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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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^{...
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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 $\...
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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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