# 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.

1,449 questions
Filter by
Sorted by
Tagged with
1 vote
48 views

### 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 ...
1 vote
53 views

34 views

### 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 ...
50 views

### 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. $a \leq b \implies H(a) \leq H(b)$...
39 views

### 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 ...
63 views

### 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 ...
1 vote
95 views

### 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 ...
17 views

### 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 ...
1 vote
34 views

### 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&...
1 vote
47 views

### 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 ...
159 views

### 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 ...
408 views

### 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 ...
1 vote
56 views

### 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 ...
19 views

26 views

### 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": ...
1 vote
197 views

### 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 ...
1 vote
80 views

### 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 ...
56 views

### 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 ...
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 ...