Questions tagged [lattice-orders]
Lattices are partially ordered sets such that a least upper bound and a greatest lower bound can be found for any subset consisting two elements. Lattice theory is an important subfield of order theory.
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Why are nets defined with directed sets (which only requires upper bound to finite subset)
The definition of net in topology is defined via a directed set, that is a set $A$ with a preorder such that every finite subset of $A$ has an upper bound.
If my understanding is correct, an ordered ...
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Generalisation of "discrete" and "indiscrete" left/right adjoint to the forgetful functor for general "ordered" Categories with monotone. morphisms.
The following is taken from "An introduction to Category Theory" by Harold Simmons
$\color{Green}{Background:}$
$\textbf{Example (Galois connection):}$ We modify the category of $\textbf{...
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Dual of an algebra in universal algebra?
Is there a universal algebraic notion of a "dual" algebra in the sense of the dual of a lattice. I.e. reversing the partial order except for any algebra in a language L.
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Applications of a theorem of M. H. Stone to general topology
Exercise 2.I of J. Kelley's book General Topology introduces a theorem by M.H. Stone concerning maximal ideals in distributive lattices. The statement is the following :
Let $A$ and $B$ be disjoint ...
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Constructing complete lattices from total functions to posets
So I ran into this statement that says that the lowest upper bound of a set Y in L, a function space built like L = {f : S → L1 | f is a total function} is somehow related to the joins of the f(s)'s ...
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Terminology for non-empty suprema preserving function
Is there an established name for a map of complete lattices $f : L \to L'$ that preserves nonempty suprema? I.e. for all $U \subseteq L$ with $U \neq \emptyset$,
$$ f( \bigvee U) = \bigvee_{u \in U} f(...
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Complemented elements of lattice of divisors of $n$.
For $n \in \mathbb N_0$, let $L = {\downarrow}n$ as a sub-lattice of $(\mathbb N_0,\leq)$.
(Here, $a \leq b$ means $a$ divides $b$.)
For which elements $m \in L$ do we have a complement?
I know that $...
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Irredundant join-representations of finite length distributive lattices is unique
Let $L$ be a lattice and $S \subseteq L$ be such that $\bigvee S$ exists in $L$.
Then $\bigvee S$ is called an irredundant join if $\bigvee(S\setminus\{s\}) \neq \bigvee S$ for all $s \in S$.
Let $J(L)...
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A special topology compatible with partial orders
Question. Are there contradictions with the following topology $\tau$? If not, is it an established topology known with a conventional name? Thanks in advance.
Suppose $(Y, <)$ is a non-empty ...
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Prove equivalence between properties of relations using fixpoint calculus
The problem
Let $a$ be a binary endorelation of some countable set $S$, i.e. $a \subseteq S \times S$ .
I need to show that the following properties are equivalent:
$ (a^*)^{-1} ; a^* \subseteq a^* ; ...
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Irredundant join of join-irreducibles
Let $L$ be a lattice and $S \subseteq L$ be such that $\bigvee S$ exists in $L$.
Then $\bigvee S$ is called an irredundant join if $\bigvee(S\setminus\{s\}) \neq \bigvee S$ for all $s \in S$.
I proved ...
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Effective algorithms for finite lattices of (higher-order) monotonous functions?
I am looking for references on effective algorithms on finite lattices or posets, and in particular on lattices of monotonous functions between two lattices, with higher-order structure -- monotonous ...
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Is the lifting of an ordering $≤$ to the a relation on the powerset $X⊑Y:=∀x∈X,y∈Y. x ≤ y$ a known construction?
Let $(A, ≤)$ be a poset. We can define a relation on $\mathcal{P}(A)$: $X⊑Y:=∀x∈X,y∈Y. x ≤ y$. My question is: Is this a known derived ordering? It's clearly not the lexicographical, since $∀X. X⊑∅$ (...
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Seeking related work to cocomplete category with "compatibility relation"
I am looking for related work that matches a set-up as follows:
Consider a cocomplete category $\mathsf X$ (i.e. a category with all colimits), and a compatibility relation $\ast$ that is a functor
$$
...
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Defining the cover relation on the product of two lattices.
Suppose I have a lattice $\langle L, \preceq\rangle$. I know that the covering relation on the lattice is that $x\lessdot y$ if $x\prec y$ and there is no $z$ such that $x \prec z \prec y$.
Now ...
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Fixed point problem involving a minimization
Let $X \subset {\Bbb R}^n$ be a compact and convex set, and let $f : X \times X \to {\Bbb R}$ be a continuous and differentiable scalar field defined by
$$
f(x,x^*)=g(x) + \sum_{i=1}^n x_ih_i(x^*).
$$
...
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What is a cone of a lattice?
I'm reading Fulton Harris group representation theory.
I encountered this:
"The virtual characters of $G$ form a lattice $\Lambda \cong \mathbb Z^{\mathfrak c} \in \mathbb C_{\text{class}}(G)$, ...
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Examples of lattice homomorphism $f:L\to K$, where $L$ is arbitrary
Definition. Let $L=\langle L, \vee , \wedge \rangle$ and $K=\langle K, \vee , \wedge \rangle$ be lattices, and let $h:L\to K$. Then $h$ is a lattice homomorphism if and only if for any $a$,$b \in L$, $...
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if f is a homomorphism from L to L', Should the image f(L) be a sublattice of L'?
I'm a beginner in the subject & my question can be meaningless, so I'm sorry from start if that's the case.
I just don't understand why all of the image f(L) can be a sublattice of L' when f is a ...
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Join of monotonic functions is monotonic
Given two monotonic functions $F$ and $G$ on a complete lattice, how would I go about proving that the function:
$$ H : x \mapsto F(x) \vee G(x) $$
is monotonic (i.e. $a \leq b \implies H(a) \leq H(b)$...
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Does there exist a lattice with a unique coatom, but whose unique coatom is not the second-from-top element?
Does there exist a (necessarily infinite) lattice $L$ which has a unique coatom, but such that the unique coatom is not the second-from-top element? By second-from-top, I mean, if the top element was ...
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Existence of paths obeying partial ordering
Consider a partially ordering on $\mathbb{R}^n$ that forms a lattice, with meet and join continuous w.r.t. the standard topology (i.e. a topological lattice). Can we choose a path $\gamma(t)$ with ...
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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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Existence of universal subnets via lattice theory
I refer systematically to J.L. Kelley's book on General Topology, in particular pages 80-81.
In exercise 2.I, the following theorem is stated:
Theorem. Let $A$ and $B$ be disjoint subsets of a ...
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Contracting a segment in a Lattice
Studying lattices, I'm looking for the following construction:
Given a finite (hence bounded and complete) lattice $L$ and $a,b \in L$ such that $a\leq b$,
obtain a lattice $L'$ by "contracting&...
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Generalized boolean algebra structure on connected subset of euclidean space
This is a curiosity question that I've been grappling with as I've been reading more about lattice theory:
Is it possible to endow some connected subset of $\mathbb{R}^n$ with a generalized boolean ...
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Lattices of clones: is $4$ worse than $3$?
For finite $k$, let $\mathscr{C}_k$ be the set of clones on a $k$-element set, viewed as a metric space by setting $d(A,B)=2^{-n}$ for distinct clones $A,B$ where $n$ is the smallest number such that ...
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Can the supremum of an uncountable family of measures be replaced by the supremum over a countable subfamily?
Consider a measurable space $(X,\mathcal{A})$. Let $\mathcal{M}$ denote the family of all countably additive measures $\mu\colon \mathcal{A}\to [0,+\infty]$. This family can be made into a partially ...
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Stone's theorem in the presence of superselection rules
Let $\mathcal L$ be a orthomodular sub-$\sigma$-lattice of the lattice ${\rm L}(H)$ consisting of all closed subspaces of the separable Hilbert space $H$,
(precisely
$\mathcal L$ is a set of closed ...
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Question regarding the Union of a chain of relations
Consider a chain of binary relations $\{a_i\}_{i \in \mathbb{N}}$, where $\forall i \in \mathbb{N} : a_i \subseteq X \times X$, where $X$ is a countable set.
Additionally the following holds:
$$ \...
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$\omega$-continuity and Scott continuity
In a recent lecture I've been presented with the concept of $\omega$-continuity, defined as "the preservation of joins in $\omega$-chains", where an $\omega$-chain is a non-empty sequence $\{...
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Understanding the dual of the tensor product of join-semilattices
The category of join-semilattices is a monoidal category with a "classical" tensor product $\otimes$ (analogous to other binary commutative algebras, e.g. abelian groups). JSL is dual to the ...
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all lattice $\{0,1\}$- homomorphisms from $L$ to $L \times K$
let K be the lattice of the chain $\{0,1\}$ and $L$ be the lattice bellow
I want to know all $\{0,1\}$-homomorphisms (homomorphism taking zero into zero
and unit into unit) from $L$ to $L \times K$.
...
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Trying to understand this note about closure of a set in Willard, Stephen General Topology definition $3.5$
For a subset $E$ of a metric space $(X,d)$, one defines the closure $\overline{E}$ as the intersection of all the closed sets that contain $E$.
Let's put aside the fact that it must be intersection of ...
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Operators with certain $\cup$-like properties
I'm wondering whether there is an operation distinct from set union that has the following properties. Consider an operation $O$ on a field of sets such that:
$O$ is commutative, associative, and ...
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Equivalent definitions of distributive lattice
A lattice is an algebraic structure $(L, \wedge, \vee)$, consisting of a set $L$ and two binary, commutative and associative operations $\wedge$ and $\vee$ on $L$ satisfying the following identities ...
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Are complete sublattices of algebraic lattices algebraic?
If $L$ is an algebraic lattice with a complete sublattice $T$, must $T$ be algebraic? I suspect that it must not be, but am struggling to find a natural counterexample.
As supremums are preserved in $...
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Distributive lattice is finite iff has finite join-irreducibles?
This is exercise 4.19 in Davey and Priestley, Introduction to lattices and order, 2nd ed.
Show that, for a distributive lattice $L$ the following are equivalent:
(1) $L$ is finite;
(2) $L$ has finite ...
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Alternative condition for modular lattices
A lattice is distributive if it holds the formula
\begin{equation}
a \vee (b \wedge c) = (a \vee b) \wedge (a \vee c).\tag{D}
\end{equation}
There are other equivalent conditions, and one is
\begin{...
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Distributive lattice freely generated by I
I am reading Lattice Theory: Foundation written by George Gratzer. In theorem 128,
he said:
Let L be a distributive lattice generated by I. The lattice L is distributive freely generated by I iff the ...
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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 $|I_{...
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Name of the monotone-like property "$f(x)\geq f(x\wedge y)$ implies $f(x\vee y)\geq f(x)$" on a lattice
Let $(X,\leq)$ be a lattice. There is a function $f$ that has the following property.
$f(x)\geq f(x\wedge y)$ implies $f(x\vee y)\geq f(x)$
where strict inequality on the left-hand side implies ...
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Name of the notion dual to disjoint union? ("Exhaustive intersection"?)
Question: Given a set $X$ and subsets $S_1, S_2 \subseteq X$, what is $S_1 \cap S_2$ called when $S_1 \cup S_2 = X$?
More generally, given a set $X$ and subsets $\{S_{i}\}_{i \in \mathcal{I}}$, with $...
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The meet of two minimal generators of a stable ideal in a polynomial ring
Let $k$ be a field and let $R$ be the polynomial ring $k[x_1,\ldots,x_n]$. Let $I$ be a monomial ideal of $R$. We say that $I$ is stable if it satisfies the following "exchange property": ...
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A lattice of integer congruence classes
Let the notation $[a]_n$ stand for $\{a + kn \mid k \in \mathbb{Z}\}$. If $n = 0$, this is $\{a\}$; otherwise, this is $\{b \mid b \equiv a \pmod{n}\}$.
We can define a lattice $L$ whose elements are ...
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Multiplicative lattice with $(a \land b)\ast(a\lor b)=a \ast b$
The natural numbers $\mathbb{N}$ carry two (order-theoretic) lattice structures: One, say $L_1$, is the division lattice (where the join is the least common multiple and the meet is the greatest ...
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Clarifications needed in an exercise about semilattice and abelian monoids in Arbib and Manes' text
The following question is taken from Arrows, Structures and Functors the categorical imperative by Arbib and Manes
Exercise: A $\textbf{semilattice}$ is a poset in which every finite subset has a ...
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Rigorously proving a lattice is sublattice of another
Consider the following two lattices, $L_1$ (top) and $L_2$ (bottom):
I apologize for the bad image arrangement.
We are asked whether $L_1$ is a sublattice of $L_2$. This can be visually observed ...
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For a distributive lattice, show $x ~\hat{\land}~ a = y ~\hat{\land}~ a$ and $x ~\hat{\lor}~ a = y ~\hat{\lor}~ a$ imply $x = y$.
Let $\hat{\land}, \hat{\lor}$ be binary operators denoting the infimum and supremum of two elements in a poset. I was given the following problem.
For a distributive lattice $\langle ~L, ~\hat{\lor}~,...
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Least fixed point and monotone mappings in complete lattice
Let $(E_1,\leq_1)$ and $(E_2,\leq_2)$ be two complete lattices. also let $f_1 : E_1 \times E_2 \rightarrow E_1$ and $f_2 : E_1 \times E_2 \rightarrow E_2$ be mappings monotonic with respect to their ...
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