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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Let $A$ be a semigroup and $X\subseteq A$. Define $X_0=X$ and $X_{n+1}=X_n\cup \{ab\, |\, a\in X,\, b\in X_n\}$. Then $\cup_{n=0}^{\infty}X_n=Sg^A(X)$

Let $A$ be a semigroup and $X\subseteq A$. Define $X_0=X$ and $X_{n+1}=X_n\cup \{ab\, |\, a\in X,\, b\in X_n\}$. Then $\bigcup\limits_{n=0}^{\infty}X_n=Sg^A(X)$, which is the subsemigroup of $A$ ...
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Consistent full Horn theories of two structures

Suppose that two structures $A$ and $B$ whose cardinality is greater than 1 (added in a revision) have the same positive primitive theory. Does it follow that the union of the full Horn theory of $A$ ...
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Can lattices which are also linear orders be characterized equationally?

There is a definition of a lattice as an algebraic structure of type $\langle 2,2 \rangle$ which satisfies commutativity and associativity for both operations and the absorption laws connecting the ...
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Is the equational theory of sine and cosine trivial?

Consider the algebraic structure $(\mathbb{R}; \sin, \cos)$. I conjecture that the only equations that hold in that structure are of the form $t=t$, for some term $t$. Is this true? If so, can someone ...
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Is there an equational theory which has non-trivial finite models but no infinite model?

This is a dual question to my previous question, here: Is there an equational theory which has infinite models but no non-trivial finite models?. My current question is, does there exist an algebraic ...
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Is there an equational theory which has infinite models but no non-trivial finite models?

Does there exist an algebraic signature $\Omega$ and an equational theory $T$ of $\Omega$ such that $T$ has infinite models, but no non-trivial finite models? Trivial means one-element models.
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Clarification of the definition of a clone

I've begun studying boolean algebra in mathematical logic, after studying a course on abstract algebra which ended with the definition of boolean lattices, so I'm familiar with some algebraic ...
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When do the (Set valued) models of a Lawvere Theory form an abelian category?

By a "Lawvere Theory", I mean a category $\mathbb{T}$ with finite products and a distinguished family of objects $(S_\alpha)$, called sorts, where every object of $\mathbb{T}$ is isomorphic ...
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disjointness of sorts of a many sorted algebra

Given a many-sroted signature sig with the sorts S1, S2, ..., Sn. A many sorted algebra alg ...
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(co)limits of models of a Lawvere theory

Somehow, I got the impression from this answer that, for a Lawvere theory $L,$ the category $\text{Mod}(L,\text{Set})$ of models of a Lawvere theory has all small limits and colimits. Assuming this is ...
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A sound and complete set of axioms and inference rules for quasi-equational logic

Equational logic has some axioms and inference rules to derive equations from other equations. What about quasi-equational logic? Is there, in some text, a set of sound and complete axioms and ...
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When are presheaves models of multi-sorted Lawvere theories?

As I understand it there is a correspondence between finitary monads and single-sorted (ordinary) Lawvere theories. My first question is, for a monoid $M$ in Set, when will it be the case that the ...
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The equational and quasi-equational theory of commutative groups in the signature $*$.

I know that the equational theory of groups in the signature $*$ is axiomatized by the associative law, so does that mean the equational theory of commutative groups in the signature $*$ is ...
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Consequences of the completeness of equational logic

It is a classical result, due to Birkhoff, that equational logic is complete; that is, if $\Sigma$ is a classical (finitary) signature in the sense of universal algebra and $s, t$ are terms ...
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Is there a finite group whose identities are the same as the group identities? [closed]

Consider the equational theory of groups in the signature $\{+,0,-\}$. The equational theory is axiomatized by the associative law, the identity laws, and the inverse laws. My question is, does there ...
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What does it mean to say that the quasi-equational theory of groups is not finitely based?

In a MathOverflow question, I asked if the quasi-equational theory of groups in the signature $\{*\}$ is finitely based. The answer was negative. However, groups in the signature $\{*\}$ can ...
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2 votes
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Is it decidable if a finite set of equations have only trivial models?

Fix an algebraic signature $\Omega$. Let $F$ be a finite set of equations in $\Omega$. Is it decidable if the set $F$ has only trivial models? By trivial models, I mean one-element models. For example,...
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3 votes
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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 ...
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Prove that surjective homomorphisms preserve satisfaction of equations

I'm currently studying universal algebra and I'm having trouble to prove the following statement: Let $\Sigma$ be a many sorted signature and A and B be $\Sigma$-algebras. Prove that if there is a ...
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Can a clone be defined as a structure in a first order theory?

Can a clone be defined as a structure in a first order theory? Clones are described in A Short Introduction to Clones by Kerkhoff, Pöschel, and Schneider. I think they're collections of graphs of ...
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Examples of a structure that is not closed under taking ultraproducts?

I simply need some more examples for a short teaching session and am out of interesting ideas, any input would be much appreciated (diagonalizable algebras and those which are similar to them are ...
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2 answers
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Left adjoint to the inclusion of Boolean algebras into distributive lattices

Let $\mathbf{Boole}$ be the category of Boolean algebras. Let $\mathbf{BDL}$ be the category of bounded distributive lattices. There is a fully faithful functor ${\mathbf{Boole} \rightarrow \mathbf{...
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Show that a distributive lattice can be embedded into product of two-element lattices

I have seen this exercise in Bergman: Universal Algebra: Fundamentals and Selected Topics. Let $L$ be a distributive lattice and let $2$ be a 2-element lattice. Show that there is a set $J$ and ...
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Do cyclic groups appear as symmetry groups of exponentiation-only terms?

Let $\mathfrak{E}=(\mathbb{N};\mathsf{exp})$ be the structure consisting of the natural numbers together with exponentiation. Given a term $t$ in the language of $\mathfrak{E}$ - that is, an ...
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What is the difference between unary relations and one-element relations?

My question is exactly like in the title: What is the difference between unary relations and one-element relations? If I start with definitions, given a set $X$ unary relation $\theta$ is a subset of $...
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1 answer
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The n-ary versions of NOR and NAND

This is really two questions in one. Let $n$ be a natural number greater than or equal to $3$. The $n$-ary NOR is defined to be the $n$-ary operation on $\{0,1\}$ which outputs $1$ when all $n$ ...
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3 votes
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Looking for 1952 paper of Higman and Neumann

I am looking for the paper Graham Higman and Bernhard Hermann Neumann, Groups as groupoids with one law, Publicationes Mathematicae Debrecen 2 (1952), 215–221. In it, the authors prove (among other ...
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Texts that talk about multi-valued functions on the reals and complexes

I am interested in texts that discuss multi-valued functions, especially on $\mathbb{R}$ and $\mathbb{C}$. For example, in many precalculus and calculus texts, they define the n-th root function. It ...
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2 votes
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How to compute a free algebra for a given algebra over a set?

This is a question from a "demo exam" a few years ago at out university course. The question: How would you compute the free algebra for $\mathcal{A}$ over $X = \{x, y \}$, where $\mathcal{...
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1 vote
1 answer
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How to prove that there is no 5-ary operation in a clone satisfying certain conditions

Question: How to prove that there is no 5-ary operation $f \in \mathcal{C}$ satisfying $f(2, 1, 3, 4, 3) = 1$ and $f(2, 1, 1, 4, 3) = 2$? The $\mathcal{C} = Clo(\textbf{A})$ is a clone of $\textbf{A}$,...
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1 vote
1 answer
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How to show that an algebra is contained in a clone

My question: If we have a be the binary operation on 2 = {0, 1}, denoted $\overline{\land}$, defined by \begin{array}{|c|c|c|} \hline \overline{\land} &0 & 1 \\ \hline 0&1 & 0\\ \...
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7 votes
1 answer
140 views

Symmetry groups corresponding to terms in an algebra

Given an algebra (in the sense of universal algebra) $\mathcal{A}$ with at least one operation of arity $>1$, let the symmetry spectrum of $A$ be the class $\mathsf{SySp}(\mathcal{A})$ of finite ...
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3 votes
0 answers
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Superposition dimension a la Kolmogorov/Arnold

(Now asked at MO.) Given a topological space $\mathcal{X}$, let $$\mathsf{Cl_C}(\mathcal{X})=\bigcup_{n\in\mathbb{N}}C(\mathcal{X}^n,\mathcal{X})$$ be the clone of finite-arity continuous functions on ...
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3 votes
2 answers
142 views

How to create multiplication table ("Cayley table") for an algebra or class of algebras?

I am studying universal algebra and getting familiar with the concept of variety of algebra. As far as I understand, a variety is just a class of all algebras satisfying given set of identities. Also, ...
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1 answer
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How to prove these algebras are subdirectly irreducible, using the claim about their congruence relations? Am I proceeding correctly?

I am solving an exercise from my class and my task is to prove that two algebras descried below are subdirectly irreducible. I have described the algebras and their non-trivial congruence relations ...
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1 answer
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The Use of Ordinals and Cardinals in Recursive Proofs

I'm reading something on ordinal numbers and I have a probably silly doubt. Well, let $Z:=\{x_i: \, i \in I\}$ be a subset of a given set $X$. Define in $X$ a sequence $Y_\alpha=\Gamma(Y_{\alpha-1})$ ...
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What is an intuition for subdirect irreducibility?

I am reading "A Course in Universal Algebra" by Burris and H.P. Sankappanavar and they provide this definition: "An algebra A is subdirectly irreducible if for every subdirect embedding ...
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How to prove these algebras are subdirectly irreducible?

During lectures, we got to show that $C_9$, $C_9 + 1$ are subdirectly irreducible, where $C_n$ := $(\{0, 1, . . . , n − 1\}, (0 1 . . . n − 1))$ $C_n + 1$ := $(\{0, 1, . . . , n − 1, n\}, (0 1 . . ...
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2 votes
0 answers
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Is there a finite basis for the equational theory of the orthocenter?

Given a topological space $\mathcal{X}$ and finite-arity partial functions $f_i$ which $(1)$ are defined on a dense open subset of (the appropriate product of) $\mathcal{X}$ and $(2)$ are continuous ...
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How to prove that normal subgroups are mapped to congruences?

I am reading a paper by Kelly Ann Chambers titled " The isomophirms of the lattice of congruence relations on a group and the lattice of normal subgroups of a group". Chambers is proving ...
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-1 votes
1 answer
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What is a discrete congruence? [closed]

I know a congruence (or congruence relation) is an equivalence relation on an algebraic structure, compatible with the operation on the structure. I am also familiar with equivalent definitions. ...
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1 vote
1 answer
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Equivalent definitions of congruences (in context of universal algebra)

I am familiar with this (informal) definition of congruence relation (or simply congruence): (1) A congruence is an equivalence relation on an algebraic structure that is compatible with the structure ...
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2 answers
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Draw lattice of all subuniverses or congruences

I am solving this exercise: Draw lattices of all subuniverses and of all congruences of $(\mathbb{N}, \star)$, where $x \star y$ is defined as $\max(x, y) + 1$ and $\mathbb{N}$ is the natural numbers ...
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1 vote
1 answer
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How to prove that $\theta_1 \vee \theta_2$ = $\theta_1 \cup (\theta_1 \circ \theta_2) \cup (\theta_1 \circ \theta_2 \circ \theta_1) \cup ...$?

How to prove that $\theta_1 \vee \theta_2$ = $\theta_1 \cup (\theta_1 \circ \theta_2) \cup (\theta_1 \circ \theta_2 \circ \theta_1) \cup ...$? This is a theorem from "Burris, Sankappanavar: A ...
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1 vote
0 answers
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How to prove something is NOT a ring isomorphism?

I have a question about rings (I assume ring without the requirement of multiplicative identity element). Suppose, I want to show, for two rings, there is NO ring isomorphism (the rings are not ...
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1 vote
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Generalization of fixpoint subgroup and Knaster-Tarski

The Knaster-Tarski theorem states that the set of fixed points of a monotone function on a complete lattice forms a complete sublattice. In group theory, the set of fixed points of an automorphism of ...
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When are direct products cancellative for finite algebras?

Suppose $\mathcal{C}$ is a variety in the sense of universal algebra with the additional stipulation that all algebras within $\mathcal{C}$ are finite. Then we'll say direct products are cancellative ...
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If $f:A \rightarrow B$ is a surjective homomorphism, is there an injective homomorphism $g:B \rightarrow A$?

So I am working on modal logic and at some point this problem arose: I have two finite modal algebras $A$ and $B$ and a surjective homomorphism $f:A \rightarrow B$, and I need to construct an ...
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0 votes
1 answer
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What is the interpretation of congruence relations?

I am studying universal algebra and read An Invitation to General Algebra and Universal Constructions by Bergman. This is a definition from the book, about congruences. My question is, what is a ...
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2 votes
2 answers
71 views

Can we derive associativity of symmetric difference from its simpler properties?

The symmetric difference $Δ: 𝒫(X)×𝒫(X) →𝒫(X)$ has a few obvious properties: $∅$ acts as the neutral element, i.e. $SΔ∅ = S$ It is commutative Every element is its own inverse. The (imo) only non-...
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