Questions tagged [lattice-theory]
The theory of lattices in the sense of order theory. For the number-theoretic notion, use the tag "lattices" instead.
510 questions
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Complete orthomodular lattices and blocks
A block of an orthomodular lattice $L$ is a maximal Boolean subalgebra of $L$. In a complete orthomodular lattice, every block is complete. I suspect the converse does not hold, but don't know a ...
- 271
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1
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Non-topological argument for the non-existence of an order-embedding $\iota: [0,1]\to ([0,1]\setminus \mathbb{Q})$
Starting point. I was toying around with the following question: If $A, B\subseteq [0,1]$ with $A\cup B = [0,1]$, does $[0,1]$ order-embed into at least one of $A, B$? Quickly I focused on $A = [0,1]\...
- 54.4k
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Logic-Compactness in complete lattice
Assume $\Gamma$ is an infinite set composed of formulas (of finite length), and $A$ is a formula (of finite length). For example, $\Gamma=\{x,y\sqcup z,x_1,x_2,x_3,\cdots\}$, $A=(x\sqcap y)\sqcup(x\...
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Directly deducing a counting formula for plane partitions in a box from another
Please show (using arithmetic) that, for $r,s,t\in\mathbb N$,
$$
\prod_{i=1}^r\prod_{j=1}^s\prod_{k=1}^t\,\frac{i+j+k-1}{i+j+k-2}
$$
equals
$$
\prod_{i=1}^r\,\frac{\binom{s+t+i-1}{s}}{\binom{s+i-1}{s}}...
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1
answer
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Posets of quotients in category of rings
Are there some general theorems which govern global structure of posets arising as posets of epimorphisms out of a (unital, associative) ring?
One may note that it is always a lattice — it is ...
- 6,030
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208
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Strong conservativity and extra structure in free objects
The background for my question is somewhat related to this; there a very interesting paper is provided, but the setting and examples are somewhat different. I can add any necessary background or ...
2
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120
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Determinant related to a union-closed family of sets
Let $\mathcal{F} = \{A_1, \ldots, A_n\}$ be a union-closed family of sets with universe $\bigcup_{i=1}^n A_i = [q] = \{1, \ldots, q\}$.
Let $M = [m_{ij}]$ be the $n \times n$ matrix with $m_{ij} = \...
- 1,186
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1
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184
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Additively idempotent semirings that are not lattices
I am looking for examples of additively idempotent semirings (which are always join semilattices) that do not have an underlying lattice structure, i.e. either meets do not exist or exist outside the ...
- 1
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2
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88
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Question on extending submodularity inequalities in lattices
On a lattice $\mathcal{L}$, I have a submodular, monotone, real-valued function $\rho$.
Submodularity means it satisfies the inequality $$\rho(x)+\rho(y)\ge \rho(x\wedge y)+\rho(x\vee y)$$ for all $x, ...
- 1,203
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Lattice from a commutative semigroup
Consider any finite commutative semigroup $S$.
Say that $x \leq y$ iff $x = y$ or $xy = y$. This is a partial order on $S$: the only nontrivial property to check is that $x \leq y$ and $y \leq z$ ...
- 1,186
3
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answer
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Cardinality of the collection of maximal antichains in ${\cal P}(\omega)$
An antichain in $\mathcal P(\omega)$ is a set $\mathcal A\subseteq \mathcal P(\omega)$ such that for all $A, B\in \mathcal A$ with $A\neq B$ we have $(A\setminus B)\neq \emptyset$ and $(B \setminus A)\...
- 54.4k
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Closed and zero-set ultrafilters on a space. Trace of closed ultrafilters
For a space $X$, consider the complete lattice $\text{CL}(X)$ of closed sets of $X$. The $\text{CL}(X)$-filters will be called $\text{CL}$-filters. Similarly consider the lattice $Z(X)$ of zero-sets ...
- 2,389
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Continuous locales and minimal (one-point) compactifications
I was reading "Stone Spaces" by Peter Johnstone. It turns out the continuous locales are exactly the ones corresponding to topologies of sober locally compact spaces. For a Hausdorff locally ...
- 813
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Reference request for an algebraic structure mimicking $\varepsilon_0$
Let $(R, +, 0, \cdot, 1)$ be a (unital) semiring equipped with a unary operation $f$ and a binary operation $\land$ such that
$f(0) = 1$
$f(x + y) = f(x) f(y)$
$x \land y = y \land x$
$x \land (y \...
- 1,845
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1
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Counting subgroups independent from a fixed subgroup of an abelian p-group
Given two finite abelian $p$-groups $H\subseteq G$. Given $n$, what is the number of subgroups $N\subseteq G$ such that $N\cap H=0$ and $\lvert N\rvert = p^n$? In particular, does it depend only on ...
- 824
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2
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322
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Shortest vector in orthant of lattice
There are many great algorithms for enumeration of vectors in a lattice such as Fincke-Pohst-Kannan, extreme pruning etc, not to mention great implementations such as fplll.
Let $L$ be high density (...
- 273
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(Non-commutative) generalisation of finite lattices
According to https://en.wikipedia.org/wiki/Absorption_law
(lets stick to finite) lattices are exactly the sets with two commutative semigroup operations satisfying the absorption law.
Question: Is ...
- 28.2k
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Seeking recent developments concerning Noether lattices
I'm currently studying some classes of distributive lattices, which I've recently found to be Noether lattices as well. I'm fairly unfamiliar with Noether lattices, beyond the definition and a theorem ...
- 11
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1
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251
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The poset of intervals of the free distributive lattice
Let $L_n$ denote the free distributive lattice on $n$ elments, which is for example defined as the lattice of order ideals of the Boolean lattice of an $n$-set.
Question: Is it true that the poset of ...
- 28.2k
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Has there been any study of normed semilattices?
First of all, I should confess that this is not my field of study. Recently, I encountered a situation where I had a (meet) semilattice $\mathcal{L}$ and a compatible valuation (positive semidefinite ...
- 1,203
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Theorem 2 of Esakia's "Topological Kripke Models"
A closure algebra is a Boolean algebra $B$ with a $\vee$-preserving closure operator $\bf C$ that sends $0$ to $0$. A Heyting algebra is a lattice $H$ with a $0$ and a binary operation $\rightarrow$ ...
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Theorem 1 of Esakia's "Topological Kripke Models"
Preliminaries
A Stone space is defined to be a compact Hausdorff space with a basis consisting of clopen sets. Let $X$ be a Stone space with a binary relation $R$ that is reflexive and transitive. A ...
- 1,979
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1
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164
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Surjective order-preserving map $f: {\cal P}(\omega)/{\text{(fin)}} \to {\cal P}(\omega)$
$\def\Pfin{{\cal P}(\omega)/{\text{(fin)}}}$The projection $\pi:{\cal P}(\omega)\to \Pfin$ is clearly surjective and order-preserving. (The quotient $\Pfin$ is defined here.)
Is there a surjective ...
- 54.4k
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103
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Heyting arrows and Pseudocomplements in partial frames
Partial frames are generalizations of frames, which are complete lattices where finite meets distribute over arbitrary joins. In partial frames however, only selected (distinguished) joins are allowed ...
- 51
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92
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Can a total order extend the 1D sort-and-match approach to multidimensional optimal transport?
I’ve been studying the optimal transport problem and I understand that in one dimension it can be solved quite easily: because $\mathbb{R}$ is totally ordered, the cumulative distribution function ...
- 105
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M-chain containing a given modular flat in a geometric lattice
Let $\mathcal{L}$ be a supersolvable geometric lattice and let $F$ be a modular element of $\mathcal{L}$. Does $\mathcal{L}$ necessarily admit a maximal chain of modular elements with $F$ in it ?
If $...
- 111
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435
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Is every multiplicative lattice isomorphic to the lattice of ideals of some ring?
A multiplicative lattice is a complete lattice
$(L, \leq)$ that is
endowed with an associative, commutative multiplication that distributes
over arbitrary joins and has $1$, the top element of $L$,
as ...
- 54.4k
8
votes
1
answer
319
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On varieties of lattices admitting "large" free complete members
Let $E$ be an equational theory (in the sense of universal algebra) in the language of lattices. Given a cardinal $\kappa$, say that $E$ is $\kappa$-cheap iff there is a set-sized complete lattice $L$ ...
- 19.6k
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382
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Are incidence algebras important?
I feel the need to explain my background before diving into this soft question, for you to understand my position.
During my undergraduate years, a Theoretical Computer Science professor asked me to ...
- 448
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179
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Every poset is isomorphic to a collection of sets ordered with inclusion, and related
Given a (finite) poset $(P,\leq)$, we can construct the poset $(\mathcal{D}, \subseteq)$ isomorphic to $P$ simply by letting $\mathcal{D}$ be the collection of all "descendant sets" i.e. $\...
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228
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Example of a certain lattice with $q$ coatoms and less than $4q+5$ elements
I would like to find an example of a lattice $L$ with the following properties:
it has $q$ coatoms $x_i$, $1 \le i \le q$;
for each coatom $x_i$ let $Y_i = \{y \in L :y \le x_i\} = (x_i]$: it is ...
- 1,186
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1
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Counter-example to a strengthening of Frankl's conjecture on lattices
The formulation of the Frankl's conjecture (union-closed conjecture) on lattices is : For any finite lattice $L$ of cardinality $n\geq 2$, there exists a join-irreducible $j$ (called an abundant ...
2
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1
answer
251
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Looking for a finite lattice example
Consider a finite lattice $L$ such that each atom has at least $|L|/2$ elements greater than or equal to it. It can be for example a boolean lattice or the following lattice:
In the boolean lattice ...
- 1,186
1
vote
1
answer
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Function f bounded whenever g is bounded
Has this been studied? Can we prove that the definition is not equivalent to a first-order statement?
Let us say $f\le^b g$ if for any set $S$, if $g$ is bounded on $S$ then so is $f$.
(Assume these ...
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Name of a specific lattice
Let $m$ be a natural number with prime factorisation $m=p_1^{a_1}\dots p_r^{a_r}$.
I'm interested in a special name for the lattice of divisors of $m$, when all exponents are equal: $a_1=a_2=\dots=a_r$...
- 28.2k
0
votes
1
answer
146
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Pseudo-complements of connected $T_2$-topologies in the lattice of all topologies
Let $\kappa \geq \aleph_0$ be a cardinal, and let $\newcommand{\Top}{\text{Top}}\Top(\kappa)$ be the complete lattice of all topologies on $\kappa$. For all $\tau\in \Top(\kappa)$ we have $\...
- 54.4k
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249
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Example of a Girard semilattice that does not satisfy a property
A Girard semilattice is an algebra $\langle A, \wedge, \to, 1\rangle$ of type $\langle2,2,0\rangle$ such that $\langle A, \wedge, 1\rangle$ is a bounded semilattice and the following six conditions ...
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Structure of a group by its lattice of subgroups
Michio Suzuki's monograph "Structure of a Group and the Structure of Its Lattice of Subgroups" discusses the relationship between a group and the lattice of its subgroups.
My question is: Is ...
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114
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Proving that $O(\Sigma L_1) \times O(\Sigma L_2)$ is a sublattice using $O(\Sigma (L_1 \times L_2))$ [closed]
I am working with a product lattice and trying to show that
$$
O(\Sigma L_1) \times O(\Sigma L_2)
$$
is a sublattice of $L_3$.
I already know that
$$
O(\Sigma L_1 \times \Sigma L_2)
$$
is a sublattice ...
- 141
3
votes
1
answer
188
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Does Priestley duality commute with finite products?
Priestley duality provides an equivalence between the category of bounded distributive lattices $\mathbf{DL}$ and the category of Priestley spaces $\mathbf{Priestley}$, which are compact, totally ...
- 141
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votes
2
answers
448
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Realising Coxeter groups as automorphism groups of lattices
The symmetric group is the automorphism group of the Boolean lattice of an $n$-set.
Question: Is there also a "canonical" nice lattice whose automorphism group is equal to the Coxeter group ...
- 28.2k
10
votes
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answer
435
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What is known about elementary equivalence of open set posets of topological spaces?
Recall from model theory that two structures are called elementarily equivalent if they satisfy the same first-order sentences. In other words, two structures $\frak A$ and $\frak B$ of the language $\...
- 1,112
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127
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Sublocales corresponding to filters
It is a well known fact that the sublocales of the locale $L$ are defined by idempoten $\wedge$-semilattice endomorphisms, known as nuclei. Each nucleus $j$ of a locale $L$ also defines a filter $\...
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101
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Which positive convex cones in $\mathbb{R}^{n}$ are closed under componentwise meets?
The vector space $\mathbb{R}^{n}$ has a natural lattice structure: for $\mathbf{a} = (a_1, \dots, a_n)$ and $\mathbf{b} = (b_1, \dots, b_n)$
$\mathbf{a} \wedge \mathbf{b} = (\min(a_1,b_1), \dots, \min(...
- 265
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189
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$\sigma$-homomorphism preserving $\sigma$-distributivity
I am basing some of my thesis on Introduction to Boolean Algebras by Givant and Halmos.
My current goal is to leverage the countable chain condition to define conditional probability measures.
In ...
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132
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Is there a characterization of monoids that distribute over each other?
Let $(M, e_1, \times_1, e_2, \times_2)$ be an algebraic structure such that
$(M, e_1, \times_1)$ and $(M, e_2, \times_2)$ are monoids
$x \times_1 (y \times_2 z) = (x \times_1 y) \times_2 (x \times_1 ...
- 1,845
4
votes
1
answer
245
views
Is the interval topology on ${\cal P}(\omega)/(\text{fin})$ connected?
If $(P,\leq)$ is a poset and $x\in X$, we let $\downarrow x = \{p\in P: p \leq x\}$, and $\uparrow x$ is defined dually. The collection $$\Big\{P\setminus (\downarrow x): x\in P\Big\} \cup \Big\{P\...
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What's the deal with De Morgan algebras and Kleene algebras?
The notion of Boolean algebras, and the corresponding classical propositional logic, is very standard, and it is easy to find information about them (for example, among many other such works, there is ...
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12
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598
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Strengthening of Frankl's union-closed sets conjecture: An algebraic approach
Let $\mathcal F$ be a union-closed family of subsets of $[n]=\{1,2,...n\}$ and $n$ real numbers $x_1,x_2,...,x_n\geq 1$.
Conjecture: There exists $k\in [n]$ such that:
$$\sum_{k\in A,A\in \mathcal F}\...
- 1,558
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0
answers
128
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Link between Carathéodory's criterion and commutation in an orthomodular lattice?
In the theory of outer measures, Carathéodory's criterion constructs from an outer measure $\mu^*$ on $X$ a $\sigma$-algebra $\Sigma$ of subsets of $X$, on which the restriction of $\mu^*$ is a ...
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