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
22 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 ...
1
vote
1answer
40 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
81 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 ...
1
vote
0answers
36 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
149 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
votes
1answer
72 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
votes
1answer
49 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? ...
3
votes
1answer
90 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 ...
1
vote
1answer
55 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 ...
2
votes
3answers
69 views

Finite free objects [closed]

Free distributive lattice with any number of generators is finite. For example with 3 generator the lattice will have 20 elements. Is there other examples of free objects that are finite and have at ...
1
vote
0answers
48 views

Universal Algebra: partial algebras and homomorphisms

I'm looking for an accessible introductory text on universal algebra, one which discusses homomorphisms in the context of partial algebras. I have been recommended the Grätzer book, "Universal ...
0
votes
1answer
73 views

Terminological conventions regarding group actions

Suppose: $G$ is a group $T$ is a Lawvere theory ("algebraic theory") and $X$ is a $T$-algebra What conventions surround the phrase: "action of $G$ on $X$"? In particular, does this mean: a ...
4
votes
1answer
81 views

Do the usual examples of “dimensive” Lawvere theories form an exhaustive list of such theories?

Call a Lawvere theory $T$ dimensive iff, letting $F_T : \mathbf{Set} \rightarrow \mathbf{Mod}(T)$ denote the free functor, we have the following. Every finitely generated $T$-algebra is free. From ...
3
votes
2answers
63 views

why only closed operations

Why does the carrier of an algebraic structure has to be closed under the operations of the algebraic structure? One could also consider $(\mathbb{N}^*, \div)$. But why isn't that an algebraic ...
4
votes
1answer
55 views

Can you find a plain aneloid?

I defined an "aneloid" to be a set endowed with two operations, adition and multiplication, with multiplication being distributive BOTH sides in relation to adition. I tried to find an example of ...
6
votes
1answer
63 views

Binomial rings closed under colimits?

A binomial ring is a ring (for the purposes of this question all rings are commutative and unital) which is torsion-free and has, for each $n$, a binomial function $\binom{x}{n}$ satisfying ...
4
votes
1answer
39 views

The identity elements of two multiplication satisfying interchange law.

Let $M$ be a set, and $\circ_1,\circ_2$ two binary operations defined on $M$ satisfying that both $(M,\circ_1),(M,\circ_2)$ are semigroups and $(a\circ_1 b)\circ_2(c\circ_1 d)=(a\circ_2 ...
5
votes
0answers
69 views

Do torsionfree abelian groups form a (possibly many-sorted) algebraic category?

By "(possibly many-sorted) algebraic category", I mean "category of models of a (possibly many sorted) Lawvere theory. I have arguments for both affirmative and negative answers, and I can't find a ...
1
vote
1answer
41 views

An Exercise From Universal Algebra and Coalgebra by Denecke and Wismath

I'm going through this book right now, and I don't understand something about exercise 9 from chapter 1: Determine all elements of t he free algebra $F_V(X_2)$ for the following varieties: (i) ...
0
votes
0answers
14 views

Definition of (minimal) domain?

Consider the following links: http://www.glottopedia.org/index.php/Domain_%28Syntax%29 http://www2.let.uu.nl/uil-ots/lexicon/zoek.pl?lemma=Minimal+domain&lemmacode=542 What kind of mathematical ...
3
votes
1answer
25 views

Ternary algebra satisfying some identities is a join-semilattice

A join-semilattice with greatest element is an algebra $(S,\vee, 1)$ of type $(2,0)$ such that $\vee$ is idempotent, commutative, and associative, and $a\vee 1=1$ for all $a\in A$. Now, let ...
3
votes
1answer
57 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 ...
0
votes
0answers
33 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 ...
4
votes
1answer
78 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 = ...
2
votes
0answers
27 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 ...
1
vote
3answers
54 views

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 ...
0
votes
0answers
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 ...
7
votes
1answer
97 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 ...
0
votes
1answer
32 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?
0
votes
0answers
39 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 ...
0
votes
0answers
24 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?
0
votes
0answers
30 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?
0
votes
1answer
40 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 ...
0
votes
1answer
45 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 ...
2
votes
1answer
74 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. ...
1
vote
2answers
76 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 ...
0
votes
0answers
42 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 $N(Z_{2}\times Z_{2})$.
1
vote
1answer
101 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 ...
1
vote
0answers
57 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. ...
0
votes
0answers
28 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 ...
0
votes
1answer
36 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 $v_1v_2$ ...
2
votes
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 ...
0
votes
0answers
49 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?
0
votes
1answer
62 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 ...
2
votes
2answers
93 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-, ...
5
votes
1answer
80 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 ...
2
votes
0answers
60 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 ...
0
votes
1answer
70 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 ...
0
votes
1answer
83 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 ...
0
votes
0answers
86 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 ...