The universal-algebra tag has no wiki summary.
6
votes
4answers
56 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 ...
0
votes
0answers
33 views
Definition of induced binary operation
I need the definition of induced binary operation on a set $T$ by $f_A: A^2 \to A$, with $T^2 \subseteq A^2 $ and $T \neq \emptyset$ and $f_A$ a binary operation on $A$..
Thanks in advance
2
votes
3answers
129 views
Are groups algebras over an operad?
I'm trying to understand a little bit about operads. I think I understand that monoids are algebras over the associative operad in sets, but can groups be realised as algebras over some operad? In ...
1
vote
1answer
60 views
Property: closure under an binary operation
We have the following definition:
Definition: let $*_A:A^2 \rightarrow A$ and $B \subseteq A $, with $B \neq \emptyset$, $B$ is closed under $*_A$ if $a *_A|_Bb \in B$ $\forall a,b \in B$.
Property: ...
1
vote
1answer
31 views
Algebraic substructure and restriction of a function
I am reading the follow pdf:
http://www.math.uwaterloo.ca/~snburris/htdocs/UALG/univ-algebra2012.pdf
in particular at pg 28 of pdf, and I think that, let $(A;f)$ an algebric structure and $B ...
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 ...
67
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 ...
0
votes
1answer
23 views
Do cancellative semigroups form a variety of algebras?
Sorry if this is a silly question.
Define that a right-cancellative semigroup is a set $G$ together with an associative operation $*$ such that for all $a,b,x \in G$ it holds that $ax=bx \Rightarrow ...
15
votes
2answers
348 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 ...
1
vote
0answers
27 views
What is an ideal-supporting algebra?
On the wikipedia page for congruence relation it mentions how for groups and rings, congruences can be identified with normal subgroups and rings respectively, and that the most general algebraic ...
2
votes
1answer
40 views
Every group and ring is congruence-permutable , but not necessarily congruence-distributive
The problem is:
Show that every group and ring is congruence-permutable , but not necessarily congruence-distributive.
I know that in group every normal sub group has permutable property and in ...
1
vote
1answer
193 views
Modus Ponens: implication versus entailment
Would it be inconsistent to write Modus Ponens using only implication, not entailment?
$(p \wedge (p \to q)) \to q$
The way I understand is that implication ($ \to$) is an operator that yields a new ...
0
votes
1answer
38 views
Represent the three element chain as a subdirect product of subdirectly irreducible lattices.
Represent the three element chain as a subdirect product of subdirectly irreducible lattices.
I know the two element chain as either a Boolean algebra or a semilattice,is subdirectly irreducible.In ...
0
votes
0answers
10 views
show that consequences of balanced identities are again balanced.
verify the claim that consequences of balanced identities are again balanced.
K is the set of identitis.
by using induction,
if the length of a formal inference is one then for all p≈q∈∑,k implise ...
0
votes
0answers
14 views
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 ...
21
votes
4answers
1k views
Simple explanation of a monad
I have been learning some functional programming recently and I so I have come across monads. I understand what they are in programming terms, but I would like to understand what they are ...
1
vote
1answer
77 views
Initial structures in the category of algebraic systems of the same type
From Handbook of Analysis and Its Foundations by Eric Schechter
9.21. Basic properties of subalgebras. We consider the category consisting of the algebraic systems of some type $(τ, \mathcal{J})$, ...
2
votes
4answers
145 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!
0
votes
0answers
25 views
Question about a property of lattice-morphism
I would like to know if there is a name for the class of commutative (i.e., $\phi(x,y)=\phi(y,x)$) lattice-morphisms $\phi : L_1\times L_{1} \rightarrow L_2$ with the following property:
$\phi(x ...
1
vote
0answers
36 views
Which “conditions” generate subalgebras?
While looking at this question I suddenly wondered about a more general question.
Think of your favorite class of algebra (groups or rings, say), and forgive me for vocabulary mistakes while I try to ...
2
votes
3answers
145 views
Free object is a coproduct: $F_{A\cup B}\cong F_A \coprod F_B$
Let $A,B$ be sets, and $A\sqcup B$ the disjoint union. Suppose that in a (concrete) category, the free objects $F_A,F_B,F_{A\sqcup B}$ exist, and that the coproduct $F_A \coprod F_B$ exists. How can I ...
0
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0answers
13 views
If V is a minimal variety of groups show that fV (x) is nontrivial, hence V = V (fV(x)).
If V is a minimal variety of groups show that fv (x) is nontrivial, hence V = V (fV (x)).
Determine all minimal varieties of groups.
0
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0answers
11 views
(Wenzel). Describe all subdirectly irreducible mono-unary algebras. [duplicate]
Possible Duplicate:
Describing all subdirectly irreducible mono-unary algebras.
(Wenzel). Describe all subdirectly irreducible mono-unary algebras. [In particular
show that they are ...
0
votes
0answers
27 views
solove of Show that for any algebra A and a, b ∈ A,Θ(ha, bi) = t∗(s({hp(a, c), p(b, c)i : p(x, y1,
Show that for any algebra A and a, b ∈ A,Θ(ha, bi) = t∗(s({hp(a, c), p(b, c)i : p(x, y1,
. . . , yn) is a term, c1, . . . , cn ∈ A}))∪A, where t∗( ) is the transitive closure operator,
i.e., for Y ⊆ A ...
0
votes
1answer
53 views
Describing all subdirectly irreducible mono-unary algebras.
(Wenzel). Describe all subdirectly irreducible mono-unary algebras. [In particular
show that they are countable.] Thanks!
0
votes
1answer
32 views
Proof for Tarski theorem in universal Algebra page 108
Given a variety V and a set of variables X, IrB(Idv(X)) is a convex set.
I need a complete proof for this theorem.
If anyone can help me it would be wonderful.
1
vote
2answers
44 views
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} $?
0
votes
0answers
76 views
the universal mapping property
let L be the four- element lattice <{0,a,b,1},⋁,⋀> where 0 is the least element,1 is the largest element, and a⋀b=0, a⋁b=1 (the hasse diagram is figure 1(c)). show that L has the universal mapping ...
0
votes
1answer
77 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
119 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 ...
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
0answers
27 views
Free Algebras and subalgebra
Show that T(X)ùP≈q iff
p = q (thus TpXq does not satisfy any interesting identities). given a type F and a set of variables X and p,qϵT(X)
1
vote
1answer
55 views
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 ...
0
votes
1answer
29 views
Equational theories & their relation to fully invarient congruences on T(X)?
The equational theories of type F over X form an algebraic lattice which is isomorphic to the lattice of fully invarient congruences on T(X).
I need the proof of above theorem which is in page 103 ...
0
votes
2answers
50 views
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 ...
1
vote
1answer
26 views
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?
1
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0answers
48 views
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 ...
0
votes
0answers
53 views
Unique operators for homomorphic functions
If there exists a homomorphic function $f: A \rightarrow B$, so that
$f(U_A(r_1, ..., r_n)) = U_B(f(r_1), ..., f(r_n))$
By given $f$, the spaces $A$ and $B$ and the operator $U_A$, how can we prove ...
3
votes
2answers
193 views
What is the smallest variety of algebras containing all fields?
A field is a ring whose nonzero elements form a commutative group under multiplication. A field is also a commutative inverse semigroup with respect to multiplication. The unique multiplicative ...
2
votes
1answer
102 views
Unification of expressions involving sets
Let's let $\def\OP#1#2{\left\langle#1,#2\right\rangle}\OP xy$ represent the set $\{\{x\},\{x,y\}\}$ as is usual, per Kuratowski. Then:
$$
\begin{eqnarray}
\OP{\OP ab}c & = & \{\{\{\{a\}, ...
4
votes
1answer
50 views
Associativity, Jacobi, and self-action representations
About a year and a half ago, I was at a talk by Martin Hyland where he suggested that the Jacobi identity is to the associative law as the anticommutative law is to the commutative law. I think this ...
1
vote
2answers
165 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 ...
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
98 views
Do filters on a Boolean algebra also make a Boolean algebra?
Let $\mathfrak{B}=(B,\bot,\top,\lnot,\wedge,\vee)$ be a boolean algebra. $B_F$ be the set of all filters on $\mathfrak B$. And for all filter $F$, $G$, $F \wedge_{B_F} G \colon= \mathbf C(F \cup G)$ ...
3
votes
2answers
109 views
Stone duality for ideals and filters (exercise)
In A Course in Universal Algebra (Burris, Sankapannavar), the exercise 4.4.7-8, p.158, says:
Let $A$ be a Boolean algebra. Denote $A^\ast:=\{\text{ultrafilters of }A\}$, and give $A^\ast$ the ...
7
votes
4answers
355 views
Are isomorphic structures really indistinguishable?
I always believed that in two isomorphic structures what you could tell for the one you would tell for the other... is this true? I mean, I've heard about structures that are isomorphic but different ...
3
votes
2answers
149 views
What is the name of the structure Z4 under subtraction?
If we consider $\mathbb{Z_4}$ under addition, then it forms a cyclic group of order 4. However if we change the binary operation to subtraction on $\mathbb{Z_4}$, we get a different structure $J$ with ...
2
votes
1answer
77 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' ...
2
votes
1answer
78 views
ideals of a ring form a modular lattice
We know that if $M$ is a left $R$-module, then $(\{\text{submodules of }M\},\subseteq)$ is a modular lattice.
Taking $M\!=\!R$, we deduce that $(\{\text{ideals of }R\},\subseteq)$ is a modular ...
