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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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Are isomorphic structures really indistinguishable?

I always believed that in two isomorphic structures what you could tell for the one you would tell for the other... is this true? I mean, I've heard about structures that are isomorphic but different with respect to some property and I just wanted…
Oo3
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A finite axiomatization of the join of the commutative and associative properties

Consider the lattice of equational theories of a single binary operation $*$. The meet of the theory axiomatized by commutativity and the theory axiomatized by associativity is simply the theory axiomatized by both of them. What about the join? Is…
user107952
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Can a single nontrivial identity imply associativity and commutativity?

As noted in this old question it's easy to see that no single identity is equivalent to the conjunction of the commutative and associative laws. Question. In the language of a binary operation, is there a single nontrivial identity which implies…
bof
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Subalgebras isomorphic to their images

Question: Given a homomorphism $f:A \to B$, where $A$ and $B$ are algebras in any variety (in the sense of universal algebra), is it true that if $f(S)$ is isomorphic to $S$ for all subalgebras $S$ of $A$ (in symbols: $\forall S \le A \ f(S) \cong…
Geoffrey Trang
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A conjecture in equational logic

In an algebra with a single binary operation g, is there a single equational identity that generates the same set of identities as the set {g(x,y)=g(y,x) , g(g(x,y),z)=g(x,g(y,z))}? My conjecture is that there isn't.
user107952
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A free algebra that is not generic

In a previous question, I was told that free algebras on sufficiently many generators are generic algebras. That raises the question, what is an example of a free algebra that is not generic? An algebra $A$ is generic for a variety $V$ iff $A$…
user107952
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Is there a HSP-like theorem for algebras that can be axiomatized by a finite number of equations?

It is a famous classic theorem that classes of algebras which can be axiomatized by equations are precisely those which are closed under homomorphisms, subalgebras, and products. Is there a corresponding theorem for those classes of algebras which…
user107952
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a result about Tame Congruence Theory

In the book The Structure of Finite Algebras, by Hobby and McKenzie, page 23, it says, in a paragraph before the exercises: The first substantial result proved as an application of tame congruence theory was that $\mathbf{Sub\,A}_{4}$ (and many…
amrsa
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Every Group is Isomorphic to the Group of Automorphisms of some unary algebra A

I am not sure the importance of A being unary here. If anyone can explain if the following is the right idea. So let A be some unary algebra, to be specific not necessarily a mono-unary algebra. I mean that A may have more than one operation but all…
oliverjones
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Subalgebras of free algebras

I am trying to prove the following statement: Problem: There is no free groups in the universal class $\mathcal{A}$ of all abelian groups satisfying $\forall x (x + x = 0) \vee \forall x (x + x + x = 0)$. Hence $\mathcal{A}$ is the union of two…
Random Jack
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Introductory universal algebra question

I've just started reading about universal algebra, and have already hit a problem (see the two bullet points at the bottom). My book gives the following definitions (paraphrased): An operational type is a pair $ (\Omega,\alpha) $, where $ \Omega…
TRY
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Is there an identity strictly between commutativity and the constant identity?

Let our language be a single binary operation symbol $\{*\}$. The commutative identity is $(x*y)=(y*x)$. The constant identity is $(x*y)=(z*w)$. I wonder, is there a single identity that is strictly between these two identities? That is, is there an…
user107952
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Universal algebra question about an infinite set of equations

Fix an algebraic signature $\Omega$. Let $F$ be a finite set of equations in the signature $\Omega$, and let $I$ be an infinite set of equations in the signature $\Omega$ which generate precisely the same equational theory as the theory generated by…
user107952
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What is the smallest possible cardinality of a non-finitely based magma?

I was told that every magma $(S,*)$ whose base set $S$ has 2 elements has a finite basis of identities. The natural question is, what is the smallest possible cardinality of a non-finitely based magma? I would be very interested to see a 3-element…
user107952
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Finite basis for the equational identities of the conditional operator

Consider the structure $(\{0,1\}, *)$, where $*$ represents the conditional operator in logic, defined by $1*1=0*0=0*1=1$ and $1*0=0$. Is there a finite basis for the equational identities of that algebraic structure, and if so, can someone exhibit…
user107952
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