Questions tagged [lattice-orders]
Lattices are partially ordered sets such that a least upper bound and a greatest lower bound can be found for any subset consisting two elements. Lattice theory is an important subfield of order theory.
1,521
questions
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
0
votes
0
answers
49
views
Poset property that an element has a unique maximal element above it
Definition:
Consider a partially ordered set $(\mathcal X,\leq)$ with a top element $\top$.
Call an element proper when it is not equal to $\top$. If $X'\leq X$ then say that $X$ is above $X'$.
Call ...
- 538
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
1
vote
1
answer
39
views
Properites of Lattices imply eachother.
Assume S is a finite poset.
A lattice is a poset S such that
S is bounded.
∀ x, y ∈ S there exists x ∧ y (existence of Meet/Infimum).
∀ x, y ∈ S there exists x ∨ y (existence of Join/Supremum).
...
- 13
0
votes
0
answers
84
views
Show that $\text{Part}(A)$ is a complete lattice
Let $A$ be a set and let $\text{Part}(A)$ denote the collection of all partitions of $A$. Define the relation $\leq $ on $\text{Part}(A)$ by $P_1\leq P_2$ if and only if for every $A_1 \in P_1$ there ...
- 4,827
3
votes
1
answer
73
views
Automorphisms of a finite partition lattice
Is the group of automorphisms of a lattice of partitions of the set $X$, where $|X| = n$, isomorphic to $S_n$? I think it is (for sure it's not 'smaller' than $S_n$), but i can't find any proof of it ...
- 33
0
votes
0
answers
128
views
How to show that lattice of subgroups D4 isn't modular lattice?
Here is a lattice of subgroups D4.
The lattice isn't modular iff there is a "pentagon" as a sublattice.
As we can see $\left \{ \rho_{0} \right\} - \left \{ \rho_{0}, \mu_1 \right\} - \left ...
- 125
0
votes
1
answer
82
views
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 ...
- 3,127
0
votes
1
answer
71
views
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 ...
- 3,127
0
votes
0
answers
40
views
Trying to understand the definition of a given lattice ($2^{(a,b,c,d)}$ ,⊆)
I'm solving this exercise that's part of my university course on models of computation and I'm trying to understand the definition of the following lattice :
Given the lattice ($2^{(a,b,c,d)}$ ,⊆). We ...
- 11
1
vote
1
answer
93
views
Distributive (sub-)lattice with a triangle 'tile'
Following from this Q+A, I have a family of lattices such that:
They are not Distributive in general;
They do not include any diamond sub-lattice M3;
They do include pentagon sub-lattice N5;
Specific ...
- 147
3
votes
3
answers
145
views
Are all complete lattices a pointed complete partial order, and vice versa?
A friend of mine asked for my help in drawing a venn diagram that includes the notions of partial orders (PO) in general, complete partial orders (CPO), pointed complete partial orders (CPPO), total ...
1
vote
0
answers
172
views
Boolean algebra is to classical logic like what is to relevant logic?
The Question:
Boolean algebra is to classical logic like what is to relevant logic?
Context:
I guess this is a terminology question, so there's not much I can add, except that I've been interested ...
- 45.7k
0
votes
1
answer
28
views
Is there any kind of distributive law for $c\cdot\max(a,b)$, allowing both signs of $c$?
Notation: For any two numbers $a$ and $b$, let the maximum be $a\sqcap b$, and let the minimum be $a\sqcup b$. (No, my symbols aren't upside-down. Compare this with floor notation; $\lfloor a\rfloor\...
- 5,726