All Questions
Tagged with lattice-orders combinatorics
105
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Number of lattices over a finite set
I'm interested in finding a general formula for the number of lattices possible over a finite set $S$ as a function of the set's cardinality.
For instance, how many lattices over $\{1, 2, 3\}$ are ...
- 3,468
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Möbius function of distributive lattice only takes values $\pm 1$ and $0$.
In this Wikipedia article, I found the statement
[...] shares some properties with distributive lattices: for example, its Möbius function takes on only values 0, 1, −1.
My question is: How it can ...
- 23.1k
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The join of two set partitions in the refinement order
Let $X$ be a set. The set $\Pi_X$ of all partitions of $X$ is partially ordered via the refinement order, which is defined by $\alpha \leq \beta$ if and only if for each block $A\in \alpha$ there is a ...
- 23.1k
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Finite distributive lattices and finite abelian monoids
A structure of semilattice over T is the same thing than a structure of finite abelian monoid such that $\forall t \in T$, $t² = t$.
Given a semilattice T, we get an abelian monoid by defining $a.b$ =...
- 302
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Relation between semiring structure of natural numbers and their hereditarily finite sets structure under bitwise operations
Some background
The natural numbers $\mathsf{Nat}=\{0,1,2,\dots\}$ has the structure of a semiring under addition and multiplication.
Write $\mathsf{HFS}$ for the set of all hereditarily finite sets (...
- 538
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81
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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 ...
- 3,127
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162
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Sublattices of rank n of the Boolean algebra and partial orders
Let $f(n)$ be the number of sub lattices of rank n the Boolean algebra $B_n$.
I want to show that $f(n)$ is also the number of partial orders of $P$ on $[𝑛]$.
I have read this question from Counting ...
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Complete the gap in a ranked poset (even number of ways?)
Let P be a ranked finite poset and suppose $x < z$ where $\textrm{rank}(z) = \textrm{rank}(x)+2$. Define the interval as
$$ I_{xz} = \{y: x < y < z\} $$
Is there a simple criterion for $|I_{...
- 5,994
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74
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L a finite distributive lattice, prove subsposet of L that cover k elements is isomorphic to the subposet of L that are covered by k elements.
Let L be a finite distributive lattice. Then I need to prove that the subposet of elements that cover k elements is isomorphic to the subposet of elements that are covered by k elements.
I know the ...
- 89
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Is there a polyhedron whose face lattice is a given lattice with the "diamond property"?
I learned that every $(k-2)$-face is contained in exactly two facets in a $k$-dimension polyhedra from 'Theory of linear and integer programming'. So every face lattice of polyhedra satisfies the &...
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If $(f, g)$ is a Galois connection between two bounded lattices, then if $T$ is an ideal we have $f^{-1}(T)$ is an ideal
Let $\mathcal{I}(L)\:$ and $\mathcal{I}(N)\:$ be the ideal lattices of the bounded lattices $L$ and $N$ and let $(f, g)$ be a Galois connection between $L$ and $N$, then show that $\:\forall \:\: T \...
- 173
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let $L$ be a bounded distributive lattice with dual space $(X:=\mathcal{I}_p(L), \subseteq, \tau)$, then the clopen downsets of $X$ are $X_a, a \in L$
11.22 Lemma, from B. A. Davey, H. A. Priestley, Introduction to lattices and order,
let $L$ be a bounded distributive lattice with dual space $(X:=\mathcal{I}_p(L), \subseteq, \tau)$ and $X_a = \{I \...
- 173
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Need a clarification of the proof that the prime ideal space of a distributive bounded lattice is compact
11.19 Theorem, from B. A. Davey, H. A. Priestley, Introduction to lattices and order,
Let $L$ be a bounded distributive lattice, then the prime ideal space $\langle \mathcal{I}_p(L); \tau \rangle$ ...
- 173
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357
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$L$ finite and distributive lattice, then $\mathcal{J}(L)$ (join-irreducible's) is isomorphic, as poset, to $\mathcal{M}(L)$ (meet-irreducible's)
Show that for any finite distributive lattice $L$, $\mathcal{J}(L)$, that is the associated poset of join-irreducible elements of $L$, is isomorphic to $\mathcal{M}(L)$, the associated poset of meet-...
- 1,133
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$P$ poset. $x = \bigvee(\downarrow x\cap U)\Rightarrow \forall x, y \in P$, with $y \lt x$, $\exists a\in U$ s.t. $a \le x $ and $a \nleqslant y$
Le $P$ be a partially ordered set, $U \subseteq P$ and $\downarrow x = \{y \in P : y \le x\}$ (a down set).
Show that if $\,\,\forall \,\,x \in P\,\,$ we have $\,\,x = \bigvee(\downarrow x\cap U)\,\,\...
- 1,133