The universal-algebra tag has no wiki summary.
68
votes
5answers
3k views
In (relatively) simple words: What is an inverse limit?
I am a set theorist in my orientation, and while I did take a few courses that brushed upon categorical and algebraic constructions, one has always eluded me.
The inverse limit. I tried to ask one of ...
1
vote
0answers
91 views
amalgam of structures
Trying to refine my question here. This is a response to the questions here:
Homomorphisms between structures
My objective is to take a set of $S-$structures and form an amalgam object out
of that ...
0
votes
1answer
54 views
Describing all subdirectly irreducible mono-unary algebras.
(Wenzel). Describe all subdirectly irreducible mono-unary algebras. [In particular
show that they are countable.] Thanks!
15
votes
2answers
350 views
A structural proof that $ax=xa$ forms a monoid
During the discussion on this problem I found the following simple observation:
If $M$ is a monoid and $a \in M$ then $\{x: ax = xa\}$ is a submonoid.
This is trivial to prove by checking ...
6
votes
4answers
62 views
Any commutative associative operation can be extended to a function on nonempty finite sets
This is a fact we use very frequently in general mathematics when we write such notations as $1+2+3+4$: since we know that $+$ is commutative and associative, we can just "drop the parentheses" and ...
6
votes
2answers
64 views
In a slice category C/A of a category C over a given object A, What is the role of the identity morphism of A in C with respect to C/A
In a slice category $C/A$ of a category $C$ over a given object $A$, what is the role of the $C$ identity morphism, $A\to A$ ($1_A$), in $C/A$, particularly with respect to composition?
I ...
3
votes
2answers
229 views
Variety generated by finite fields
Let $K_1,\dotsc,K_n$ be finite fields and let $V$ be the variety of rings, generated by the $K_i$ (rings aren't necessarily unital). I want to figure out what $V$ looks like. By a theorem of Tarski, ...
8
votes
1answer
186 views
An exercise in infinite combinatorics from Burris and Sankappanavar
Exercise 6.7 in chapter IV of Burris and Sankappanavar's A Course in Universal Algebra starts as follows:
Show that for $I$ countably infinite there is a subset $S$ of the set of functions from ...
3
votes
1answer
117 views
Introductory universal algebra question
I've just started reading about universal algebra, and have already hit a problem (see the two bullet points at the bottom).
My book gives the following definitions (paraphrased):
An operational ...
2
votes
4answers
146 views
why is a nullary operation a special element, usually 0 or 1?
Does a nullary operation mean an operation not taking any argument?
Then why is a nullary operation a special element, usually 0 or 1, in an algebraic structure?
Thanks!
2
votes
1answer
133 views
correspondence for universal subalgebras of $U/\vartheta$
Let $U$ be a universal algebra of type $T$, and denote $\mathrm{Con}(U)\!=\!\{\text{congruence relations on }U\}$ and $\mathrm{Sub}(U)\!=\!\{\text{subalgebras of }U\}$. Let "$\leq$" mean "subalgebra".
...
2
votes
1answer
79 views
$M_3$ is a simple lattice
I'd like to prove (exercise 9.5 in Roman's Lattices and Ordered Sets, p.203) that the lattice $M_3$
is simple, meaning that the only congruences on $M_3$ are the trivial ones (the 'equality' ...
1
vote
2answers
166 views
Existence of universal enveloping inverse semigroup (similar to “Grothendieck group”)
Context
In its simplest form, the Grothendieck group construction associates an abelian group to a commutative semigroup in a "universal way".
Now I'm interested in the following nilpotent ...
0
votes
0answers
41 views
Identities, Free Algebras
Given a type F and a set of variables X and p; q P TpXq show that TpXq |ù p q iff
p q (thus TpXq does not satisfy any interesting identities).
EDITED VERSION:
Exercise 11.1 from ...
0
votes
1answer
78 views
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 $A$?
-2
votes
1answer
121 views
presentation of the direct sum of commutative rings / algebras
If $I,J$ are index sets, $R$ a commutative unital ring, $\mathfrak{a},\mathfrak{b}$ ideals of polynomial rings $R[x_i; i\!\in\!I]$, $R[y_j; j\!\in\!J]$, and $\langle\langle\ldots\rangle\rangle$ is the ...
