# 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 ...
1 vote
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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?
1 vote
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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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1 vote
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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 ...
56 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 ...
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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 ...
1 vote
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1 vote
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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 ...
1 vote
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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 ...
1 vote
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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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1 vote
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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 ...
117 views

### 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 ...
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 $E$...
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 \$[\...