Questions tagged [lattice-theory]
The theory of lattices in the sense of order theory. For the number-theoretic notion, use the tag "lattices" instead.
504 questions
0 votes
1 answer
143 views
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
0 votes
2 answers
84 views
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
2 votes
1 answer
132 views
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 votes
1 answer
122 views
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)\...
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1 vote
1 answer
74 views
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 ...
- 2,287
1 vote
0 answers
53 views
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 ...
- 803
7 votes
0 answers
170 views
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 \...
- 1,703
3 votes
1 answer
105 views
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
3 votes
2 answers
311 views
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
5 votes
1 answer
257 views
(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 ...
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0 votes
0 answers
151 views
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
4 votes
1 answer
250 views
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 ...
- 28.1k
2 votes
0 answers
125 views
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
1 vote
0 answers
110 views
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 $\rightarrow$ ...
- 1,903
3 votes
1 answer
233 views
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 ...
- 1,903
3 votes
1 answer
162 views
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 ...
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0 votes
0 answers
103 views
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
0 votes
1 answer
89 views
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
3 votes
0 answers
80 views
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
8 votes
1 answer
417 views
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 ...
- 53.5k
8 votes
1 answer
317 views
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 $L$ ...
- 19.5k
2 votes
1 answer
355 views
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 ...
- 444
1 vote
0 answers
172 views
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. $\...
3 votes
1 answer
227 views
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
6 votes
1 answer
319 views
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 votes
1 answer
246 views
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
175 views
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 ...
- 25.2k
2 votes
0 answers
175 views
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: $a_1=a_2=\dots=a_r$...
- 28.1k
0 votes
1 answer
140 views
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 $\...
- 53.5k
3 votes
2 answers
247 views
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 ...
- 63
4 votes
3 answers
454 views
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 ...
- 243
1 vote
0 answers
112 views
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
184 views
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
6 votes
2 answers
444 views
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 ...
- 28.1k
10 votes
1 answer
409 views
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 $\...
- 1,092
2 votes
0 answers
125 views
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 $\...
- 803
0 votes
0 answers
98 views
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
0 votes
1 answer
187 views
$\sigma$-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 ...
- 216
1 vote
1 answer
129 views
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 ...
- 1,703
4 votes
1 answer
241 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 $$\Big\{P\setminus (\downarrow x): x\in P\Big\} \cup \Big\{P\...
- 53.5k
0 votes
0 answers
108 views
Order-convergence and interval topology on ${\cal P}(\omega)/(\text{fin})$
On any poset $(P, \leq)$ we can consider two different topologies that arise directly from the ordering relation. 1) Order convergence topolog $\tau_o(P)$ : By a set filter $\mathcal{F}$ on $P$ we ...
- 53.5k
13 votes
2 answers
1k views
What's the deal with De Morgan algebras and Kleene algebras?
The notion of Boolean algebras, and the corresponding classical propositional logic, is very standard, and it is easy to find information about them (for example, among many other such works, there is ...
- 38.1k
12 votes
0 answers
597 views
Strengthening of Frankl's union-closed sets conjecture: An algebraic approach
Let $\mathcal F$ be a union-closed family of subsets of $[n]=\{1,2,...n\}$ and $n$ real numbers $x_1,x_2,...,x_n\geq 1$. Conjecture: There exists $k\in [n]$ such that: $$\sum_{k\in A,A\in \mathcal F}\...
- 1,558
3 votes
0 answers
101 views
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 ...
- 203
3 votes
1 answer
232 views
Do idempotents in an abelian category constitute a lattice?
Consider an object $X$ in an abelian category $\mathcal{C}$. We define $\text{Idem}_{\mathcal{C}}(X)$ as the set of idempotent endomorphisms $a$ in $\text{End}_{\mathcal{C}}(X)$, meaning that $a \circ ...
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9 votes
1 answer
493 views
Is the partial order of all equations in the signature of magmas a lattice?
$\newcommand\Eq{\mathrm{Eq}}$I asked this question on math stack exchange, here, but there were no comments or answers. So, I am asking it here on mathoverflow. Consider the signature of a single ...
- 2,183
1 vote
0 answers
132 views
Simplicial complexes on $[n] := \{0,\ldots,n\}$ that are identical under any contraction of consecutive vertices
For $n\in\mathbb{N}$, let us denote by $\Omega(n)$ the set of all (possibly empty) “abstract” simplicial complexes on $[n] := \{0,\ldots,n\}$ (“on $[n]$” means “labeled by the elements of $[n]$”). To ...
- 38.1k
9 votes
1 answer
686 views
The reals: a topological lattice in more than the obvious way?
Define a topological lattice as a (not necessarily bounded) lattice in $\textbf{Top}$, i.e. meet and join are continuous maps $X^2 \rightarrow X$. There are two obvious topological lattice structures ...
- 1,703
6 votes
0 answers
270 views
Can one naturally transform Tamari lattices into distributive lattices with the same number of elements?
Many of the zillions of combinatorial objects counted by Catalan numbers come with various lattice structures. The $n$th Tamari lattice $T_n$, as originally defined, is the lattice of all those maps $...
- 19.4k
6 votes
1 answer
238 views
Do projections in an $AW^\ast$-algebra form an orthomodular lattice?
I’m currently studying orthomodular lattices arising out of operator algebras. One of the most standard examples is the projection lattice of a von Neumann algebra - if $M$ acts on a Hilbert space $H$,...
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