All Questions
Tagged with lattice-orders graph-theory
25
questions
0
votes
1
answer
60
views
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 ...
- 3,127
-1
votes
1
answer
52
views
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
views
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
views
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
1
vote
1
answer
36
views
Infinite lattice with every totally ordered set finite
Construct a lattice L such that L is infinite but every totally ordered subset of L is finite? I really don't know how to proceed because i don't see how every totally ordered set would be finite ...
- 79
2
votes
1
answer
81
views
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 ...
- 3,127
2
votes
0
answers
69
views
Is there a name for graphs that appear as Hasse diagrams of finite lattices?
Hasse diagrams
are mathematical diagrams used to represent finite partially ordered sets,
and may be seen as a kind of graph.
Apparently, there are some relations between particular kinds of lattices ...
- 1,148
1
vote
1
answer
56
views
If you add one edge at a time to an infinite path will you eventually get a less-than operator?
Let $P = \left \{\,\,(\,k \,\,,\, k+1): k \in \mathbb{N} \cup \{0\}\,\,\right\}$
Alternatively, $P =\left\{\,\,(0 \,\,,\, 1), \,\,(1 \,\,,\, 2), \,\,(2 \,\,,\, 3), \,\,(3 \,\,,\, 4), \,\,(4 \,\,,\, 5),...
- 1,104
0
votes
1
answer
67
views
A subgraph relation from an algebraic identity
Let $G_1, G_2$ and $G_3$ be undirected simple graphs such that $(G_1\cap G_2)\cup(G_2\cap G_3)\cup(G_1\cap G_3)=G_2$. How to show that $G_1\subseteq G_2\subseteq G_3$?
Note that the graph's union $\...
- 1,352
2
votes
0
answers
184
views
Dual definition of self-duality
We can define the notion of self-duality on lattices as follow :
A lattice $L$ is self-dual if there is a permutation $\pi$ such that : for all $a \in L$, $\ \downarrow \pi(a)$ is antiisomorphic ...
- 1,310
1
vote
0
answers
114
views
Is there any connection between distributive lattices and planar graphs?
I'm asking this because there are very similar theroems describing these properties: A lattice is distributive if and only if it does not contain N5 (pentagon) or M3 (diamond) and according to ...
- 19
6
votes
1
answer
765
views
Improvement on the concept of separating families for the union-closed sets conjecture?
The union-closed sets conjecture states the following. Let $F$ be a finite family of finite sets that is union-closed and let $\cup (F)$ be the union of all sets in $F$. Then we can find an element in ...
- 635
2
votes
0
answers
47
views
Finding the greatest lower bound of a finite lattice
For the lattice below. What is the Greatest Lower Bound of $a_2$ and $a_4$ (i.e., $a_2 \land a_4$) and $a_3$ and $a_4$ (i.e. $a_3 \land a_4$)?
I thought it was $a_2 \land a_4 = a_2$ and $a_3 \land ...
- 865
2
votes
0
answers
113
views
A problem in proving isomorphism classes of cores forming a lattice
I am reading the book "Algebraic Graph Theory" by Chris Godsil and Gordon Royle and I got confused with Lemma 6.3.3:
The set of isomorphism classes of cores, partially ordered by "$\rightarrow$", ...
- 4,485
2
votes
1
answer
129
views
What does distributive property translate into in an non directed graphs?
The distributive property of distributive lattice according my intuition should bring in some restriction so as to the kind of links we can have in our hasse diagram. Is it so? Earlier I thought that ...
