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Questions tagged [universal-algebra]

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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Does the variety of Boolean Algebras contain no proper nontrivial subvarieties/subquasivarieties?

Consider the variety, in the sense of universal algebra, of Boolean Algebras in the language $\{\cup,\cap,',0,1\}$, where $'$ represents complementation, and the other symbols are well known. I ...
user107952's user avatar
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Is there a Jacobson congruence - the universal-algebraic generalization of the Jacobson ring.

What do you call the universal algebra generalization of the Jacobson ring and where can I read more about it? This question is a follow-up to this question that I asked about an hour ago, more ...
Greg Nisbet's user avatar
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Which varieties admit Boolean product representation?

In the popular textbook A Course in Universal Algebra it is mentioned the following open problem (p. 289 of the Millennium Edition): "For which varieties $V$ is every algebra in $V$ a Boolean ...
Mockingbird's user avatar
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9 votes
2 answers
543 views

What is the "closest approximation" to fields by an equational variety?

Consider the equational theory obtained by starting from the theory of commutative rings and adding a unary operator $(-)^{-1}$, the weak inverse, obeying the following equational axioms: $a \cdot a^{...
Qiaochu Yuan's user avatar
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1 answer
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Tensor product of algebraic theories

The following is a theorem from Borceux, Handbook of Categorical Algebra 2: Theorem 3.11.3 Let $\mathcal{T}$ and $\mathcal{R}$ be algebraic theories. There exists an algebraic theory, written $\...
Nick Mertes's user avatar
2 votes
1 answer
62 views

Show that the group with this presentation has order $12$ and is isomorphic to the alternating group $A_4$ [closed]

Show that the group $$G=\langle a, b, c; a^3, b^2, c^2, ab=ba^2, ac=ca, bc=cb\rangle$$has order $12$ and is isomorphic to the alternating group $A_4$. I did manage to show that this group is of order ...
Jovana Rechkoska's user avatar
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1 answer
50 views

Congruences on the pentagon lattice $\mathcal{N}_5$

Let $\mathcal{N}_5$ refer to the Pentagon lattice, or the lattice generated by the set $\{0, a, b, c, 1\}$ subject to $1 > a$, $1 > c$, $a > b$, $b > 0$ and $c > 0$. My aim is to find ...
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Two ways to produce natural transformations

Let $\mathcal{C},\mathcal{D}$ be two categories. Let $F_1,F_2\colon\mathcal{C}\to\mathcal{D}$ be functors from $\mathcal{C}$ to $\mathcal{D}$, $G_1,G_2\colon\mathcal{D}\to\mathcal{C}$ be functors from ...
Estwald's user avatar
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3 votes
1 answer
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Counting homomorphisms from non-free Boolean algebras to the free Boolean algebra on $n$ generators

I have been foraying a bit into belief revision theory and formal epistemology recently, and that has ended up at me having to explore some universal algebra and combinatorics. I'll cut straight to ...
safsom's user avatar
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What is free (Baxter) algebra over empty set?

In the article "Baxter Algebras and Hopf Algebras" it is stated that divided power algebra is a free Baxter algebra of weight 0 over the empty set. I don't understand this statement. In 1969 ...
Daigaku no Baku's user avatar
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1 answer
102 views

Is there a $\forall$-statement in the language of magmas which is not equivalent to any semi-quasi-equation?

Let our signature be that of a single binary operation $*$, in other words, the signature of magmas. In my previous post, here: Has this generalization of quasi-equations been studied in the ...
user107952's user avatar
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1 answer
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Has this generalization of quasi-equations been studied in the mathematical literature?

In universal algebra, there is a lot of talk of quasi-equations, which are conditionals, where the antecedent is a conjunction of finitely many (possibly even $0$) equations, and the consequent is a ...
user107952's user avatar
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Equational basis for Cartesian product, union, intersection, and the empty set

Let $M$ be a model of ZFC set theory, and consider the algebraic structure $(M;\times,\cup,\cap,\emptyset)$, where $\times$ represents Cartesian product based on the Kuratowski ordered pair, and the ...
user107952's user avatar
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2 votes
1 answer
95 views

Smallest possible cardinality of finite set with two non-elementarily equivalent magmas which satisfy the same quasi-equations?

This is a natural follow-up to my previous question, here: Examples of two finite magmas which satisfy the same equations but not the same quasi-equations?. In the answer to that question, Keith ...
user107952's user avatar
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3 votes
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Examples of two finite magmas which satisfy the same equations but not the same quasi-equations?

Does there exist two binary operations $+$ and $*$ on $\{0,1\}$ such that $+$ and $*$ satisfy the same equations, but not the same quasi-equations? If not, are there such binary operations on a finite ...
user107952's user avatar
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Smallest cardinality of finite algebraic structure whose quasi-equational theory is not finitely based?

I understand that, for the set $\{0,1\}$ and any finite number of constants and operations on that set, the resulting algebraic structure has a finite basis of identities. But, does there exist a ...
user107952's user avatar
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Is addition by a specific nonzero natural number a term function in this structure?

Consider the structure $(\mathbb{N};+,\times,0)$. I know that every nonzero natural number $k$ is definable by a first-order formula in that structure, and hence, so is the unary function $x+k$. ...
user107952's user avatar
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Is addition a term-function in this structure?

This is a follow-up to my previous model theory question, here: Is addition definable from successor and multiplication?. I asked whether addition is definable by a first-order formula in the ...
user107952's user avatar
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Are there any new identities when we go from subtraction to subtraction with a nonzero constant?

This is the subtraction counterpart to my previous universal algebra question on addition with a nonzero constant, here: No simplifying identities for any single nonzero number under addition.. I know ...
user107952's user avatar
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0 answers
39 views

Universal enveloping algebra of a Lie algebra $\mathfrak{g}$

I'm studying the existence and the uniqueness of the universal enveloping algebra of a Lie algebra $\mathfrak{g}$ but i don't understand why $\mathfrak{U(g)}$ is generated by $i(\mathfrak{g})$, where $...
Elianna 's user avatar
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Is it possible to express and prove a theorem only with commutative diagrams?

By saying "only", I mean that no words are involved except "commutative". Just draw a commutative diagram, and make some arrows dotted lines to mean "the other arrows already ...
ZhenRanZR's user avatar
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A Mal'cev algebra $\mathbf{A}$ have typ$\{\mathbf{A}\} \subseteq \{\mathbf{2},\mathbf{3}\}$

A Mal'cev algebra is an algebra $\mathbf{A}$ with a ternary term $t$ such that $\mathbf{A} \models t(x,x,y) \equiv y, t(x,y,y) \equiv x$. For the remaining of the question, refer to the definitions ...
Mockingbird's user avatar
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Does every variety of locally finite closure algebras have a unification type?

An algebra has one of four unification types - unitary, finitary, infinitary or nullary. A variety $\mathbf{V}$ has type unary, if every member has unary type, finitary, if every member has finitary ...
Ohbi's user avatar
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4 votes
1 answer
112 views

Computational Complexity of Equational Logic

Equational logic uses a surprisingly small set of axioms to prove all algebraic identities (algebraic in the sense of universal algebra, so things like field theory fall beyond this scope). This makes ...
Thomas Anton's user avatar
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80 views

Why universal enveloping algebra U$(\mathfrak{g})$ has the same representation theory as that of $\mathfrak{g}$?

I am new to concepts from Lie algebra theory, so I was reading Lectures on Lie groups ad Lie algebras by Carter, Macdonald and Segal. At the third chapter we are introducing universal enveloping ...
Mahammad Yusifov's user avatar
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0 answers
55 views

Passing an involution to a quotient algebra

This question is inspired by the discussion under this MO answer. I hope I have captured correctly what is going on in the below. Let $A_0$ be a finitely generated universal unital complex algebra $...
JP McCarthy's user avatar
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6 votes
1 answer
197 views

Identities using only binomial coefficient

There are many known combinatorial identities involving the binomial coefficient, although essentially all of the well known ones involve some function or constant other than the binomial coefficient ...
volcanrb's user avatar
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1 vote
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Further resources on category theory

I'm currently looking for sources on category theory that take a more intuitive approach and delves deeper into the nature of the idealogy leading to the theory. I read some elementary books on the ...
Aryan's user avatar
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4 votes
2 answers
99 views

Construction of free objects in an arbitrary variety using only the HSP operations

According to Birkhoff's HSP theorem, a class of algebraic structures is a variety if and only if it is closed under the taking of homomorphic images, subalgebras, and arbitrary products. It is also ...
M. Sperling's user avatar
1 vote
1 answer
47 views

If the direct product of two semilattices exists, what does its Hasse diagram look like in terms of its constituent semilattice Hasse diagrams?

This is likely to be a quick question. Definition: A semilattice $(L,\lor)$ is a commutative, idempotent semigroup. The Hasse diagram $H$ of $L$ is with respect to the order $x\le y$ iff $x\lor y=y$....
Shaun's user avatar
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1 vote
1 answer
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Can you determine the polymorphisms of this relation?

Let $A\neq \emptyset$ be arbitrary set. Let us define a relation on $A$ in the following way: $R=\{(a;b;c)\in A^3| a=b \lor b=c\}$. Show that the clone of polymorphisms of $R$ denoted as $Pol(\{R\})$ ...
Björn's user avatar
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5 votes
1 answer
292 views

What is the minimum number of axioms necessary to define a (not necessarily commutative) ring (that doesn't have to have a one)?

Motivation: According to this, the minimum number of axioms required to define a group is one. What can we say about rings (that are not necessarily commutative nor do they have to have a one)? The ...
Shaun's user avatar
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4 votes
2 answers
155 views

Describe the class of groups that satisfy $(x^2y^2)^2 \approx 1$

I have trouble with the following assignment: Let $A$ denote the class of all Abelian groups satisfying the identity $x^2 \approx 1$. Show that the class $$\{G \mid \exists N: N \trianglelefteq G \...
Björn's user avatar
  • 140
2 votes
1 answer
132 views

Does cartesian product satisfy any nontrivial equational identity?

This is a question concerning the model theory of set theory. Let $M$ be a model of ZFC set theory, and consider the algebraic structure $(M;\times)$, where $\times$ is (the Kuratowski implementation ...
user107952's user avatar
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1 vote
1 answer
144 views

How do I prove $U(\mathfrak{g} \oplus \mathfrak{h})\cong U(\mathfrak{g})\otimes U(\mathfrak{h})$ using the universal properties?

Reading lecture notes on Hopf algebras, I came across a statement (which was heavily used by the author without a proof) that, given Lie algebras $\mathfrak{g}, \mathfrak{h}$, the universal enveloping ...
Daigaku no Baku's user avatar
1 vote
2 answers
122 views

Why coproduct in the algebra of linear operators is actually a coproduct?

Coproduct in a coalgebra $V$, where $V$ is also a vector space over a field $\mathbb{K}$, is defined as a $\mathbb{K}-$linear map $\Delta : V \rightarrow V\otimes V$ satisfying two diagrams obtained ...
Daigaku no Baku's user avatar
0 votes
1 answer
54 views

Prove that in every algebra the union of any chain of subalgebras is a subalgebra

Prove that in every algebra the union of any chain of subalgebras is a subalgebra. Let we have an $\left \langle A, \Omega \right \rangle$ and a chain $A_1 \subset A_2 \subset ... \subset A_n$ of ...
Miganyshi's user avatar
  • 125
13 votes
3 answers
1k views

Group is to ring as ring is to...

I'm wondering if there is a structure that builds on the ring axioms analogously to how rings build on the group axioms. For example, $(X, +)$ is a group and $(X, +, \times)$ is a ring with identity. ...
Adam Brown's user avatar
1 vote
1 answer
89 views

Is there a finite generating set for the clone of all operations on a finite set with at least 3 elements?

Let $S$ be a finite set with at least $3$ elements, and let $O(S)$ be the set of all finitary operations on $S$. That is, it is the set of all unary, all binary, all ternary, etc operations collected ...
user107952's user avatar
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1 vote
0 answers
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Techniques for showing that sole sufficient operators of a given arity do not exist for a given algebra

I came across this question yesterday which is interesting. It asks whether a binary sole sufficient operator for the modal logic K exists. I tried to find an example of a sole sufficient operator for ...
Greg Nisbet's user avatar
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2 votes
0 answers
122 views

Subrings and quotients of finite semigroup algebras

Let $F = \Bbb F_p$ be a prime finite field and $R$ an arbitrary finite-dimensional associative (+ let's say unital) algebra over $F$. Then $R$ is a subalgebra (=subring here) of a matrix algebra $M_n(...
Amateur_Algebraist's user avatar
2 votes
0 answers
39 views

Free algebra over the intersection of varieties

I am wondering if given to varieties $V,W$ in the sense of Universal algebra once could describe free algebras over $V\cap W$ in some meaningful way? Now what I mean by meaningful is not even ...
TdotA's user avatar
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6 votes
0 answers
64 views

When does a modified distributive law, like $a\cdot(b+c)=(a\cdot a)+(b\cdot c)$, allow polynomials to be written in the standard form?

Suppose we have two unary operations $f,g$, related by the law $$\forall a,\quad g(f(a))=f(f(g(g(a)))),$$ that is, $gf=ffgg$. We might then wonder whether any expression involving these operations can ...
mr_e_man's user avatar
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2 votes
1 answer
83 views

Every variety has a simple algebra -- proof

I'm having a hard time understanding the proof (as presented for example in A Course in Universal Algebra of the GTM series) of the theorem by Magari which states that a variety $V$ with a nontrivial ...
Mockingbird's user avatar
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1 vote
0 answers
34 views

Can I use ''exist'' predicate in the definition of algebra in equationally definable class?

I'm reading Algebraic Theories by Ernest G. Manes, and I'm wondering about equationally-definable class. For example, group is a typical example of equationally-definable algebra, but the definition ...
ぼっけなす's user avatar
-1 votes
1 answer
140 views

Provide an example of a universal algebra A that decomposes into a nontrivial direct product in which each factor is not isomorphic to the algebra A.

A subalgebra A of a direct product $\prod_{k} A_{k}$ is a subdirect product of algebras $A_{k}$ if the projection A on each factor $A_{k}$ coincides with the factor itself. The subdirect product of ...
Miganyshi's user avatar
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2 votes
1 answer
141 views

Prove that function composition is subalgebra

Let $\varphi \subseteq A \times B; \psi \subseteq B \times C$. Then $\varphi \circ \psi = \left \{ (a, c)| \exists b: (a,b) \in \varphi, (b,c) \in \psi \right \} \subseteq A \times C$. Task: Let $\...
Miganyshi's user avatar
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1 vote
1 answer
136 views

Maximal subalgebras of $F(\Omega,X)$

Let $X$ be a non-empty set, $\Omega$- is some signature. $\Omega$ - terms are all elements from $X$, $0$-arity symbols from $\Omega$, and $\forall \omega\in\Omega$ of arity $n$ and $a_1,\ldots,a_n\in ...
Miganyshi's user avatar
  • 125
2 votes
1 answer
104 views

Does this algebra have a name?

I thought of viewing $\bar{\mathbb{R}}$ as a relative subalgebra of a total algebra $\mathbb{R}^\ast := \bar{\mathbb{R}} \cup \{\ast\}$, wherein an output of an operation, if not already defined in $\...
joeb's user avatar
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2 votes
1 answer
55 views

What is a basis for the common equational identities of $(\mathbb{R};*,0)$, $(\mathbb{R};*,1)$, and $(\mathbb{R};*,-1)$?

Consider the three algebraic structures $(\mathbb{R};*,0)$, $(\mathbb{R};*,1)$, and $(\mathbb{R};*,-1)$, where $*$ denotes multiplication. What is a basis for the equational identities those ...
user107952's user avatar
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