3
votes
0answers
49 views
+100

Compositions of filters on finite unions of Cartesian products

Let $\Gamma$ be the lattice of all finite unions of Cartesian products $A\times B$ of two arbitrary sets $A,B\subseteq U$ for some set $U$. See this note for other equivalent ways to describe the set ...
4
votes
4answers
68 views

Number of join-irreducible elements of a lattice: is it monotonic?

Let $\mathcal L$ be a sub-lattice of $\mathcal P(X)$, where $X$ is a finite set. Denote by $\mathcal I(\mathcal L)$ the set of union-irriducible elements of $\mathcal L$ (i.e. $A\in \mathcal ...
0
votes
1answer
120 views

Is every sub-lattice of $\mathcal P(X)$ isomorphic to a sub-lattice of $\mathcal P(X')$ containing singleton sets?

Let $X$ be a finite set. $\mathcal{P}(X)$ denotes the set of all subsets of $X$. Let $\Gamma$ be a sub-lattice of $\mathcal{P}(X)$, i.e. $\Gamma$ is a collection of subsets of $X$ closed under union ...
0
votes
0answers
26 views

Operation table of Hasse diagram

Consider the following Hasse diagram: My book gives the following join and meet operation tables for this diagram: $$\begin{array}{|c || c | c|} \hline Subset & x \wedge y & x \vee y \\ ...
0
votes
1answer
23 views

Going down from filters to sets for specific relations

Let $\mathfrak{A}$ be a bounded lattice. I call a $2$-staroid a relation $f\in\mathscr{P}(\mathfrak{A}\times\mathfrak{A})$ such that $i\sqcup j \mathrel{f} b\Leftrightarrow i\mathrel{f} b\vee ...
0
votes
0answers
17 views

Certain constructs on filters and on principal filters

Let $\mathfrak{X}$ be a lattice. I will call a set $S\in\mathscr{P}\mathfrak{X}$ a free star when the least element of $\mathfrak{X}$ is not in $S$ and $X\sqcup Y\in S\Leftrightarrow X\in S\vee Y\in ...
4
votes
2answers
39 views

disconnected ordered set

Is there a totaly ordered infinite set $A$ with the least element $a$ and the greatest element $b$ such that for any sequence $\{\alpha_n\}$ and $\{\beta_n\}$ in $A$ which satisfies ...
0
votes
1answer
14 views

A criterion for complete lattice.

Is there an infinite partially ordered set $(X,\le)$, in which for each $A\subseteq X$, either $\inf A$ or $\sup A$ exists but for some $A\subseteq X$ either $\inf A$ or $\sup A$ does not exist.
1
vote
1answer
86 views

Every countable lattice has a cofinal totally ordered subset?

If a lattice is countable, prove that it has a subset that is both totally ordered and cofinal in the lattice. Cofinal means that for each $l$ in the lattice, there is some $a$ in the subset such that ...
4
votes
3answers
296 views

Where can I learn about the lattice of partitions?

A set $P \subseteq \mathcal{P}(X)$ is a partition of $X$ if and only if all of the following conditions hold: $\emptyset \notin P$ For all $x,y \in P$, if $x \neq y$ then $x \cap y = \emptyset$. ...
5
votes
2answers
2k views

Given the Hasse diagram tell if the structure is a lattice

Let's consider the following Hasse diagram: I need to tell whether this is a lattice. By lattice definition I can prove the above shown structure $M_5$ to be a lattice if and only if $\forall x,y ...
3
votes
1answer
200 views

What is a complete lattice?

In a lecture of real analysis (this course is about Lebesgue measure) the lecture said: For a set $X$ - $P(X)$ (the power set) is a compele lattice: For every $S\subseteq P(X)$ there exist $\cup ...
1
vote
2answers
313 views

Draw Hasse diagram as two elements have same image

Let $S=\{1,2,3,4,5,6,7,8,9,10\}$, $P=\{y \in \mathbb N : y \text { is a prime number}\}$, consider the map $f$ defined as follows: $$\begin{aligned} f:x\in S \rightarrow f(x) \in \wp (P) ...
5
votes
4answers
2k views

Simplest Example of a Poset that is not a Lattice

A partially ordered set $(X, \leq)$ is called a lattice if for every pair of elements $x,y \in X$ both the infimum and suprememum of the set $\{x,y\}$ exists. I'm trying to get an intuition for how a ...
2
votes
4answers
1k views

difference between join and union

Is there a difference between the join operator, $\wedge$, and the union of a set? In particular, what is the join of $a \wedge b $ and $b \wedge c$? Is it $a\wedge b \wedge c$ or is it $0$? I seem ...
5
votes
1answer
1k views

limsup and liminf of a sequence of points in a set

My ways to define/write limsup and liminf of a sequence of points in a set $X$: They come from what I have understood. If you have other ways of understanding, really appreciate if you can reply ...
6
votes
2answers
2k views

limsup and liminf of a sequence of subsets of a set

I am confused when reading Wikipedia's article on limsup and liminf of a sequence of subsets of a set $X$. It says there are two different ways to define them, but first gives what is common for ...