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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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Is the partial order of equational theories of a single binary operation dense?

Let our signature be a single binary operation $\{*\}$. Consider the set of equational theories of that signature, partially ordered by inclusion. Is that partial order dense?
user107952
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Apparent counterexample in the theorem that term functions form a clone.

Consider the structure $(\mathbb{R};)$, that is, just the set $\mathbb{R}$ with no operations, constants, or relations. We have a countably infinite set of variables $x_1, x_2, x_3, ...$. A term function, in this structure, is simply represented by…
user107952
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In the language of two binary operations, is there a single identity that is equivalent to both distributive laws?

Let our language be $\{+,* \}$. $*$ is said to be left-distributive over $+$ iff $x * (y + z)=(x*y)+(x*z)$, and is said to be right-distributive over $+$ iff $(x+y)*z=(x*z)+(y*z)$. That raises the question, in the language of $\{+,*\}$, is there a…
user107952
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Two finitely based binary operations whose union is not finitely based, and vice versa

Does there exist a set $S$ and two binary operations $+$ and $*$ on that set, such that both the structures $(S;+)$ and $(S;*)$ have a finite basis of identities, but the conjoined structure $(S;+,*)$ does not have a finite basis? Also, what about…
user107952
  • 20,508
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Is this definition equivalent to the definition of clone of functions?

A clone of functions on a set $S$ is a collection of n-ary functions on $S$ which contain projection functions and is closed under composition of functions. But I have a different definition. My definition of a clone is a collection of functions…
user107952
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Congruences on the lattice $N_5$.

I am studying Universal Algebra using the book of Clifford Bergman and I encounter trouble in one of the exercises. Task (2.12 - 3.) Let $L$ be a copy of $N_5$ with elements $\{a,b,c,u,v\}$ and $u
dmk
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Existence of a closure of a subset

Let $M$ be a set and $\circ$ an operation defined on subsets of $M$ in the following way: $A \circ B \subseteq M$ for any $A \subseteq M$ and $B \subseteq M$; If $A' \subseteq A$ and $B' \subseteq B$, then $A' \circ B' \subseteq A \circ B$. Let us…
Alex C
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Does every equational theory have an independent equational axiomatization?

An equational theory is a theory axiomatized by a set of equations. Does every such theory have an independent equational axiomatization? Independent means no axiom in the set can be deleted without loss of theorems.
user107952
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The Frattini subuniverse is the intersections of all proper maximal subuniverses

From "Universal Algebra: Fundamentals and Selected Topics" of Clifford Bergman. An element $a$ of an algebra $A$ is called a non-generator of $A$ if for every $X \subseteq A$, $A = Sg(X \cup \{a\})$ implies $A = Sg(X)$. (a) Prove that the…
Giovanni Barbarani
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Are there two different binary operations on a set with at least two elements that do not enmesh?

Consider an algebraic structure $(S,+,*)$, where $+$ and $*$ are simply two different binary operations, not necessarily addition or multiplication. We say that $+$ and $*$ do not enmesh if the equational identities of $(S,+)$ united with the…
user107952
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On the algebraic theory of Boolean algebras

I have a question which (I think) should be easy for the experts: Is the Lawvere theory of Boolean algebras commutative, i.e. are its operations "algebra homomorphisms under any interpretation"?
Marti M.
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Notation for many-sorted term algebras

I'm writing a computer science paper, and need to give a formal definition of many-sorted term algebras (which correspond to the abstract syntax of programming languages). I'm having trouble finding a standard development of the theory to use. So…
James Koppel
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Which "conditions" generate subalgebras?

While looking at this question I suddenly wondered about a more general question. Think of your favorite class of algebra (groups or rings, say), and forgive me for vocabulary mistakes while I try to frame an unfamiliar question. For an algebra $A$,…
rschwieb
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free algebras and K-free algebras

Given a set $X$ and a class $K$ of universal algebras of the same type $F$, an $F$-algebra $U$, generated by $X$, is said to be free over $X$ for the class $K$ if, for every $F$-algebra $A$ in $K$ and every function $\alpha\colon X\to A$ there…
bateman
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What should be the definition of Mal'cev algebra in Wikipedia's kernel page?

In the Wikipedia article about kernels, universal algebra section, Mal'cev algebras subsection (https://en.wikipedia.org/wiki/Kernel_(algebra)#Mal.27cev_algebras) appear some so-called Mal'cev algebras (different from the nonassociative ones) which…
Jose Brox
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