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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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Is this a mathematics field?

When I'm studying, I often find myself searching for the minimal conditions to define something. A simple example is: to define a circumference in a plane you need 3 points or 1 point and a distance, or even 4 points, but this last one is not…
Luis Dias
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Every discriminator variety has equationally definable principal congruences

A variety $V$ has equationally definable principal congruences (EDPC) if there is a conjunction $\Phi(x, y, z, w$) of finitely many equations on four variables such that for all A $\in V$ and all $a, b, c, d \in$ A,$ (c, d) \in \theta(a, b)$ iff A…
wasatar
  • 353
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Varieties with finitely cocomplete category of finite algebras

For which varieties $\mathbf{V}$ (in the sense of universal algebra) do the finite algebras in $\mathbf{V}$ form a finitely cocomplete category, or more strongly, are closed under finite colimits in $\mathbf{V}$? Some examples: Sets (Left or…
Geoffrey Trang
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verify the claim that consequences of balanced identities are again balanced.

verify the claim that consequences of balanced identities are again balanced. An identity is p≈q balanced if each variable occurs the same number of times in p as in q.if ∑ is balanced set of identities then using induction on the lenght of a formal…
Masoomeh Mohammadi
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homomorphisms and congruence relations

Do compositions of homomorphisms in universal algebra correspond to joins of congruence relations? That is- is the congruence relation $g \circ f(a ) = g \circ f( b) \Leftrightarrow a \sim b $ the join of $f(a ) = f( b) \Leftrightarrow a \sim b $…
dje33
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Quotients of quotients in universal algebra

In universal algebra, when is the quotient of a quotient of an algebra $\mathcal{A} $, a quotient of $\mathcal{A} $?
h99yt8
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Why fully invariant congruence is an algebraic closure operator?

If we have an algebra $A$ of type $F$ then congruence of fully invariant is an algebraic closure structure operator on $A\cdot A$. Actually it's in Universal Algebra Sankappanavar page $100$ (Lemma $14.4$). And specially I'm asking why the fully…
MohammadSadegh YazdanParast
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My question is why the set of fully invariant congruence of a set A is closed under arbitrary intersection?

Why ($\operatorname{Con}_{FI}(A))$ is closed under arbitrary intersection?
MohammadSadegh YazdanParast
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Another way of saying that algebraic objects are isomorphic

From a universal algebraic perspective, let's say we have two isomorphic groups. Then can I speak of their isomorphic nature by saying the binary operations of multiplication of the two groups are "isomorphic" in that they encode the same structure?
jf9rj4
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For two classes $K_1,K_2$ of similar type, show $P_U(K_1 \cup K_2) = P_U(K_1) \cup P_U(K_2)$

For two classes $K_1,K_2$ of similar type, show $P_U(K_1 \cup K_2) = P_U(K_1) \cup P_U(K_2)$. One direction seems fairly straight forward. Namely, $P_U(K_1) \subseteq P_U(K_1 \cup K_2)$ and $ P_U(K_2) \subseteq P_U(K_1 \cup K_2)$. We can see this…
oliverjones
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Let $A,B$ be finite SI-algebras and $V(A)$ be congruence-distributive. Then $V(A) = V(B) \iff A \simeq B$

Let $A,B$ be finite SI-algebras and $V(A)$ be congruence-distributive. Then $V(A) = V(B) \iff A \simeq B$ $"\Rightarrow"$ Assume that $V(A) = V(B)$. By a corollary by Jónsson, all the SI-algebras from $V(A)$ are in $HS(A)$. and $V(A) = IP_S(HS(A))$.…
oliverjones
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Show the only (non-trivial) subdirectly irreducible right-zero semi-group is the 2-element one.

Show the only (non-trivial) subdirectly irreducible right-zero semi-group is the 2-element one. (I only need to restrict myself to the finitely generated ones). A right-zero semi group is a semi-group + $xy=y$. Let $RZ$ be a finitely generated…
oliverjones
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Prove every onto homomorphism is a co-equalizer for a pair of homomorphisms

Let $K$ be the collection of algebras of similar type with all the homomorphisms between them. Show that every onto homomorphism is a coequalizer of a pair of homomorphism. Preliminaries Onto Homomorphism : self explanatory Coequalizer : is a pair…
oliverjones
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show $h : \underline{A} \to \underline{B}$ is a homomorphism iff $h$ is a subuniverse of $A \times B$

show $h : \underline{A} \to \underline{B}$ is a homomorphism iff $h$ is a subuniverse of $\underline{A} \times \underline{B}$ where $\underline{A}$ and $\underline{B}$ are similar algebras $\Rightarrow$ Assume $h$ is a homomorphism, $h \subset A…
oliverjones
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What can we say about generating subsets if all free subsets are finite

Definitions An (universal) algebra is a pair $\mathcal A=(A, (f_1,\dots, f_n))$ where $A$ is a non-empty set and $(f_1, \dots, f_n)$ is a family of finitary operations on $A$. The notation $o(f_i)$ will be used for the arity of $f_i$. Given a subset…
xavierm02
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