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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).

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Why would the 'naive' definition of homomorphism not work?

What would be a counterexample to the condition in the definition of a homomorphism that goes like down below? $h(f^\mathfrak{A}(a_1,...,a_n)) = h(f^\mathfrak{B}(a_1,...,a_n))$ [as opposed to the original condition, $h(f^\mathfrak{A}(a_1,...,a_n)) =…
user332328
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Free Objects and Equational classes

I can't give a proof of this fact: "Given a non-empty set $X$ and an equational class $V$, then $V$ contains a free object on $X$." $V$ is a class of algebras of a certain language of algebras $L$ and $V$, qua equational class, is $M(\Sigma)$ where…
SilvioM
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Is the intersection of any collection of equational theories itself an equational theory?

An equational theory, under my definition, is the set of all equational consequences of a set of equations. For example, the set of equational consequences of the commutative property is an equational theory. Certainly, the union of two equational…
user107952
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Is there such a thing as a free quasivariety?

I have heard in universal algebra there is such a thing as a free variety, but is there such a thing as a free quasivariety? I would assume, that, for instance, in the language of a single binary operation symbol $*$, a free quasivariety is a free…
user107952
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Are there any congruence relations on $(\mathbb{Z}, *)$ besides mod n?

Consider the structure $(\mathbb{Z}, *)$. Are there any congruence relations on that structure, in the sense of universal algebra, that are not of the form mod n for some integer n? In fact, is there a classification of all the congruence relations…
user107952
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What is the largest variety of ring-like algebras in which the additive identity is absorbing, without it being an axiom?

When I say "ring-like algebra" I mean two binary operations, "addition" and "multiplication", such that multiplication distributes over addition on at least one side, and addition has at least a one-sided additive identity, but no other requirements…
Seth Schmidt-O'Hainle
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Why is an isomorphism between (total) algebras required to have an inverse which is a homomorphism?

Let us consider homomorphisms between partial algebras as defined in https://planetmath.org/homomorphismbetweenpartialalgebras There, an isomorphism from $A$ to $B$ is a bijective homomorphism from $A$ to $B$ such that its inverse is a…
user572708
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Why $Z(A)$ is an equivalence relation on $A$?

For every algebra $A$, the center $Z(A)$ is a congruence on $A$. Why is $Z(A)$ an equivalence relation on $A$?
MIRABBASI
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Proof for Tarski theorem in universal Algebra page 108

Given a variety V and a set of variables X, IrB(Idv(X)) is a convex set. I need a complete proof for this theorem. If anyone can help me it would be wonderful.
MohammadSadegh YazdanParast
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Equational theories & their relation to fully invarient congruences on T(X)?

The equational theories of type F over X form an algebraic lattice which is isomorphic to the lattice of fully invarient congruences on T(X). I need the proof of above theorem which is in page 103 (Corollary14.10) of Universal Algebra sankappanavar.
MohammadSadegh YazdanParast
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Embed an algerbra A into an ultraproduct of its finitely generated subalgebras.

The question is to prove that any algebra $A$ is embeddable into a ultraproduct of its finitely generated subalgebras. In the case of boolean algberas, this seems rather intuitive. I don't know what to use here, say the algebra is $A$, well then…
oliverjones
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if $A$ is free for a class $\mathcal{K}$ over $X$ then $A$ is free over $HSP(\mathcal{K})$ over $X$

if $A$ is free for a class $\mathcal{K}$ over $X$ then $A$ is free over $HSP(\mathcal{K})$ over $X$ So letting $\mathcal{K}$ be a class of algebras and $X$ be a set of generators such that $A \in \mathcal{K}$ is free over $X$. Then we know for any…
oliverjones
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Show coequalizers always exist in the collection $K$ of algebras of similar type with all homomorphisms between them

Show coequalizers always exist in the collection of algebras of similar type with all homomorphisms between them Definition: A coequalizer of two homomorphisms $h,k : A \to B$ in $K$ consists of another algebra $C$ and homomorphism $s : B \to C$…
oliverjones
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Does this algebra whose signature is (1,1) have a name?

Let there be an algebra $(S,f,t)$ with the laws: $$ f(t(x)) = t(x) \\ t(f(x)) = t(x) $$ or, put another way, $$ f \circ t = t \\ t \circ f = t. $$ Does that particular algebra have a name? Does a set with two unary operations (algebras whose…
edom
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why $(\theta _{1}\circ \theta _{2})^{\vee }\subset(\theta _{2}\circ \theta _{1})^{\vee } $

my questiones at this theorem: i coud not undrestand $a\Rightarrow b$ and why $(\theta _{1}\circ \theta _{2})^{\vee }\subset(\theta _{2}\circ \theta _{1})^{\vee } $ please guide me?
zahra
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