Mathematics Stack Exchange
Mathematics Stack Exchange

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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Finite equational basis for the identities of the field of real numbers

Consider the structure $(\mathbb{R},+,-,\times,0,1)$, where the $-$ is the unary additive inverse function, not binary subtraction. Can someone exhibit a finite set of identities that can be used to derive all the rest of the identities of that…
user107952
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Converse of a proposition on BE-algebras

A BE-algebra $(B,*,1)$ is a type $<2,0>$ algebra satisfying the identities: $x*x=1$ 2.$x*1=1$ 3.$1*x=x$ 4.$x*(y*z)=y*(x*z)$. Define a relation $R(x,y)$ that holds iff $x*y=1$. It can be proven that $R$ is a partial order if the identity…
user107952
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Set of generating equations for subtraction.

Consider the algebraic structure $(\mathbb{R}, -)$. Is there a finite generating set of equations for subtraction, and if so, can someone exhibit a list of equations.
user107952
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how are vector spaces viewed as universal algebras?

Hey I have this question from Universal Algebra texts where you can see groups, rings, lattices and other structures as Universal Algebras, but I still don't have clear how vector spaces can be viewed in this way (taking into account that all the…
user9398
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Why does "$[\beta,\beta]\leq\alpha$" imply that "$\alpha$ has $\beta,\beta$-term condition" in general algebras?

Let $\mathbf A$ be an (universal) algebra and let us denote $\mathrm{Clo}(\mathbf A)$ the set of term operations of $\mathbf A$. Let $\alpha, \beta$ be congruences of $\mathbf A$. We say that $\alpha$ has $\beta,\beta$-term condition iff $$ \forall…
Ondrej Draganov
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What is non-algebraic structure

If an algebraic structure is a set of operations on a set of elements, what is a non-algebraic structure? https://en.wikipedia.org/wiki/Outline_of_algebraic_structures#Algebraic_structures_with_additional_non-algebraic_structure
Lance
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Characterization of demi-semi-primal algebras

Let me start by introducing some terminology which might not be seen as standard for some people. The discriminator function on an algebra $\mathbf{A}$ is the function $f:A^3 \to A$ defined by $$ f(x,y,z) = \begin{cases} x &, \text{ if } x \neq…
amrsa
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Finitely approximable variety and the finite model property

Call an algebra $A$ finitely approximable if $\forall a, b \in A$ with $a \neq b$ there is a congruence $\theta_{ab} \in \text{Con}(V)$ with $(a, b) \notin \theta_{ab}$ and $A / \theta_{ab}$ is finite. Proposition: $A$ is finitely approximable iff…
Random Jack
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Can you find a plain aneloid?

I defined an "aneloid" to be a set endowed with two operations, adition and multiplication, with multiplication being distributive BOTH sides in relation to adition. I tried to find an example of "plain" aneloid, a such one that adtion and…
Rodrigo Tavares
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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 set of higher cardinality, and if so, what is…
user107952
  • 20,508
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Is there a converse to Baker's theorem?

Suppose that V is a variety of finite signature. Baker's Theorem says that every finite algebra is finitely based if V is congruence distributive. Conversely, if V is locally finite, then every finitely based algebra in V is finite. Are there other…
Ohbi
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Is this a basis for the equational identities of the structure $(\mathbb{N};+,\cdot,0,1)$?

Consider the equational identities of the algebraic structure $(\mathbb{N};+,\cdot,0,1)$. I believe that the following identities are a basis for it: The commutative properties, of both addition and multiplication. The associative properties, of…
user107952
  • 20,508
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A question about Birkhoff's Theorem and subdirectly irreducible algebras

By Birkhoff's Theorem, we know that every algebra is a subdirect product of subdirectly irreducible algebras. So every finite algebra is a subdirect product of finitely many subdirectly irreducible algebrass. Is it known that for which infinite…
khers
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Is it true that every quasi-variety exends to a variety?

I am wondering if the following is true: If $Q$ is a quasivariety in an algebraic signature $S$, must there be a variety $V$ in an extended signature $S′⊇S$ such that $Q$ is the class of algebras isomorphic to $S$-reducts of algebras in $V$?
A. D.
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Equational identities of involutions

Let $S$ be a set, and let $f$ be an involution on $S$ that is not the identity. Consider the algebra $(S;f)$. I conjecture that the equational identities of that algebra are generated by the single equation $f(f(x))=x$. Is this true, or is there a…
user107952
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