Questions tagged [lattice-theory]
The theory of lattices in the sense of order theory. For the number-theoretic notion, use the tag "lattices" instead.
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Finite lattices that are Koszul
Let $L$ be a finite lattice and $A=KL$ the incidence algebra of $L$.
It should be true that $L$ is modular if and only if the algebra $KL$ is quadratic (since being modular is equivalent to having no ...
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Book and article recommendations with the purpose of studying the intersection between probability theory and lattice theory
Lately, I have been studying probability theory and lattice theory separately and I would like to investigate ideas which relate both subjects together. Having said that, I would like to know if ...
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Interesting uniform posets
A sequence $(P_0,P_1,\ldots)$ of finite posets is called uniform if: each $P_n$ is graded of rank $n$ with a minimum $\hat{0}_n$ and a maximum $\hat{1}_n$; for any $p \in P_n$ with $\mathrm{rank}(p)=n-...
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Classification of multiplicative lattices
Question 1:Is there a classification of finite lattices which admit a multiplication making them into a finite multiplicative lattices? (see https://encyclopediaofmath.org/wiki/Multiplicative_lattice ...
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poset of lattice properties
Is there a good overview of the dependencies between properties that a (finite) lattice poset can have?
To give a practical example, I was looking for a property weaker than congruence uniform and ...
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Are there finitely many sieves on each object in the distributive lattice cube category?
Define the distributive lattice cube category or Dedekind cube category $\square_{\land\lor}$ to be the full subcategory of the category of posets and monotone maps consisting of objects of the form $[...
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Is $(\omega+1)^\omega/{\cal U}$ complete for ${\cal U}$ free ultrafilter?
Let ${\cal U}$ be a free ultrafilter on $\omega$. Is the linearly ordered set $(\omega+1)^\omega/{\cal U}$ complete?
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Upper bound of rank of a matrix
Given a set $U=\{1,\ldots,c\}$, we have $t$-dimensional vector $v_i=(x_1,x_2,\ldots,x_t)$, where $x_j$ is a subset of $U$. We can check that there are $2^{ct}$ above vectors.
Then we construct the ...
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Constructive lattice completion
The Dedekind–MacNeille Completion is the generalized way of completing an arbitrary lattice $L$. We will call $C$ the Dedekind–MacNeille completion of $L$ (I will not go into the details of the ...
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Dedekind-MacNeille completion of ${\cal P}(\omega)/({\rm fin})$
Let ${\cal P}(\omega)/({\rm fin})$ be the quotient of the Boolean algebra ${\cal P}(\omega)$ where two sets are considered to be equivalent if they differ by a finite number of elements.
It turns out ...
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Medians in lattice theory
John Isbell defined a notion of a median algebra (although the original idea is due to Birkhoff). A median algebra is a set $S$ with a ternary operation $[x,y,z]$ satisfying a set of axioms. The ...
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Ideals of $F_2[x_1, x_2, \cdots, x_n]/(x_1^2, x_2^2, \cdots x_n^2)$
I am interested in the poset of all ideals of the local ring
$$R_n = \mathbb{F}_2[x_1, x_2, \cdots, x_n]/(x_1^2, x_2^2, \cdots x_n^2).$$
$n=1$ is trivial. $n=2$ takes little work and it is shown below....
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Lattice structure (wrt dominance order) on the set of Young diagrams appearing in the decompositions given by the Littlewood-Richardson rule
The irreducible decomposition of the tensor product of two irreducible representations of GL(n) is described by the Littlewood-Richardson rule. This same rule also governs the decomposition of the ...
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Which Boolean lattices have a left-to-right symmetric drawing?
This question is inspired by a similar MSE question about partition lattices.
Question: Which finite Boolean lattices have a symmetric drawing on the 2D plane?
By a symmetric drawing of a lattice, I ...
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"Pseudo-Boolean" lattice (almost every element has several complements)
If $(L,\leq)$ is a lattice with bottom element $0$ and top element $1$ and $x\in L$ we say that $y$ is a complement of $x$ if $x\vee y = 1$ and $x\wedge y = 0$.
Is there a lattice $(L,\leq)$ with more ...
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What is constructive complete distributivity minus frame distributivity
Let $L$ be a complete lattice. It is well known that $L$ is a frame, i.e., $x\wedge \bigvee S=\bigvee \{x\wedge s\mid s\in S\}$, iff $\bigvee\colon \mathcal D(L)\to L$ preserves finite meets (here $\...
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In a constructive order/lattice theory are the arbitrary join and the weak suprema the same?
For a poset $X$ we define an upperbound $w \in X$ of a subset $S \subseteq X$ to be a weak-supremum if
$(\forall a \in S (a \leq b)) \implies w \leq b$.
While a supremum is defined more carefully (in ...
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Hyperplane separation of a concave functional and a point, in Domain Theory
Problem:
Let $D$ be an $\omega$-BC domain, and $[D\to[0,\infty]]$ be the space of Scott-continuous nonnegative functions on $D$, equipped with the obvious ordering and the Scott-topology.
EDIT: I don'...
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A definition in poset theory
I am working on a article in poset theory. In that article, I am defining a subposet of a poset. The definition is following:
Let $P$ be a finite poset. A subposet $P'$ of $P$ is called closed under ...
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Finding a subclass of lattices in the literature
Assume you have an algebraic problem that outputs a list of (finite) lattices on $n$ points for a given number $n$.
Question 1: Is there a way to search the internet/literature to see what exactly ...
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How to compute the intersection of an ideal with the maximal order of a subfield?
I asked this earlier on math.stackexchange but I think this is a better place for this question.
Computing the intersection of ideals belonging to the same maximal order of a number field $K$ can be ...
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Partial orders on $\mathbb{N}^m$ compatible with addition
I'm looking for a classification (or just non-trivial examples) of partial orders on monoid $\mathbb{N}^{m}$ that are compatible with addition. That is, partial orders $\leq$ satisfying two additional ...
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Objects in bijection with integer partitions (and lattices)
A partition of $n$ is a non-increasing sequence of positive integers of sum $n$. Several lattices are defined over integer partitions, in particular the dominance order and the Young lattice.
Several ...
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A combinatorial characterization of the central inversion of a polytope?
Given a convex full-dimensional polytope $P\subset\Bbb R^d$ (convex hull of finitely many points and not contained in any proper affine subspace) and a symmetry thereof (a linear map $\smash{T\in\...
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Does every finite lattice embed into a finite Eulerian lattice?
A finite Boolean lattice is a lattice isomorphic to the subset lattice of a finite set. Every Boolean lattice is Eulerian, namely, a graded lattice $L$ such that $\mu(a,b) = (-1)^{|b|-|a|}$ for all $a,...
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closure operator on a complete lattice arising from adjunction on lattice itself
Define a closure operator on a complete lattice $L$ as a function $f:L \to L$ which is order preserving and idempotent and satisfies $x \leq fx$.
Every closure operator arises from an adjunction ...
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Sober spaces vs. spatial frames-a big picture
For any topological space $X$ one can consider the so called frame of all open subsets of $X$ to be denoted by $\mathcal{O}(X)$. If $f:X \to Y$ is continuous taking the inverse image we get the ...
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Does every geometric lattice of rank $r$ contain the Boolean $B_r$ as a sublattice?
A finite lattice is geometric if it is semimodular and atomistic. Geometric lattices can have arbitrarily high rank $r$, as evidenced by the Boolean lattice $B_r$ (power set of $r$ elements with the ...
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Can one characterize maximal antichains in terms of distributive lattices?
This is inspired by the recent question Verification of a maximal antichain
The celebrated duality between finite posets and finite distributive lattices has several nice formulations. One of them ...
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Ordering preserved by an inverse frame homomorphism
Recall that a frame homomorphism $h:L\to M$ is called ($L$ and $M$ are frames):
Dense if, for any $x ∈ L$, $h(x) = 0$ implies $x = 0$.
Codense if, for any $x ∈ L$, $h(x) = 1$ implies $x = 1$.
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Do Frobenius algebras have a lattice basis and what lattices do appear?
Let $K$ be for simplicity be the field with two or three elements (or alternatively we could restrict to ideals containing only the field elements $-1$ or $1$ as coefficients).
A (commutative) ...
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Euler characteristic of the simplicial complex of sets of elements in a semilattice with non-zero meet
In a combinatorial computation, I came across the following quantity:
Consider a finite meet semilattice $L$, that is, a finite poset which is closed under $\min$. Denote the least element of $L$ by $...
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Differential posets, the Plancherel state $\varphi_\mathrm{P}$, and minimality
$\newcommand\rank[1]{\lvert#1\rvert}$Let $\Bbb{P}$ be a 1-differential poset with a unique bottom element $\emptyset \in \Bbb{P}$. With some minor abuse in terminology, The
Plancherel measure state $...
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A homological algebra approach to the Union-closed sets conjecture
I noted a while ago that there is a nice homological formulation using incidence algebra of the Union-closed sets conjecture (https://en.wikipedia.org/wiki/Union-closed_sets_conjecture). It might just ...
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irreducible compact set vs prime compact set
Let $X$ be a topological space.
A compact set $K$ is called irreducible if for any two compact subsets $K_1$, $K_2$ of $K$ with $K$ is equal to the union of $K_1$ and $K_2$, then $K$ is equal to $K_1$ ...
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A congruence relation on the projection lattice
This question is a continuation of what I asked here. Tristan Bice showed the following nice result there:
Let $A$ be a von Neumann algebra and $P$ its projection lattice, ordered by $p\leq q\...
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On free lattices
Free distributive lattices on a finite set exist and are finite, while free modular lattices on a finite set exist but are not finite when the set has at least 4 elements.
Question: Is there a class (...
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Are there axioms satisfied in commutative rings and distributive lattices but not satisfied in commutative semirings?
Consider the language of rigs (also called semirings): it has constants $0$ and $1$ and binary operations $+$ and $\times$. The theory of commutative rigs is generated by the usual axioms: $+$ is ...
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A meet-semilattice with top element that is not a lattice?
I am reading Francis Borceux’s “Handbook of Categorical Algebra I” and on page 135 it says
In particular a finite version of 4.2.5 does not hold: a finitely complete and well-powered category ...
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Periods of Coxeter transformation associated to root posets
$\DeclareMathOperator\Co{Co}$Let $P$ be the root poset associated to a simple Lie algebra.
Let $L=L(P)$ denote the distributive lattice of order ideals of $P$ and let $\Co_L$ denote the Coxeter matrix ...
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Classification of posets that are quotient posets of the Boolean lattice
Quotient posets of the Boolean lattice $B_n$ have interesting properties and are for example discussed in chapter 5 of Stanley's book on algebraic combinatorics.
$B_n/G$ for a subgroup $G$ of the ...
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Monster group as automorphism group of a distributive lattice
It is known that every finite group is the automorphism group of a finite distributive lattice.
Question: What is the minimal order of a distributive lattice $L$ such that the automorphism group of $...
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Distributive lattices with periodic Coxeter matrix
Let $L$ be a finite distributive lattice and $U$ its incidence matrix with entries $u_{i,j}=1$ iff $i \leq j$ and $u_{i,j}=0$ else.
Then $U^{-1}$ is the Moebius matrix of $L$ and $C_L:=- U^{-1} U^{T}$ ...
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Rowmotion for general lattices
Let $L$ be a finite lattice and $x \in L$ with covers $r_1,...,r_l$ in $L$.
One can define $row(x):= \min \{ y | y \leq r_1 \lor \cdots \lor r_l $ and $ y \nleq r_1 \lor \cdots \lor \overline{r_t} \...
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lattice suprema vs pointwise suprema
What is the difference between the lattice supremum and the pointwise supremum of a family of functions? I mean, given a family of real valued functions $\mathcal{F}$, is the function $\sup\mathcal{F}:...
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A multiplication for distributive lattices via rowmotion
Let $L$ be a finite distributive lattice. In case $L$ is Boolean one can define a multiplication by $a*b:=(a \cup b) \setminus (a \cap b)=(a \cup b) \cap (a \cap b)^c$, where $(-)^c$ denotes the ...
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Which complete orthomodular lattices arise from von Neumann algebras?
Let $A$ be a von Neumann algebra. Then a classic observation is that the set of projections $\Pi(A)$ is naturally a complete orthomodular lattice.
Question 1: Is the construction $A \mapsto \Pi(A)$ a ...
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How to construct a lattice having a subset of a given relations?
I am given a (smallish, say $n=14$ element) set $X$, and a set $R$ of (a few hundred) quadruples of elements $(a, b, c, d)$ with $a,b,c, d\in X$.
I want to construct lattices on $X$, such that for all ...
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Group structure for distributive lattices
On the (finite) Boolean lattice there is a group structure given by the symmetric difference and this group is an elementary abelian 2-group.
Question: Does there exist a natural group structure on ...
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Lattices on classical combinatorial families
I am asking for examples of lattices defined on classical combinatorial families, such as Permutations, Catalan objects, set partitions or integer partitions, graphs.
I am mosty interested in lattices ...
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