# 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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### A generalized Boolean algebra gives rise to an implication algebra

A generalized Boolean algebra $G$ is relatively complemented distributive lattice with largest element 1. That is, an element $a\in G$ has a complement in any interval $[x\,,\,1]$ that contains $a$. ...
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### Optimal object in category? Metric, objective function on category objects. Optimization over category?

Is there notion about optimal object in category (that can be found by some algorithms, or - more importantly - that can be constructed (if unknown) by some algorithms), about metric and objective ...
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### Every model of an algebraic theory is a quotient of a free model

I have stumbled upon the following proposition from Borceux: Proposition 3.8.9 Let $\mathcal{T}$ be an algebraic theory. Every $\mathcal{T}$-model $M$ is a quotient of a free model. More precisely,...
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### Are pseudoheaps and heaps the same thing?

An exercise in a category textbook asked me to show that the category of pointed heaps and the category of groups are isomorphic. But my proof somehow didn't use the most unintuitive of the defining ...
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### How can I prove that $\ cong(R)‎\cong‎ Id(R)$?

If $R$ is an arbitrary ring, $\ cong(R)$ is the set of all congruence of $R$ and $Id(R)$ is the set of all Ideals of $R$, How can I prove that $\ cong(R)‎\cong‎ Id(R)$?
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### Nonexistence of infinite subdirectly irreducible algebras

I am trying to prove a theorem of Quackenbush (Theorem 3.8 in Chapter V of Burris & Sankappanavar): If $V$ is a locally finite variety with only finitely many finite subdirectly irreducible ...
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### Image-preimage adjunction induced from regular epi respects regular monos?

Concretely, every set function $f:A\rightarrow B$ induces an adjunction $f_\ast \dashv f^\ast$ between image and preimage. For rings groups, the image and preimage along a surjective ring group ...
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### What equational properties of a group only need to be checked on a generating set?

Let $G$ be a group and $S\subset G$ a generating set. Let $P$ (short for $P(x_1,\dots,x_n) = 1$) be an equational property that may or may not be satisfied by all $n$-tuples of elements of $G$. My ...
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### In which algebraic theories do 'free' and 'projective' coincide?

Free models of algebraic theories are always projective objects in the category of models, but the converse is not always true. For instance, some (actually, all) projective modules are direct ...
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### Definition of finitely generated model

Let $M$ be a model of an algebraic theory $\mathcal T$. $M$ is said to be finitely generated if it is the quotient of a free model $F(n)$ over a finite set $n$. Here, quotient means there exists a ...
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### Are there any preservation theorems for quotients of subalgebras?

Let $X$ denote an algebraic structure. Then: Every subalgebra of $X$ satisfies each quasi-identity that is satisfied by $X$. In other words, taking subalgebras preserves quasi-identities. Every ...
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### What is the standard term for the property of an order-embedding $h \colon P \to P$ such that for all $a \in P$, $a \leq h(a)$?

In an essay, I want to talk about order-embeddings $h \colon P \to P$ on a partially ordered set P such that for all $a \in P$, $a \leq h(a)$. Does somebody know the standard term for a function ...
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### What properties are shared by isomorphic universal algebras?

There is a consensus, that "isomorphy" (the "is isomorphic to"-relation) is the right kind of sameness between universal algebras (say groups, (single-sorted) vector spaces, lattices, ...), because ...
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### This is just the Eilenberg-Moore category, right?

Let $F$ denote a monad on $\mathbf{Set}$. Write $\mathbf{Set}_F$ for the corresponding Kleisli category and $\mathbf{Set}^F$ for the Eilenberg-Moore category. On the train home today, it occured to me ...
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### Polynomial algebras

What is a polynomial algebra? I cannot find any definition of this concept. Is it a set of n-ary polynomial symbols defined over an algebra, as defined on p.29 of this document? http://www.illc.uva....
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### Are there any free or fascist boolean algebras?

A boolean algebra is an algebra with the binary operations $\wedge$, $\vee$, an unary operation $\neg$, and constants $0$, and $1$, satisfying axioms. A heyting algebra is an algebra with the binary ...
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### Free idempotent semigroup with 3 generators

On Finite free objects I asked examples of finite free objects. I got answer "The free idempotent semigroup satisfies $x^2=x$ for each element $x$. And I confirm it is finite if it is finitely ...
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### Finite free objects [closed]

Free distributive lattice with any number of generators is finite. For example with 3 generator the lattice will have 20 elements. Is there other examples of free objects that are finite and have at ...
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### Universal Algebra: partial algebras and homomorphisms

I'm looking for an accessible introductory text on universal algebra, one which discusses homomorphisms in the context of partial algebras. I have been recommended the Grätzer book, "Universal ...
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### Terminological conventions regarding group actions

Suppose: $G$ is a group $T$ is a Lawvere theory ("algebraic theory") and $X$ is a $T$-algebra What conventions surround the phrase: "action of $G$ on $X$"? In particular, does this mean: a ...