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2
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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
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
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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!
3
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0answers
35 views
What is a simple axiomatisation of groups using division?
I recall from an old exercise I did as an undergrad that groups can be axiomatised using division rather than multiplication:
A group is a non-empty set equipped with a binary division operator / ...
1
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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 ...
1
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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 ...
1
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0answers
49 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 ...
1
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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
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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
0
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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
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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 ...
0
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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 ...
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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.
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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
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0answers
79 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
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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)
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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
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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 ...
0
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0answers
67 views
direct product of different algebras?
Is it possible to define a "direct product" of two algebras with different signatures? For example, boolean lattice $\otimes$ monoid? Perhaps we need to take some quotient to make sense of it (e.g Q ...
