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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123 views

What is a simple axiomatisation of groups using division?

I recall from an old exercise I did as an undergrad that groups can be axiomatised using division rather than multiplication: A group is a non-empty set equipped with a binary division operator / ...
4
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1answer
119 views

Reference for understanding coalgebra

I am trying to read this paper, but I have no knowledge of coalgebra and have just started to learn Category Theory so I am struggling to understand it. Are there any references that can explain ...
4
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1answer
79 views

The “closed” subspaces of topological algebraic structures

Every set-theoretic model of an algebraic theory gives rise to notion of (algebraically) "closed subset" in a canonical fashion; namely, the closed subsets are those that cannot be escaped via the ...
2
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1answer
138 views

Every group and ring is congruence-permutable , but not necessarily congruence-distributive

The problem is: Show that every group and ring is congruence-permutable , but not necessarily congruence-distributive. I know that in group every normal sub group has permutable property and in ...
1
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1answer
74 views

Textbook question on variety

Suppose a variety V is defined by an infinite minimal set of identities. Show that V is a subvariety of at least continuum many varieties.
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1answer
47 views

Variety satisfying an identity.

$V$ is a variety of commutative semigroup satisfying the identity $x^2 = x^3$. I need to prove that: $|F_V(\{x_1,\dots,x_n\})|$ = $3^n -1$. Any hints on this ? $F_V$ is V-free algebra.
1
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1answer
144 views

homomorphisms and congruence relations

Do compositions of homomorphisms in universal algebra correspond to joins of congruence relations? That is- is the congruence relation $g \circ f(a ) = g \circ f( b) \Leftrightarrow a \sim b $ the ...
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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? ...
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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 $v_1v_2$ ...
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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 ...
5
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0answers
91 views

Functorial approach to Ideals and Quotients, Multiplicative Sets and Localizations

I have been playing with substructures of commutative rings today and noticed that there is a strong analogy between the formation of quotients and kernels with the formation of localizations with ...
5
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0answers
49 views

Is there a name for the algebra of substructures?

Let $X$ denote an entropic algebra (see here), which just means that all the operations of $X$ are homomorphisms $X^n \rightarrow X.$ Abelian groups are the classic example. Then for any operation of ...
3
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0answers
60 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 ...
3
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0answers
93 views

Prove that $\Pi_{i \in I} \mathbf{A}_i \cong \Pi_{j \in J} (\Pi_{i \in I_j} \mathbf{A}_i)$ where $\{I_j \ | \ j \in J\}$ is a partition of I.

My problem is following: prove that $\Pi_{i \in I} \mathbf{A}_i \cong \Pi_{j \in J} (\Pi_{i \in I_j} \mathbf{A}_i)$, where $\langle \mathbf{A}_i \ | \ i \in I \rangle$ is an indexed set of similar ...
3
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0answers
48 views

Seeking information about (category-theoretic) varieties, quasivarieties, and universal Horn classes.

I'm looking for a list of basic facts regarding (category-theoretic) varieties, quasivarieties, and universal Horn categories, as well as information about which forgetful functors preserve what. In ...
3
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0answers
89 views

What big families of theories/structures are there?

To gain a big picture of (pre-categorical) mathematics, is it correct to divide mathematical theories resp. structures in two big families? universal algebra: classes of objects with arbitrary ...
2
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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 ...
2
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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 ...
2
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0answers
187 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 ...
2
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0answers
80 views

Which algebraic identities survive the journey to the powerset?

Given an algebraic structure $A$ (call its underlying set $U$) we can obtain a new algebraic structure $B$ with underlying set $V=\mathcal{P}(U)$ in the obvious way. In particular, if $f : U^n ...
2
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0answers
84 views

A subalgebra of $P(M, \Delta)$ is a Peano Algebra

Let $M$ be any set and let $\Delta=(n_i)_{i \in I}$ be an algebraic type. Let $P=P(M, \Delta)$ be the algebra such that the generalized Peano Axioms hold: (P1) $f_{i}(a_0,\dots,a_{n-1}) \notin M$ ...
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23 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 ...
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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 ...
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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 ...
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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. ...
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110 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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43 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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0answers
55 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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0answers
32 views

Question on HSP and SHPS inquality.

In the screenshots attached above George Bergman outlines his way of proving $HSP \ne SHPS$ I understand the first definition as the group of affine transformations and each element of the group ...
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0answers
161 views

Collections of Homomorphic (defined) structures via $f$

Long ago I read a text about a collection of algebraic sturctures all homomorphic (or isomorphic) via a unique homomorphism An Example similar to the construction I found was this: Lets take define ...
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0answers
54 views

What is the $K$-free algebra for the class of implication algebras, over a finite set

I suppose the title is pretty self explanatory. I have been struggling with the concepts of $K$-free algebras, where $K$ is some class of same-type algebras, over some set $X$. So, in trying to ...
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0answers
78 views

Congruence lattice of a partial algebra is algebraic

A partial operation on a nonempty set $A$ is a map $f:\mathrm{dom}(f,A)\to A$ where $\mathrm{dom}(f,A)\subseteq A^n$ for some $n\in\mathbb{N}$. A partial algebra is an ordered pair $(A,P)$ where $A$ ...
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0answers
71 views

Which “conditions” generate subalgebras?

While looking at this question I suddenly wondered about a more general question. Think of your favorite class of algebra (groups or rings, say), and forgive me for vocabulary mistakes while I try to ...
1
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0answers
80 views

Another way of saying that algebraic objects are isomorphic

From a universal algebraic perspective, let's say we have two isomorphic groups. Then can I speak of their isomorphic nature by saying the binary operations of multiplication of the two groups are ...
1
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0answers
85 views

direct product of different algebras?

Is it possible to define a "direct product" of two algebras with different signatures? For example, boolean lattice $\otimes$ monoid? Perhaps we need to take some quotient to make sense of it (e.g Q ...
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0answers
163 views

amalgam of structures

Trying to refine my question here. This is a response to the questions here: Homomorphisms between structures My objective is to take a set of $S-$structures and form an amalgam object out of that ...
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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 ...
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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 ...
0
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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
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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
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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
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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 $N(Z_{2}\times Z_{2})$.
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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 ...
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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?
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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 ...
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97 views

A Course in Universal Algebra (Millennium edition), page 74

The line before Theorem 10.12 says that "In general $F_K(\overline{X})$ is not isomorphic to a member of K (for example, let K={L} where L is a two-element lattice, then $F_K(\bar{x}, \bar{y}) \notin ...
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0answers
65 views

What do we call functions that are definable by expressions?

Let $X$ denote a model of an algebraic theory $T$. What do we call the functions $f : X^n \rightarrow X$ that are definable by some expression in the language of $T$? e.g. If $S_3$ is the symmetric ...
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0answers
73 views

Medium-strong (graph) homomorphisms

Weak (graph) homomorphisms are mappings $f: V(G) \rightarrow V(G')$ such that the images of connected nodes $x,y$ (in the source graph) are connected: $$R(x,y) \rightarrow R(f(x),f(y)) = R(x',y')$$ ...
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0answers
21 views

verify the claim that consequences of balanced identities are again balanced.

verify the claim that consequences of balanced identities are again balanced. An identity is p≈q balanced if each variable occurs the same number of times in p as in q.if ∑ is balanced set of ...
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0answers
43 views

Question about a property of lattice-morphism

I would like to know if there is a name for the class of commutative (i.e., $\phi(x,y)=\phi(y,x)$) lattice-morphisms $\phi : L_1\times L_{1} \rightarrow L_2$ with the following property: $\phi(x ...