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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 there a finite generating set for the clone of all operations on a finite set with at least 3 elements?

Let $S$ be a finite set with at least $3$ elements, and let $O(S)$ be the set of all finitary operations on $S$. That is, it is the set of all unary, all binary, all ternary, etc operations collected in one big set. Does there exist a finite set of…
user107952
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Can I use ''exist'' predicate in the definition of algebra in equationally definable class?

I'm reading Algebraic Theories by Ernest G. Manes, and I'm wondering about equationally-definable class. For example, group is a typical example of equationally-definable algebra, but the definition of group uses ''for-all'' sentence: $\forall…
ぼっけなす
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Is $s=t$ always strictly stronger than $s+s=t+t$?

This question was inspired by a comment on one of my previous questions. As before, let our signature be that of a single binary operation $+$. Suppose $s$ and $t$ are two distinct terms in this signature. Then, is it always the case that the…
user107952
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Closed Set Sytem: closed under union of countable chains implies closed under union of chains

I'm reading from "ALGEBRAS, LATTICES, VARIETIES" by Ralph N. McKenzie, George F. McNulty and Walter F. Taylor and had a question about closure systems introduced in Chapter 2 on Lattices. Let $A$ be a set. We say countable chain here to mean a…
Josstopher
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$\Omega$-algebra is finitely-generated iff the union of any directed set of proper subalgebras is proper.

I'm working through Jacobson's Basic Algebra II and a problem in the chapter on Universal Algebra is to prove that an $\Omega$-algebra $A$ is finitely-generated iff the union of any directed set of proper subalgebras is proper. It was easy to prove…
ComplexManifold
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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 signature $\Omega$ and an equational theory $T$…
user107952
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No independent generating set of equations for addition with cardinality greater than 2.

Consider the structure $(\mathbb{R};+)$. Now, I know that it's equational theory can be generated by the commutative and associative equations. I also know that the commutative and associative equations are independent of each other. My question is…
user107952
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An algebra whose equational theory is finitely based but whose quasiequational theory is not, and vice versa.

Is there an algebraic structure $K$ such that its equational theory has a finite basis, but that its quasiequational theory does not have a finite basis? Also, what about vice versa? That is, is there an algebraic structure $K$ such that its…
user107952
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Is there a finite equational basis for this metric space structure?

Consider the algebraic structure $(\mathbb{R};d,0)$, where $d$ is the distance $|x - y|$ between two real numbers. Is there a finite equational basis for the identities of that structure? I conjecture that the commutative law and the equation…
user107952
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Equational identities of the $(+,*,0,1)$ reducts of rings

This is a follow up to my previous question on the equational identities of the $(+,*,0,1)$ reducts of commutative rings. In this question, I want to consider the equational identities of the $(+,*,0,1)$ reducts of rings, not necessarily…
user107952
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Tarski's Theorem: V=HSP

I'm trying to show the Tarski's Theorem in Universal algebra. The theorem states that V=HSP, where V,H,S,P are operators between classes of algebras, V(K) is the smallest variety containing K, H(K) is the class of algebras which are homomorphic…
SilvioM
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Neither NAND nor NOR generates all n-ary functions on $\{0,1\}$

It is said that NAND alone can generate all $n$-ary functions on $\{0,1\}$, for all natural numbers $n$. However, I think the correct statement is that it can generate all $n$-ary functions, where $n\geq 1$. I don't think it generates the nullary…
user107952
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Is this an axiomatization of the meet of the commutative and associative theories?

This is related to a question I asked before. In that question, I asked if the meet of the commutative and associative properties can be axiomatized by the equation $(x * y) * x = x * (y * x)$. The answer was no. So, now I am asking, if we add also…
user107952
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Are "quasi-injective homomorphisms" necessarily injective?

Given two algebras $A$ and $B$ in any fixed variety $\mathbf{V}$ of algebras and any $\mathbf{V}$-homomorphism $f:A \to B$, define $f$ to be "quasi-injective" if for any $\mathbf{V}$-algebra $C$ and any map $g:C \to A$, if $f \circ g$ is a…
Geoffrey Trang
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Is the class of anticommutative magmas a variety?

A magma is simply a set $S$ with a single binary operation $*$. An anticommutative magma, under my definition, is one where $x*y=y*x \implies x=y$. This is certainly a quasivariety, simply by definition. Is it in fact a variety?
user107952
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