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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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 ...
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65 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 ...
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34 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 ...
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66 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 ...
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65 views

Are subvarieties just full subcategories that happen to be algebraic categories?

Suppose $T$ is a Lawvere theory. Suppose furthermore that $\mathbf{C}$ is a replete full subcategory of $\mathbf{Mod}(T)$ that is equivalent (as a category) to $\mathbf{Mod}(S)$ for some Lawvere ...
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64 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 ...
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61 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 ...
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52 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 ...
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62 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 ...
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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 ...
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68 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 ...
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39 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) ...
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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 ...
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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 ...
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56 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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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 ...
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77 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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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 ...
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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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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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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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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 ...
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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?
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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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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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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 ...
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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. ...
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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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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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99 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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50 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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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 ...
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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 $v_1v_2$ ...
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1answer
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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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48 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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58 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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92 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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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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57 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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69 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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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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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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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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87 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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108 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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1answer
47 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: $\{ b \in \mathbb{Z}_{p^n} \mid b^2 =1\} = \{-1, 1 \}$, ...
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391 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
59 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 ...
3
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2answers
73 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 ...