# 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 ...
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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 ...
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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 ...
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### 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 ...
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### 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 ...
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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 ...
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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 ...
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### 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 ...
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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 ...
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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 ...
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### 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 ...
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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 ...
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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 ...
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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 $x+k$. ...
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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 ...
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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 ...
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### 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 ...
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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 ...
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### 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 ...
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### 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 $x\lor y=y$....
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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 $Pol(\{R\})$ ...
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### 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 ...
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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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