppose we ...
108
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6answers
7k 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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28 views

Pro-completion of finite algebras as Stone algebras

Recall that a profinite algebra (e.g. group, monoid, or whatsoever) is a cofiltered/inverse limit of finite algebra. In Johnstone's Stone space, he showed that finite discrete algebras are finitely ...
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50 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 ...
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26 views

Where can I find Kan's paper “On c.s.s. categories” from 1957.

Does anyone know how I can find the following article by Daniel Kan: Kan, D. M. On c.s.s. categories Bull. Soc. Math. Mexicana (1957), 82-94. Quillen lists it as a reference in his paper Rational ...
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1answer
68 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 = ...
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0answers
21 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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2answers
81 views

Is there a standard name for a set equipped only with an idempotent binary operation?

Is there a name for an idempotent magma, or do they not arise often enough to warrant a special name? (By idempotent binary operation, I mean an operation $+$ such that $x + x = x$ for any $x$.)
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38 views

Show Sg$(X) = X\cup E(X) \cup E^{2}(X)\cup …$

Given an algebra $A$ define, for every $X\subset A$, $$\text{Sg}(X)=\bigcap\{B\mid X\subset B \text{ and } B \text{ is a subuniverse of } A\},$$ $$E(X)=X\cup \{f(a_1,\ldots,a_n)\mid f \text{ is ...
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1answer
80 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 ...
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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 ...
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9 views

Closure systems [duplicate]

Let A be any set. A system $\mathscr{C}$ of subsets of A is said to be a closure system if $\mathscr{C}$ is closed under intersections, i.e. $$\textrm{for any subsystem ...
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1answer
27 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?
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28 views

why $P$ is not a sublattice of the lattice $\{a,b,c,d,e\}$

I could not understand why $P$ is not a sublattice of the lattice $\{a,b,c,d,e\}$ how to find?
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0answers
20 views

how to find N(G) is a modular lattice?

answer: $N_{1}\leq N_{2}$ so $N_{1}\vee(N_{2}\wedge N_{3})=(N_{1}\vee N_{2})\wedge (N_{1}\vee N_{3})=N_{2}\wedge (N_{1}\vee N_{3})$ how to continue?
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45 views

Algebraic systems

Are there any books about algebraic systems without having Mal'cev book? Are there books in general about the varieties and quasi-varieties?
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39 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 ...
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1answer
71 views

Dual atoms of the lattice of varieties

I'm reading Jaroslav Ježek's "Universal algebra". There is a Theorem. For a signature containing at least one symbol of positive arity, the lattice of varieties of that signature has no coatoms. ...
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1answer
43 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 ...
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1answer
65 views

Factor congruences of non trivial Lattices

A pair of congruences $\theta$ and $\theta^*$ are called factor congruences if $\theta \vee \theta^*$ = full congruence. $\nabla$ $\theta \wedge \theta^*$ = trivial congruence. $\triangle$ I need ...
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0answers
52 views

Congruence lattice of $N_5$

I calculated the the congruence lattice of $N_5$ using hit and trial and then verified it with Universal Algebra calculator. But I need to prove that it is the congruence lattice of $N_5$ How should I ...
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1answer
103 views

Subdirect embedding of a quotient algebra

If $A$ is an algebra and $\theta_i$ $\in$ $Con(A)$, i $\in$ $I$, let $\theta$ = $\cap \theta_i$. Show that $A/\theta$ can be subdirectly embedded in $\prod$$A/\theta_i$. What intuitively I think of ...
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40 views

If $G$ is a group, show that the lattice $N(G)$ of normal subgroups of $G$ is a modular lattice

If $G$ is a group, show that the lattice $N(G)$ of normal subgroups of $G$ is a modular lattice. Does the same property holds for the lattice $S(G)$ of all subgroups? Describe $N(Z_{2}\times Z_{2})$.
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69 views

Categories of relations over a fixed category $\mathcal{C}$

Let $\bf{Set}$ be the category of sets and functions. We have an associated category $\bf{Set}_\bf{Rel}$, whose objects are also sets but whose morphisms are relations, i.e. a morphism ...
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1answer
87 views

Lawvere algebraic theories as presentation-invariant

We can read in a lot of papers, included Lawvere's PhD thesis, that algebraic theories are "an invariant notion of which the usual formalism with operations and equations may be regarded as a ...
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41 views

Definition of $\Omega$-algebra

I'm studying universal algebra. I have this definiton: given a signature $\Omega$, an $\Omega$-algebra is comprised of a "carrier" for the algebra and an "interpretation" for every operation symbol. ...
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1answer
33 views

Lemma concerning compatibility of words (formed by a term algebra)

I need to prove the next lemma regarding compatibility of words in term algebras, that includes 3 parts: $u,v$ are compatible iff $u^ \smallfrown w_1= v^ \smallfrown w_2$. If $u_1u_2$ and $v_1v_2$ ...
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25 views

Quoting a quasi-variety by a set of quasi-equations

Given a variety $V$, an algebra $A \in V$ and a set of equations $E$ one may form, in the standard way, the quotient algebra $A/E$. The algebra $A/E$ has the property that is satisfies all the ...
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1answer
30 views

If $B$ is an algebra, $X \subseteq B$ and $A$ the smallest subset that extends $X$, then $A= \bigcup_{n < \omega} A_n$

I need to prove the next thing: Let $\textbf{B} = \langle B, (b_i)_{i \in I}, (g_j)_{j\in J} \rangle$. Let $X \subseteq B$ and let $A$ be the smallest subset of $B$ which extends $X$ and is closed ...
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1answer
67 views

Is the existence of finite biproducts a strengthening of commutativity?

Let $T$ denote a monosorted Lawvere theory (call its distinguished object $G$) equipped with a distinguished constant $0 : G \leftarrow 1$ that is "idempotent" in the following sense: for all arrows ...
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1answer
50 views

The congruence class of a distributive lattice

I need to prove the next thing: For every non-empty ideal $I$ of a lattice $L$ consider the relation $\theta(I)$ defined by: $ \theta(I) = \{⟨a, b⟩ : ( \exists c \in I) a \vee c = b \vee c\}$ Prove ...
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1answer
70 views

Showing finite lattices are isomorphic to their sets of ideals.

I'm working out a problem from Gratzer '71. We are to show that any finite lattice $L$ $\simeq$ $I(L)$. My attempt was as follow: prove the contrapositive. So assume $L$ is not isomorphic to $I(L)$. ...
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2answers
89 views

Relating categorical properties of arrows

Diagram is from some book on categories shelf (ID?)- It's a good visual to rembember. What are some obvious extensions of categorical properties? Ie valid in every category, eg: strong-, extremal-, ...
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0answers
155 views

Algebraic theory and definition of multiplication: tensor product v.s. Cartesian product

In a monoidal category, one can define multiplication on a object $M$ as a morphism \begin{equation} M\otimes M\longrightarrow M, \end{equation} or a morphism \begin{equation} M\times ...
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2answers
143 views

Algebraic Structures: Does Order Matter?

(I want to link to similar question with a very good answer: Question about Algebraic structure?) An algebraic structure is an ordered tuple of sets. One of these is called the underlying set, and ...
2
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0answers
56 views

Name for classes of algebras closed under products and quotients

A class of algebras closed under products, quotetiens and subalgebras is a variety. Is there a name for a class of algebras closed under products and quotients? Could you refer me to any theorems ...
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1answer
62 views

Rel is a concrete category over Sets, but how to concretize that?

The traditional denotation of a structured set object is something like $(1)\qquad(X,\mathcal S_1,\dots\mathcal S_n)$ for some "structures" $\mathcal S_1,\dots\mathcal S_n$ on a set X. The modern ...
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1answer
82 views

Common conditions on functions to be morphisms. [closed]

When coming in contact with the concept of morphism one may start to wonder what makes different structured objects of the same kind to be similar in a "morphical" way. At least I did. Below ...
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83 views

An example of a free algebra in SP(K)

The following theorem is found in the book "Universal Algebra, Fundamentals and Selected Topics" by Clifford Bergman (pp.98). Theorem 4.28. Let $U$ be free for $K$ over $X$. Then, $U/\lambda_k$ is ...
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0answers
106 views

Is this a general structure for constructs?

Here a construct is a category where the objects are sets and the morphisms are structure preserving functions. Common examples are groups, graphs and topological spaces. As far as I can see there is ...
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4answers
283 views

Preserving structures

Category theory abstracts the notion of the preservation of structure by means of morphisms. Is there a description of what it means to preserve structure of different types of mathematical structures ...
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55 views

When do (universal-algebraic) varieties have (enough) injectives?

Following Zhen Lin's suggestion in my previous question, I ask this question separately. Suppose $\mathcal{V}$ is a variety in the sense of universal algebra. considered as a category. When does ...
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2answers
80 views

Are varieties cocomplete?

Consider a variety $\mathcal{V}$ in a sense of universal algebra, i.e. algebras of some fixed signatures described by a set of identities. Then $\mathcal{V}$ can be thought of as a category with ...
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4answers
299 views

Which texts do you recommend to study universal algebra and lattice theory?

As I'm planning to study some algebraic logic (a lot of!), I found that some knowledge of universal algebra, lattice theory and boolean algebras is a must. I wonder if you have any recommendation to ...
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1answer
42 views

Prove the identity in Ring of Integers Modulo Prime

I have many study tasks, but I do not have any example. Therefore, I do not know, how to solve these tasks. For example, I need prove, that: $\{ b \in \mathbb{Z}_{p^n} \mid b^2 =1\} = \{-1, 1 \}$, ...
3
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1answer
55 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 ...
2
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3answers
79 views

Class of algebras defined by quasi-identities is not closed under taking the quotients

Let $\mathbf{A}$ be an algebra (in the sense of universal algebra) of some signature $\Sigma$. By quasi-identity I mean the formula of the form $$(\forall x_1) (\forall x_2) \dots (\forall x_n) ...
3
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2answers
71 views

Are $e : 1 \rightarrow X$ and $\mathrm{id}_X : X \rightarrow X$ the only operations of $\mathsf{Grp}$ that are homomorphisms?

Let $\mathsf{Grp}$ denote the Lawvere theory of groups. (For concreteness, let us say that $\mathsf{Grp}$ is presented by the generators $c : X \times X \rightarrow X$ $e : 1 \rightarrow X$ $i : X ...
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42 views

Is there accepted terminology for algebraic structures whose every subalgebra is free?

Is there accepted terminology for algebraic structures whose every subalgebra is free? Examples: Any free group Any vector space More generally, any free module over a PID. In fact, this ...
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1answer
81 views

Existence of countably generated free algebra in the universal class

I'm trying to solve the following exercise (from Smirnov's "Varieties of algebras"): Problem: Let $K$ be the universal class of $\Omega$-algebras, i.e. $K = Mod(\Sigma)$, where $\Sigma$ is the ...