All Questions
Tagged with algebraic-combinatorics lattice-orders
13
questions
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20
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Lift and frame matroids.
I want to read more about lift matroid and frame matroid and their flats and relations to signed graphs, do you know any basic resources for this?
- 2,087
1
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23
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Characteristic polynomial and bounded regions.
I know that the number of bounded regions of a homogeneous hyperplane arrangement $\mathcal{A}$(a collection of n hyperplanes) in $\mathbb R^d$is ${ n - 1 \choose d}$ but how can this be expressed in ...
- 3,139
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1
answer
53
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Understanding contraction in hyperplane arrangements.
Here are two figures that shows a hyperplane arrangement's contraction (this is from McNulty book, "Matroids, a geometric introduction"):
I am not sure why a became a line in the right ...
- 3,139
0
votes
1
answer
60
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Contraction, loops and flats.
This idea is being used a lot, but I cannot justify why it is correct:
If M is a matroid and $T$ a subset of $E(M).$ Then $M/T$ has no loops iff $T$ is a flat of $M.$
I know how to proof that in a ...
- 3,139
0
votes
1
answer
46
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The basis of a regular matroid.
I know that a regular matroid is one that can be represented by a totally unimodular matrix.
I also know that a rank r totally unimodular matrix is a matrix over $\mathbb R$ for which every submatrix ...
- 3,139
-1
votes
1
answer
52
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what will happen if we contract an element in a uniform matroid? [closed]
Are the parallel elements in a matroid just behaving like loops? If so, why?
For example, in $U_{2,3}$ if we contract an element what will happen? In $U_{2,2}$ if we contract an element what Will ...
- 95
2
votes
1
answer
87
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Partition lattice properties and an invariant.
I am trying to guess the value of the beta invariant of the partition lattice $\pi_4$ if I know the following information:
For any matroid $M,$ I know that
1- $\beta(M) \geq 0.$
2- $\beta(M) > 0$ ...
- 95
1
vote
1
answer
82
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what will happen to the uniform matroid $U_{2,m}$ if we remove an element from it?
I am trying to figure out what will happen to the uniform matroid $U_{2,m}$ if we remove an element e from it, where e is neither a coloop nor a loop. I am guessing that it will become disconnected ...
- 95
2
votes
1
answer
82
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Showing the isomorphism between (the geometric lattice)$\pi_n$ and the lattice $\mathcal{L}(M(K_n)).$
Here is the question I am trying to solve:
Show that the lattice of flats of $M(K_n)$ is isomorphic to the partition lattice $\pi_n$. Definition: $\pi_n$ is the set of partitions of $[n]$, partially ...
- 3,139
0
votes
1
answer
82
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Proving that $ \beta(M) = \beta (M - e) + \beta (M /e).$
Here is the statement I am trying to prove:
If $e \in E$ is neither a loop nor an isthmus, then $$ \beta(M) = \beta (M - e) + \beta (M /e).$$
Here are all the properties I know about the Crapo's beta ...
- 3,139
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votes
1
answer
71
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Why always the Crapo beta invariant value greater than or equal zero?
Here are the definitions of the Crapo beta invariant I know:
My definition of the Crapo's beta invariant of a matroid from the book "Combinatorial Geometries" from page 123 and 124 is as ...
- 3,139
1
vote
0
answers
68
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How to prove Crapo's Lemma
Crapo's Lemma states:
Let $X$ be a subset of a lattice $L$, and let $n_k$ be the number of $k$-element subsets of $X$ with join equal to $\hat{1}$ and meet equal to $\hat{0}$. Then
$$\sum_k (-1)^{...
- 2,085
2
votes
0
answers
130
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Proving Crapo's Lemma
Let $L$ be a finite lattice with least and greatest elements $0, 1$, respectively, and let $X\subseteq L$. Let $n_k$ be the number of $k$-element subsets of $X$ with join $1$ and meet $0$. I want to ...
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