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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sum i = 1 to ∞ ((sqrt(2) + i * sqrt(2)) ^ n)/(z ^ n) find the summation domain of the given sequence [closed]
I am seeking assistance in analyzing the convergence of the following series:
sum i = 1 to ∞ ((sqrt(2) + i * sqrt(2)) ^ n)/(z ^ n)
I am interested in understanding the convergence behavior of the ...
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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 ...
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How to find and draw the congruence lattice of N5? [closed]
How to find the congruence classes and draw the congruence lattice of N5?
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How to draw the congruence lattice of a lattice? [closed]
How to draw the congruence lattice of a lattice with a given diagram of the lattice?
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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 ...
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Count idenpotences (length k) in symmetric inverse semigroup.
Task: Count idenpotences (length k) in symmetric inverse semigroup $IS_{n}$
$IS_{n} \subset PT_{X}$, where $PT_{X}$ is a partial transformation semigroup on X.
Semigroup $IS_{n}$ has $\sum_{k=0}^{n}r!...
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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(...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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When equivalence relation on algebra is congruence.
Task: Prove that equivalence relation on algebra A will be congruence $ \leftrightarrow $ it is a sub-algebra of $A^2$.
My thoughts: A stable equivalence relation on algebra is a Congruence
Let $\sim -...
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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 $\...
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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 ...
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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 $\...
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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 ...
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Equational axiomatization and first-order axiomatization of the class of fields united with the class of Boolean algebras
This is really two questions in one, one about universal algebra and one about model theory. Let our signature be $\{+,*,-,0,1\}$. I have recently realized that both fields and Boolean algebras share ...
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Equivalent equational identities which are not alphabetical variants of each other.
Let our signature be that of a single binary operation $+$. Suppose $E$ and $E'$ are equivalent equational identities. Suppose also that neither $E$ nor $E'$ are equivalent to the trivial identity $x=...
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Equational identities equivalent to the associative identity
This is a natural follow-up to my previous question, here: Is only the commutative identity equivalent to the commutative identity?. As usual, let our signature be that of a single binary operation $+$...
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Identities equivalent to the reflexive identity
This is a follow-up to my previous question, here: A characterization of the identities which are equivalent to the trivial identity. As in that question, let our signature be that of a single binary ...
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Universal algebra: Fundamental group of a topological group is abelian
I am working through 'Notes on logic and set theory' by P.T. Johnstone. Below, $\Omega$ is a set of operation symbols with $\alpha \colon \Omega \to \mathbb{N}$ assigning to each symbol its arity. ...
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Relations on free algebra
If we have any class $K$ of algebras (sets with operations and constants) of given type, then there is the well-known construction of free algebra $F_K(X)$ with respect to $K$ generated by given set ...
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Is it decidable if a finite set of identities imply the commutative identity?
This is a follow-up to my previous question, here: Is it decidable if a finite set of equations have only trivial models?. Let our signature be that of a single binary operation symbol $*$. Suppose I ...
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What is precisely an anti-linear anti-involution?
Given an arbitrary $\mathbb{C}$-algebra $A$, what should be required from a function $f:A\to A$ for it to be an anti-linear anti-involution?
As I understand “anti-linear” for $A$ as a vector space ...
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What maps preserve and reflect basic Horn formulas?
A function $f : M \to N$ preserves and reflects a formula $\phi$ if $M \models \phi(\overline a) \iff N \models \phi(f(\overline a))$. For many fragments of first-order logic, there is a clear ...
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A characterization of the identities which are equivalent to the trivial identity
This is a follow-up to my previous question, here: Is only the commutative identity equivalent to the commutative identity?. As in that question, let our signature be that of a single binary operation ...
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Research monographs and open problems in universal algebra
I am someone who is very interested in the mathematical subfield of universal algebra. I want to know, what are some significant open problems in universal algebra? I would like a list of such ...
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Is $s=t$ always strictly stronger than $s+s=t+t$?
This question was inspired by a comment on one of my previous questions. As before, let our signature be that of a single binary operation $+$. Suppose $s$ and $t$ are two distinct terms in this ...
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Is only the commutative identity equivalent to the commutative identity?
Let our signature be that of a single binary operation $+$. Suppose I have an equational identity $E$ such that $E$ is equivalent to the commutative identity $x+y=y+x$. In other words, $E$ implies and ...
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Atoms and coatoms of the partial order of equations
Let our signature be that of a single binary operation symbol $*$. Consider the set of equations in that signature. I define a preorder on that set by saying $E \geq E'$ if and only if the equation $E$...
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Subuniverse generated by preimage equals preimage of generated subuniverse?
Let $\mathfrak{A} = \langle A, \mathcal{F}^\mathfrak{A} \rangle$ and $\mathfrak{B} = \langle B, \mathcal{F}^\mathfrak{B} \rangle$ be algebras of the same type $\mathcal{F} \to \omega$, and let $\...
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Basis for the equational identities of the algebraic structure $(\mathbb{R};-,abs)$
Consider the algebraic structure $(\mathbb{R};-,abs)$, where $-$ is the additive inverse unary function, and $abs$ is the absolute value function. What is a basis for the equational identities of that ...
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What is this quotient of the free product called?
I've started thinking about a particular "equation-preserving" quotient of the free product of groups (or more generally, coproduct of appropriate algebraic structures). Let $\mathcal{G},\...
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A basis for the equational identities of unordered $n$-tuples
This is somewhat related to my previous question, here: Unordered $n$-tuples can't define unordered $(n+1)$-tuples. As before, suppose we are working in the model theory of ZFC set theory. Also as ...
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Is this identity equivalent to the constant identity?
Let our signature be that of a single binary operation $*$. I define $*$ to satisfy the constant identity if $x*y=z*w$. But, I am interested in another identity, this one: $x*y=y*z$. I want to know if ...
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Decidability of the universal algebra consistency problem of the bi-unary signature
This is yet another follow up to my previous question on universal algebra and decidability of consistency, here: Follow up to a previous universal algebra question on decidability of consistency. In ...
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Lattices of clones: is $4$ worse than $3$?
For finite $k$, let $\mathscr{C}_k$ be the set of clones on a $k$-element set, viewed as a metric space by setting $d(A,B)=2^{-n}$ for distinct clones $A,B$ where $n$ is the smallest number such that ...
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Follow up to a previous universal algebra question on decidability of consistency
This is a follow up to my previous question on universal algebra and decidability, here: Is it decidable if a finite set of equations have only trivial models?. In that question, the answerer said ...
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Writing down a finite axiomatization of a group given by finite presentation
I am working on a problem and I am wondering if what I am trying can even be done.
Assume that we are given a finite group $G$ by it's presentation via generators and relations. For example $$G = \...
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Can an infinite-dimensional Lie algebra be always embedded inside the universal enveloping algebra?
Let $L$ be a Lie algebra and $U(L)$ be the corresponding universal enveloping algebra. If $L$ is finite-dimensional then by the virtue of PBW theorem we know that $L$ is embedded in $U(L).$ Can we ...
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Question about primitive elements of the universal enveloping algebra.
Is there any non-category theoritic argument of the fact that the primitive elements of the universal enveloping algebra of a Lie algebra are precisely those which belong to the underlying Lie algebra?...
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How is the Kronecker Delta decomposed in Sierpinski's proof that every n-ary operation is a finite composition of binary operations?
I came across the following answer that proofs that every $n$-ary operation on a finite set is a finite composition of binary operations:
A proof is especially simple for operations on a finite set $\...
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Smith commutator and other commutators
Smith introduced a notion of commutators for congruence permutable (that is, Malcev) varieties, developed in the monograph "Mal'cev Varieties". This notion was generalised by Hagemann and ...
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Is there a clone on a three element set which is not finitely generated?
Let 3 be the three element set $\{0,1,2\}$. Is there a clone (in the sense of universal algebra) on 3 which is not finitely generated? I know that every clone on a two element set is finitely ...
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Algebraic structures with opposite subalgebra lattices and congruence lattices that are not groups with operators
Are there any algebraic structures where the congruence lattice and subalgebra lattice are opposites that do not look like groups with operators?
My motivation for this question is noticing that ...
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Is there an infinite version of the brute-force lattice congruence representation algorithm that succeeds?
I am interested in the congruence lattices of algebraic structures.
Is there a brute-force way to produce an infinite algebraic structure whose lattice of congruences is isomorphic to some given ...
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Adjoining a function to a ring: what is this called?
There's a kind of construction of an extension of a ring, and that I've seen used e.g. here (although that may not be the best example) which is essentially adding a function into the ring, and taking ...
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Is the equational theory generated by this set of equations the same as the set itself?
This is related somewhat to my previous question, here: What is a finite equational basis for this equational theory?. As before, our signature is a single binary operation $*$. Also as before, let $E$...
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Commutator of congruences of a ring
I am trying to prove the following fact that should be mostly trivial to see:
If $\alpha, \beta$ are congruences of a ring $R$, corresponding to ideals $I$ and $J$ respectively, then the commutator $[\...
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