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.
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Would the Following Table Strategy Work as an Intuitionistic Decision Procedure?
I had previously sought some insight for handling logical operators in the Rieger-Nishimura lattice and, with assistance here, was able to work out a fairly rigorous way. To the best of my ability, I ...
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Lattice defined on poset vs. Lattice defined on group?
I've seen two different definitions of the term lattice, one is defined on poset, the other one is defined on group. I believe these two are fundamentally different mathematical objects. But I'm not a ...
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Are There Universal Entailments Under the Rieger-Nishimura Lattice for Conditionals When the Antecedent is Higher on It?
I'm working on a bottom-up (atomics-to-proposition) intuitionistic decision procedure, and I encountered some fruits with the Rieger-Nishimura lattice. Specifically, I am looking at this article from ...
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Show that the set of all partitions of a set S with the relation refinement is a lattice.
This one may be one duplicate of QA_1, but its example $\{\{a,d\},\{b,c\}\}\wedge\{\{a\},\{b,c,d\}\}$ seems to not meet the definition in the book because $(\{a,d\} \not\subseteq \{a\}) \wedge (\{a,d\}...
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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?
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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 ...
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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 ...
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When are projective, affine geometries uniform matroids?
I am trying to understand the following corollary in James Oxley book:
A simple rank-r matroid M that is representable over $GF(q)$ has at most $\frac{q^r - 1}{q - 1}$ elements. Moreover, if $|E(M)| = ...
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Help undsersanding matroid closure, loops, contraction and duality.
These ideas are being used a lot, but I cannot justify why they are correct:
If M is a matroid and $T$ a subset of $E(M).$ Then $$(a)\ cl(T) = T \cup \{e \in E(M) - T: e \text{ is a loop of M/T}\}.$$ ...
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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 ...
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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 ...
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Why is this description of the Dedekind–MacNeille completion never mentioned?
There are various ways to describe the Dedekind–MacNeille completion of a poset, the minimal complete lattice in which the poset can be embedded. I’ll first state the ones I’ve seen and then one I ...
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Does this relation partially order partial partitions?
By a partial partition of $X$, I mean a collection of non-empty subsets of $X$ that are disjoint. Given partial partitions $\mathcal{A}$ and $\mathcal{B}$ of $X$, define $\mathcal{A} \leq \mathcal{B}$ ...
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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 ...
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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- $\beta(M) > 0$ ...
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