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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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Follow up to a previous question on universal algebra

I know that if $r$ is a transcendental constant, then $(\mathbb{R},+,*,r)$ has no identities beyond commutativity, associativity, and distributivity. My intuition tells me that the converse is false. So, given an arbitrary algebraic constant $r$,…
user107952
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The variety of rings which contain a fixed ring as a subring

From Barnes and Mack's "An Algebraic Introduction to Mathematical Logic" S is a commutative ring with 1. Show that the class of commutative rings R with 1_R = 1_S and which contain S as a subring is a variety (in the sense of universal algebra). I…
Duncan Ramage
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Jónsson's lemma : A variety generated by finitely many finite lattices has finitely many subvarieties.

I need to show that a variety generated by finitely many finite lattices yields only finitely many subvarieties. I think Jónsson's lemma is the key but I am not sure how to use it. For those that do not know here is the lemma: If $V$ is a congruence…
oliverjones
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Show every algebra can be embedded into each of it's ultrapowers

Show every algebra can be embedded into each of it's ultrapowers. Let $I$ be any set and and consider $\mathcal{P}(I)$. Let $U$ be an ultrafilter over $I$. Then, we define the ultrapower for some algebra $A$ to be $\prod_{i \in I} A_i/U$ where for…
oliverjones
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If a variety of algebras V has the CEP then $HS(K) = SH(K), K \subseteq V$

If a variety of algebras V has the CEP then $HS(K) = SH(K), K \subseteq V$ CEP := congruence extension property $SH(K) \le HS(K)$ is always true; I have a lemma about it, not going to list it here. Just need to get $HS(K) \le SH(K)$ Let $A \in…
oliverjones
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Show given bi-unary algebra is sub-directly irreducible.

Show given bi-unary algebra is subdirectly irreducible. Let $A$ be the the set of functions from $\omega$ to {$0,1$} and define the bi-unary algebra $\langle A, f, g \rangle $ as follows: $f(a)(i) = a(i +1) \\ g(a)(i) = a(0)$ A definition states…
oliverjones
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Cokernel in universal algebra

Let $(S,f_1,\ldots,f_n)$ be an algebra of some variety and $(T,g_1,\ldots,g_n)$ be another algebra of the same variety. Next let $\varphi:S\to T$ be a homomorphism. I understand well that $\ker\varphi=\{(x,y)\in S^2:\varphi(x)=\varphi(y)\}$, all…
Zelos Malum
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Definition of $\Omega$-algebra

I'm studying universal algebra. I have this definiton: given a signature $\Omega$, an $\Omega$-algebra is comprised of a "carrier" for the algebra and an "interpretation" for every operation symbol. The carrier of $A$ is a set (written $|A|$). The…
Danae Kissinger
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amalgam of structures

Trying to refine my question here. This is a response to the questions here: https://math.stackexchange.com/questions/110530/homomorphisms-between-structures My objective is to take a set of $S-$structures and form an amalgam object out of that set…
atat
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K-free algebra over $\overline{X}$

This is Definition 10.9 of the book "A Course in Universal Algebra" by Burris and Sankappanavar (page 73, Millennium Edition). http://www.math.uwaterloo.ca/~snburris/htdocs/UALG/univ-algebra.pdf Definition 10.9 Let $K$ be a family of algebras of…
Tim Lee
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A Course in Universal Algebra (Millennium edition), page 74

The line before Theorem 10.12 says that "In general $F_K(\overline{X})$ is not isomorphic to a member of K (for example, let K={L} where L is a two-element lattice, then $F_K(\bar{x}, \bar{y}) \notin I(K))$." (The definition of $F_K(\overline{X})$…
Tim Lee
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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.
Alvis
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Construct an algebra from its finitely generated algebras

In the general sense of an algebra (a set with some operations, as in Universal Algebra courses), is it always possible to construct any full algebra (up to isomorphism) just from its finitely generated subalgebras, by the taking of suitable direct…
FPP
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What is the $K$-free algebra for the class of implication algebras, over a finite set

I suppose the title is pretty self explanatory. I have been struggling with the concepts of $K$-free algebras, where $K$ is some class of same-type algebras, over some set $X$. So, in trying to grasp a more intuitive understanding, I am trying to…
FPP
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Can you determine the polymorphisms of this relation?

Let $A\neq \emptyset$ be arbitrary set. Let us define a relation on $A$ in the following way: $R=\{(a;b;c)\in A^3| a=b \lor b=c\}$. Show that the clone of polymorphisms of $R$ denoted as $Pol(\{R\})$ is the clone of all operations of the form…
Björn
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