# Tagged Questions

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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### Are algebraic structures required to satisfy axioms?

Is it a requirement for algebraic structures, when studying universal algebra, to satisfy axioms? The reason I ask is because algebraic structures are only defined by a underlying set, a signature, ...
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### Varieties of groupoids which aren't definitionally equivalent

Here is the exercise from Smirnov's book "Varieties of algebras" (in Russian). Problem: Let $\mathcal{U}$ be the variety of all groupoids $(A, \cdot)$ and $\mathcal{V}$ be the variety of all ...
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### Weak Amalgamation Property for Boolean algebras

I'm trying to study universal algebra and lattice theory by myself. Just got stuck with an exercise from Gratzer's "General Lattice Theory" and it seems to me that I don't fully understand the notion ...
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### A question about commutative algebraic theories and free elements on one generator

Let $T$ denote a commutative algebraic theory with a constant symbol. (We definitely need to assume that $T$ has a constant symbol, otherwise the algebraic theory of idempotent Abelian semigroups is ...
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### Embeddings $A → B → A$, but $A \not\cong B$?

Are there any nice examples of structures (groups, modules, rings, fields) $A$ and $B$ such that there are embeddings $A → B → A$ while $A \not\cong B$? I would especially like to see an example for ...
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### Reducing a set of generators to a free base

Given an algebraic structure $A$ which is free on some subset of its underlying set, does every generating subset of $A$ contain a free generating set? For vector spaces, this is true, what about ...
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### Congruence lattice of $N_5$

I calculated the the congruence lattice of $N_5$ using hit and trial and then verified it with Universal Algebra calculator. But I need to prove that it is the congruence lattice of $N_5$ How should I ...
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### Textbook question on variety

Suppose a variety V is defined by an infinite minimal set of identities. Show that V is a subvariety of at least continuum many varieties.
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### Variety satisfying an identity.

$V$ is a variety of commutative semigroup satisfying the identity $x^2 = x^3$. I need to prove that: $|F_V(\{x_1,\dots,x_n\})|$ = $3^n -1$. Any hints on this ? $F_V$ is V-free algebra.
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### Suggestions for a learning roadmap for universal algebra?

I think a useful combination of resources for universal algebra would ideally, when taken together: Provide ample motivation behind the various developments in the field. Either provide powerful ...
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### Is there a name for the algebra of substructures?

Let $X$ denote an entropic algebra (see here), which just means that all the operations of $X$ are homomorphisms $X^n \rightarrow X.$ Abelian groups are the classic example. Then for any operation of \$...
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### Why are particular combinations of algebraic properties “richer” than others?

Pedagogically, when students are exposed to algebraic structures it seems standard for the major emphasis, if not all the emphasis, to be on groups, rings, R-modules, and categories. These are rich ...
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### How is modularity a weakened form of distributivity?

While reading an essay Lattice Theory- Its Birth and Life, the following line confused me: modularity is a weakened form of distributivity Just to be clear, here modularity and distributivity of ...
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### Universal Algebras for Pseudovarities and their cardinality

A Birkhoff variety is a class of algebras closed under division and arbitrary products, a pseudovariety is a class of algebras closed under division and finite products. Now for each type of variety,...