Questions tagged [lattice-theory]
The theory of lattices in the sense of order theory. For the number-theoretic notion, use the tag "lattices" instead.
425
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Word problem for finitely presented bounded lattices
There is a solution to the word problem for finitely presented (non-bounded) lattices, as well as a solution to the word problem for free bounded lattices. I am assuming that there is a solution to ...
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2
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Is ${\cal P}(\omega)/\mathrm{(fin)}$ order-isomorphic to its intervals?
Let $a, b \in {\cal P}(\omega)/\mathrm{(fin)}$ with $a<b$. Do we have ${\cal P}(\omega)/(fin)\cong [a,b]$?
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3
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1
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Order-preserving surjective map $f: {\cal P}(\omega)/(fin) \to [0,1]$
Is there an order-preserving surjective map $f: {\cal P}(\omega)/(fin) \to [0,1]$? Or from ${\cal P}(\omega)/(fin)$ onto $[0,1]\cap \mathbb{Q}$?
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12
votes
2
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What is known about ideal and divisibility lattices of GCD domains and their generalizations?
The divisibility relation "$a$ divides $b$", or concisely, $a \vert b$ defined over a commutative integral domain $R$ with identity induces a partial order on the multiplicative semigroup $R/R^{\times}...
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2
votes
1
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Is $({\cal P}(\omega), \leq_{\text{inj}})$ a distributive lattice?
For $A, B \subseteq \omega$ we set $A \leq_{\text{inj}} B$ if there is an injective and order-preserving map $f:\omega\to \omega$ , such that $f(A)$ is a down-set of $B$. It is easy to see that $\leq_{...
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2
votes
1
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272
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Infima in the Rudin-Keisler ordering
Let $\text{NPU}(\omega)$ be the set of non-principal ultafilters on $\omega$. The Rudin-Keisler preorder on $\text{NPU}(\omega)$ is defined by
$${\cal U} \leq_{RK} {\cal V} :\Leftrightarrow (\exists f:...
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4
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Can infinite bounded distibutive lattices be "arbitrarily wide"?
I was always thinking, in an informal way, that the powerset lattices ${\cal P}(X)$ (where $X$ is an infinite set) are the "widest" bounded distributive lattices with respect to their height. (In ${\...
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2
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1
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Chains of maximum cardinality in distributive lattices
It's quite straightforward to construct a (complete) lattice in which no chain has maximum cardinality: for each $n\in \omega\setminus\{0\}$ let $C_n$ be a copy of $n$ with the chain ordering ...
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3
votes
1
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262
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Antichains of maximum cardinality: posets vs lattices
The following construction gives a poset such that no antichain has maximum cardinality: For $n\in\mathbb{N}\setminus\{0\}$, let "layer" $n$ consist of an antichain of $n$ points, and as for the ...
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Covering property of complete distributive lattices
Let $(L,\land,\lor)$ be a complete distributive lattice. Given $x\neq y \in L$, is there a finite set ${\cal I}$ of closed intervals in $L$ such that
no member of ${\cal I}$ contains both $x$ and $y$,...
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13
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2
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What does first-order co-intuitionistic logic look like (and does it have an equivalent type theory)?
So, this is where I'm at so far:
Heyting algebras model propositional intuitionistic logic (IL)
so do Cartesian closed categories which also model the simply typed lamda calculus
co-Heyting algebras ...
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1
answer
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Hausdorff interval topology on distributive lattices
Given a poset $(P,\leq)$ the interval topology $\tau_i(P)$ on $P$ is generated by
$$\{P\setminus\downarrow x : x\in P\} \cup \{P\setminus\uparrow x : x\in P\},$$
where $\downarrow x = \{y\in P: y\leq ...
- 45.5k
0
votes
1
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63
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Order-preserving surjections on the Dedekind MacNeille completion
Suppose $L$ is a complete lattice, $P$ is a poset, and $f: L \to P$ is a surjective order-preserving map. If ${\bf DM}(P)$ is the Dedekind MacNeille completion of $P$, is there necessarily a ...
- 45.5k
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Order-preserving surjection ${\mathbb N}^{\mathbb N}\to [0,\infty)$
This is kind of a continuation of a recent (closed) question.
Is there an order-preserving surjective function $f:{\mathbb N}^{\mathbb N}\to [0,\infty)$ (where for $a,b\in {\mathbb N}^{\mathbb N}$ we ...
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1
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1
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134
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Is an Eulerian subgroup lattice boolean?
Let $G$ be a finite group and $\mu$ the Möbius function of the subgroup lattice $\mathcal{L}(G)$.
The reduced Euler characteristic of the order complex of the coset poset $\{ Kg \ | \ K<G, \ g \...
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4
votes
1
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Existence of a non-Eulerian atomistic lattice with this property on the Möbius function
Let $L$ be a finite lattice with least element $\hat{0}$, greatest element $\hat{1}$, and Möbius function $\mu$.
Question 1: What class of lattices the following property characterizes? $$\mu(\hat{0},...
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votes
2
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Bound on number of nxn grids with lexicographical ordering / poset structure
Given $n\in\mathbb{N}$, consider the numbers $\{1,\ldots,n^2\}$ and a permutation $\pi\in S_{n^2}$. It induces pairs $(1,\pi(1))$, $\ldots$, $(n^2,\pi(n^2))$.
Consider an $n\times n$ grid. How many ...
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1
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Definition of an Orlicz modular space
In Nowak (1989), a modular $\rho$ on a vector lattice is defined by the following properties
(N1) $\rho(x)=0\implies x=0$;
(N2) $\lvert x\rvert \le \lvert y\rvert\implies \rho(x) \le \rho(y)$;
(N3) ...
- 138
2
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1
answer
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Equations satisfied by finite modular lattices
I found this very interesting paper by Freese, The Variety of Modular Lattices is Not Generated by its Finite Members, which shows that finite modular lattices satisfy an identity that is not ...
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1
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Atomicity of blocks in a Hilbert lattice
Where can I find the proof that any block (maximal boolean subalgebra) $\mathbf{B}$ of the orthomodular lattice $\mathcal{L}$ of closed subspaces of a separable Hilbert space $\mathcal{H}$ is atomic?
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An inequality in complete lattices
Let $(\Gamma,\leq)$ be a complete lattice and put $\bigwedge_{x\in \Gamma}x =0$. Assume $A$ and $B$ are two subsets of $\Gamma$ with $a\wedge b=0$ for every $a\in A$ and $b\in B$.
Q. True or false:
...
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6
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342
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Functions from a lattice to $[0,1]$
Let $\mathcal L$ be a bounded modular lattice. Suppose also that $\mathcal L$ is complete and upper-continuous (i.e. for any directed subset $\{x_i:i\in I\}$ of $\mathcal L$ and any $x\in \mathcal L$, ...
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5
votes
1
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254
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Divisibility labeling on a boolean lattice and nonzero Euler totient
Let $B_n$ be the subset lattice of $\{1,2, \dots , n \}$, also called the boolean lattice of rank $n$.
A labeling $f: B_n \to \mathbb{N}_{\ge 1}$ is called acceptable if $\forall a,b \in B_n$:
...
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16
votes
1
answer
786
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Is this Wikipedia article linking to the wrong notion of coherent space
I'm reading up on infinite generalizations of the fundamental theorem of distributive lattices. Wikipedia (June 15, 2017) says that there is a duality
between distributive lattices and coherent ...
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2
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2
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Bounded lattices with lattice surjections but no injections between them
What is an example of two bounded lattices $L, K$ such that there exist surjective lattice homomorphisms $f:L\to K$ and $g:K\to L$, but there are no injective lattice homomorphisms between $L, K$?
- 45.5k
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An inequality between idempotents
Let $R$ be a unital ring. Let us denote $I_R$ by the set of all idempotent ($p^2=p$) in $R$. For given $p,q$ in $I_R$, we write $p\leq q$ if $pq=qp=p$.
Assume that both $p\vee q$ (the supremum of $...
- 3,992
2
votes
1
answer
216
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$2^\omega$ vs $({\omega+1})^\omega$
It is easy to see that there is a surjective lattice homomorphism $s:({\omega+1})^\omega \to 2^\omega$ (construction see after the horizontal line below). Is there a surjective lattice homomorphism $...
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165
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A boolean representation of the Möbius function on a finite lattice
Let $(L,\wedge , \vee)$ be a finite lattice with minimum $\hat{0}$ and maximum $\hat{1}$.
Consider the Möbius function $\mu$ on $L$ defined inductively by $$\mu(\hat{1}) = 1 \text{ and } \mu(a) = - \...
- 25.9k
3
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Distributive lattices -> left regular bands -> Atomistic lower semimodular lattices
Consider the following construction : let $(L,\vee,\wedge)$ be a finite distributive lattice, and let $(\mathrm{Int}(L),\star)$ be the monoid defined on the set of non empty intervals of $L$
$$\mathrm{...
- 2,673
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3
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The set of complements equal to the complement of set
Consider $A \subset \{0,1\}^n$
I want $A$ to have two properties.
$1.$ $A$ is increasing, i.e., If $x \in A$ and $x \subseteq y$ then $y \in A$ too.
[$x \subseteq y$ means that every coordinate ...
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2
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Proper coverings as a quotient of the coverings
Let $X\neq \emptyset$ be a set. We say that $U\subseteq {\cal P}(X)\setminus \{\emptyset\}$ is a covering of $X$ if $\bigcup U = X$. We call a covering proper if for $a\neq b\in U$ we have $a\not\...
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2
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Quotients of $\text{Part}(X)$
Let $\text{Part}(X)$ denote the collection of all partitions of $X$. For $A, B\in \text{Part}(X)$ we set $A\leq B$ if $A$ refines $B$, that is for all $a\in A$ there is $b\in B$ such that $a\subseteq ...
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lattice with Voronoi cell inside a circle
This considers real-valued lattices in two dimensions.
I need to find the densest lattice $\Lambda$, i.e., the one with the smallest determinant of its generator matrix, such that the Voronoi cell of ...
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Distance measures between Boolean algebra homomorphisms
Is there a natural way to define the 'distance' between two Boolean algebra homomorphisms $f, g: B \rightarrow B'$? I'm thinking of something like the Kullback leibeler divergence for probability ...
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0
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251
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Transforming reaction-diffusion equations to random walk processes
I have a two species reaction-diffusion system which is a Turing-type (activator-inhibitor) equation. I am trying to transform my reaction-diffusion system into a system of multiple walkers on a ...
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Does every non-type-I factor's projection lattice admit a dense embedding of the standard continuum-collapsing poset?
Let $R$ be a non-type-I factor acting on a separable Hilbert space.
Let $P(R)$ be the set of $R$'s projections with the usual ordering ($x \leq y \iff$ range$(x) \subseteq$ range$(y)$) under which it ...
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1
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A property on meet of coatoms in a finite modular lattice
Let $L$ be a finite lattice with $\hat{1}$ its maximum and $c_1, \dots, c_n$ its coatoms. Let $E_n=\{1, \dots, n \}$.
For any subset $I \subset E_n$ we define $$C(I) := \bigwedge_{i \in I} c_i$$ then ...
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How much of a factor's structure is determined by the order-type of its projection lattice?
H. A. Dye showed that a type II or III factor $R$ is determined, up to *-algebraic isomorphism or anti-isomorphism, by the ortholattice-isomorphism type of its projection lattice ("ortholattice-...
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2
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Finite-join antichains in lattices
Let $(L,\vee,\wedge,0,1)$ be a lattice with unique least and greatest elements $0$ and $1$, respectively. I'll say that an antichain $A$ in $L$ is a subset of $L\setminus\{0\}$ such that for every $a,...
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1
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179
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Uniquely complemented but not Boolean
What is an example of a lattice $(L,\leq)$ that is uniquely complemented, but not Boolean?
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Sup preserving maps between distributive lattices
I have been looking at categories of sup semilattices and sup preserving maps. If $A$ and $B$ are two such, the set I denote $[A,B]$ is sup preserving homomorphisms between is also a sup semilattice ...
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Intermediate lattices $C\mathbb{Z}^n \subseteq \Lambda \subseteq \mathbb{Z}^n$
Let $C \in \mathfrak{gl}(\mathbb{Z},n)$ be a symmetric full rank integer valued matrix (in my case its the symmetric part of a Cartan matrix).. Let $\Lambda \subseteq \mathbb{Z}^n$ be a full rank ...
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2
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453
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Divisibility labeling on a boolean lattice and positive Euler totient
Let $B_n$ be the rank $n$ boolean lattice (i.e. the subset lattice of $\{1,2, \dots , n \}$). Let $\hat{0}$ and $\hat{1}$ be the minimum and the maximum of $B_n$. Let $f: B_n \to \mathbb{N}$ be a ...
- 25.9k
1
vote
1
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106
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Does the lattice of coverings embed in the lattice of partitions?
Let $X\neq \emptyset$ be a set. We say that $U\subseteq {\cal P}(X)\setminus \{\emptyset\}$ is a proper covering if
$\bigcup U = X$, and
for $a\neq b\in U$ we have $a\not\subseteq b$.
Let $\text{Cov}...
- 45.5k
2
votes
1
answer
247
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What lattices are isomorphic to $R^{N}$ for some $N$, equipped with the product order?
What lattices are isomorphic to $\mathbb{R}^{N}$ for some $N\in \mathbb{N}$, equipped with the canonical order?
Remark:
When I say $\mathbb{R}^N$, I don’t mean it to be a vector space. Instead, I ...
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votes
1
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171
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Is there an atom K of [H,G]≃B2 with |K:H|≡|G:H|(mod 2)?
Let $[H,G]$ be a rank $2$ boolean interval of finite groups.
Statement 1: There is an atom $K$ of $[H,G]$ such that $|K:H|≡|G:H|($mod $ 2)$.
The following picture illustrates the statement.
...
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1
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3
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185
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A problem with an edge labeling on the boolean lattices
Let $B_n$ be the boolean lattice of rank $n$. Let $\hat{0}$ and $\hat{1}$ be the minimum and the maximum, respectively.
We identify the notion of edge with the notion of interval $[a,b]$ of cardinal $...
- 25.9k
2
votes
2
answers
441
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An example of a frame homomorphism which does not preserve Heyting implication
A frame is a complete lattice $\langle L,\mathord{\leqslant}\rangle$ which satisfies the following distributivity law:
$$a\wedge\bigvee_{i\in I}b_i=\bigvee_{i\in I}a\wedge b_i\,.$$
A frame ...
- 1,102
7
votes
1
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168
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Enumerative characterisation of boolean lattices II
This is a sequel of this post.
The boolean lattice $B_n$ is graded with rank numbers $\binom{n}{0}, \binom{n}{1}, \dots, \binom{n}{n}$, and $n2^{n-1}$ edges.
Question: Is a graded lattice with the ...
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5
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1
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243
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Enumerative characterisation of boolean lattices
The boolean lattice of rank $n$ (noted $B_n$) is the subset lattice of $\{1,2, \dots , n \}$.
See the Hasse diagram of $B_3$ below:
The Hasse diagram of $B_n$ is of length $n$, with $2^n$ vertices ...
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