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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).

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).

891 questions
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Two finitely based equational theories whose meet is not finitely based.

Consider the lattice of equational theories of a single binary operation $*$. The join of two finitely based equational theories is of course finitely based. Do there exist two finitely based equational theories whose meet is not finitely based?
user107952
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The definition of $K$-free algebras

This is Definition 10.9 of the book "A Course in Universal Algebra" by Burris and Sankappanavar (page 73, Millennium Edition). Definition 10.9. Let $K$ be a family of algebras of type $\mathscr{F}$. Given a set $X$ of variables, define the…
Suhaib Afzal
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If an algebra has a discriminator term then so does every SI in the variety

Prove that if an algebra A has a discriminator term then every subdirectly irreducible algeba in V(A) has the same discriminator term. Does this follow directly from the fact that every variety is generated by its SI algebras? Since if we suppose…
wasatar
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If V is a congruence-permutable variety...

If V is a congruence-permutable variety such that every subdirectly irreducible algebra is simple, show that every finite algebra in V is isomorphic to a direct product of simple algebras. I'm not sure how to get started on this. Any help is…
wasatar
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Describe the free algebra on one generator in the variety of all algebras with two unary operations

Describe the free algebra on one generator in the variety of all algebras with two unary operations $f, g$. Do the same for the subvariety axiomatized by $f(g(x)) = g(f(x))$. I'm not sure how to get started with this. Any insight would be greatly…
wasatar
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Show that the variety of semilinear Heyting/Brouwerian algebras is not generated by any single finite Heyting/Brouwerian chain

Show that the variety of semilinear Heyting/Brouwerian algebras is not generated by any single finite Heyting/Brouwerian chain where a Brouwerian algebra is a Heyting algebra which does not include a bottom element and semilinear means that it is a…
wasatar
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Show that the class of Heyting algebras has the CEP.

An algebra A has the congruence extension property (CEP) if for every B ≤ A and θ ∈ ConB there is a φ ∈ Con A such that θ = φ ∩ (B x B) . A class K of algebras has the CEP if every algebra in the class has the CEP. Show that the class of Heyting…
wasatar
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Show that every finite distributive lattice is a reduct of a Heyting algebra

Show that every finite distributive lattice is a reduct of a Heyting algebra. I was thinking the following claim could be helpful but I haven't figured out how to prove this either: The class of bounded distributive lattices (L, ∨, ∧, 0, 1) such…
wasatar
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Why an alternative magma need not even be power-associative?

I don't understand well this situation Any associative magma (i.e., a semigroup) is alternative. More generally, a magma in which every pair of elements generates an associative submagma must be alternative. The converse, however, is not true, in…
Jack
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Show that every variety of mono-unary algebras is defined by a single identity.

Show that every variety of mono-unary algebras is defined by a single identity. Intuitively this makes sense but I am having trouble showing it. An example of a mono-unary algebra is $\langle \mathbb{N}, f \rangle$ where $f(n) = n + 1, \forall n \in…
oliverjones
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Examples of Arithmetical Varieties, and Varieties that admit a majority term.

I am looking for examples of varieties of Universal Algebras that admit a ternary majority term $p$. For example, boolean-algebras have such a term: $p(x,y,z) = (x \wedge y)\vee (x \wedge z) \vee (y \wedge z)$ After looking through standard…
Mike
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Do you lose any more equational identities when you go past sedenions?

Every Cayley-Dickson algebra can be viewed as a $(+,-,*,0,1)$ algebra. The reals and the complexes share the same equational identities. The quaternions have a subset of the equational identities, because they lose commutativity of multiplication.…
user107952
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Finite free objects

Free distributive lattice with any number of generators is finite. For example with 3 generator the lattice will have 20 elements. Is there other examples of free objects that are finite and have at least ten elements?
Jori Mäntysalo
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why only closed operations

Why does the carrier of an algebraic structure has to be closed under the operations of the algebraic structure? One could also consider $(\mathbb{N}^*, \div)$. But why isn't that an algebraic structure?
asdfusername
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Existence of countably generated free algebra in the universal class

I'm trying to solve the following exercise (from Smirnov's "Varieties of algebras"): Problem: Let $K$ be the universal class of $\Omega$-algebras, i.e. $K = Mod(\Sigma)$, where $\Sigma$ is the set of sentences which are the universal closures of…
Random Jack
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