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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A normal intermediate subgroup in L30 lattice with an additional index condition?

This post is a sequel of: A normal intermediate subgroup in L30 lattice? Let $G$ be a finite group and $H$ a subgroup. Let $\mathcal{L}(H \subset G )$ be the lattice of intermediate subgroups ...
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Properties of distributive lattices and congruences.

Let $L$ be a lattice and let $a,b,c,d \in L$. Show that: $\theta(a,b) \subseteq \theta(c,d)$ iff $\langle a,b\rangle \in \theta(c,d)$ $\theta(a,b)=\theta(a \wedge b, a \vee b)$ Where $\theta$ is ...
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The congruence class of a distributive lattice

I need to prove the next thing: For every non-empty ideal $I$ of a lattice $L$ consider the relation $\theta(I)$ defined by: $ \theta(I) = \{⟨a, b⟩ : ( \exists c \in I) a \vee c = b \vee c\}$ Prove ...
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A normal intermediate subgroup in L30 lattice?

Let $G$ be a finite group and $H$ a subgroup. Let $\mathcal{L}(H \subset G )$ be the lattice of intermediate subgroups between $H$ and $G$. An intermediate subgroup $H \subset K \subset G$ is a ...
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Is there a simple group and a subgroup with intermediates lattice L30?

Let $G$ be a finite simple group and $H$ a subgroup. We consider the lattice of intermediate subgroups between $H$ and $G$, noted $\mathcal{L}(H \subset G )$. Let $\mathcal{L}_n = ...
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How can I prove that the set of $A$-invariant subspaces forms a lattice?

How can I prove the following proposition: the set of $A$-invariant subspaces forms a lattice.
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25 views

Are the maps f(n)=3n and h(x) = the greatest natural number ≤ x residuated?

I have issues with proving the next things: Let $P = ⟨N,\le⟩$ the poset of the natural numbers with the standard order. Consider the map $f : N → N$ defined by $f(n) = 3n$. Is $f$ residuated? Let $P ...
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whether given lattice is distributive or complemented or both?

whether given lattice is distributive or complemented or both? a /|\ / | \ b | c | d | e | f \ | / \ |/ g ...
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What is the universe of a sub algebra generated by $ \{(a \wedge b)\vee(c \wedge b') : a,c \in C\}$ ?

I need to prove the next thing, Let $B$ be a Boolean algebra and $C$ a proper subalgebra of $B$. Let $b ∈ B−C$. Prove that the set $ \{(a \wedge b)\vee(c \wedge b') : a,c \in C\}$ is the universe of ...
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23 views

Distributivity of lattices and prime ideals.

I need help proving that, if $L$ is a lattice with the property that for every nonempty proper filter $F$ and every ideal $I$ such that $F \bigcap I = \emptyset$ then there is a prime filter $G$ such ...
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Proving lattice distributive property - discrete math [closed]

I'm studying discrete math. I'm stuck on this question. Thank you for solution. I don't have any idea to solve it. ...
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Hasse diagram of finite linearly ordered set

What form does the Hasse diagram of a finite linearly ordered set take? I think the linearly order set is nothing but totally ordered set which usually takes lattice form since every element is ...
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example of a monotone non-continuous map.

Let me start by defining some terminology to be sure I made no errors there. Parts of this are translated freely from my mother tongue so feel free to correct terminology or the definitions themselves ...
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30 views

Example of a map between two lattices such that the map is an order embedding but not a lattice embedding

Is there an example of a map $h$ between two lattices such that $h: L_1 \rightarrow L_2$,that is order embedding from $L_1$ to $L_2$ but not an embedding when they are taken as algebras? I would ...
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22 views

Upper bound, lower bound, distributive lattice and complimented lattice

I do know how to find the below properties. But am not able to understand the practical application of these concepts. Upper bound lower bound, If the lattice is a distributive lattice ...
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30 views

Explain the notation for the Cartesian product of a family of sets.

In "Lattices and Ordered Sets" author S. Roman defines the Cartesian product of a family of sets. I understand the concept. What I don't understand however is the notation he has used. He says, "for ...
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Text for study of subgroup lattices of finite abelian groups.

I want to study the subgroup lattice of a finite abelian group. I have found a text on the subject: Subgroup Lattices of Groups by Roland Schmidt, de Gruyter 1994. This book is about subgroups of any ...
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31 views

Lattice of interval

I would like to create a lattice of intervals of integers, but I don't know how to 'draw'(Hasse diagram) it. An interval looks like: [1,4] or ]-Inf,3] for example. I'm having difficulties deciding ...
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A property of the subgroups lattices

Let $G$ be a finite group. Consider all the subgroups $H$ such that its subgroups lattice $\mathcal{L}(H)$ is distributive (i.e. $H$ cyclic), and among them, let $(H_{i_1})$ be the sequence of ...
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unique children of a point in a boolean lattice

I am working with two-element boolean algebra, e.g. points composed of strings of $0$s and $1$s and bit-wise $AND$ and $OR$ to find maxima and minima. In the domain I'm working in, I need to assign ...
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58 views

Scott continuity on powerset

I am looking for the name of the class of functions $f:\mathcal P(A)→\mathcal P(A)$ that are monotone and that are characterised by their image on finite subsets, i.e. the functions $f$ satisfying the ...
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How to formalize a lattice in a graph

Given directed a graph: G = (V, E). How to use algebra symbols to express a lattice in G? where reachability stands for partial order, i.e. in the lattice ...
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Generalized semilattice morphism

Join-semilattice morphism from a join-semilattice $\mathfrak{A}$ to a join-semilattice $\mathfrak{B}$ is a function $\alpha$ conforming to the formula $\alpha(X\sqcup Y) = \alpha X\sqcup\alpha Y$ ...
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Does this combinatorial object have a name?

I have come across the need to with subsets of meet-semilattices. Specifically, my setting is that I need posets that have a meet operation that is unique when defined, but is not necessarily defined ...
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A question on bounded, distributive lattice. [closed]

Is every bounded distributive lattice is complemented? Can anyone explain with an example if possible?
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Show natural numbers ordered by divisibility is a distributive lattice.

I need a proof that the set of natural numbers with the the relationship of divisibility form a distributive lattice with gcd as AND and lcm as OR. I know it can be shown that a AND (b OR c) >= (a ...
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Complete lattice without greatest element

Is there any term for "complete lattice without greatest element" (because the lattice is too big to have the greatest element). A typical example would be the lattice of all small (in Grotendieck's ...
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38 views

Proving distributivity of Heyting algebras with the Yoneda lemma.

How can one prove distributivity of a Heyting Algebra via the Yoneda lemma? I'm able to prove it using the Heyting algebra property $(x \wedge a) \leq b$ if and only if $x \leq (a \Rightarrow b)$. ...
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How many unique crystal faces does a given unit cell have?

I am not sure how to best approach this problem. "A diamond crystal is composed of an enormous number of cubic unit cells that are stacked to produce crystal faces. Stacking of cubes to produce an ...
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Why is this not a lattice??

The solution says it's not a lattice. I can't figure out why the following is not a lattice...I think I checked the meet (join) of almost every pair.
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Joins in lattices and sublattices

Let $A$ be a lattice, and $B$ be a sublattice of $A$. Why is the join of $A$ included in the join of $B$? That is, why is $\bigcup_{t\in T}^{A} a_t\leq\bigcup_{t\in T}^{B} a_t$? (I am tempted to ...
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Least and greatest element of the $(\mathbb{N}, |)$

Consider the relation | on $\mathbb{N}$, where $\mathbb{N} = \{0,1,2,... \}$ and $n|m$ means $n$ divides $m$. I know that the pair $(\mathbb{N}, |)$ is a partial order, : (1) Find the least and ...
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Directed Graph (Lattice) Integration

Let $G=(V,E)$ be a directed graph of vertices $V$ and edges $E$, with $|V|=n$. Assume $G$ is simple (no self loops or multiple edges) and planar (for simplicity). Let $V=\{v_1,\cdots,v_n\}$ be the ...
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Von Neumann and Hausdorff continuous dimensions are related?

Von Neumann in his book Continuous Geometry introduced (in a suitable lattice) a dimension function that has a continuous range. The definition of a dimension function is axiomatic: see Continuous ...
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A question about cofinal totally ordered sets.

Let $A$ be an uncountable set, and let $L$ be the poset consisting of all finite subsets of $A$ (the ordering on $L$ is inclusion). Show that $L$ does not have a totally ordered cofinal subset. I am ...
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mapping functions for power set graphs

Let $G(\mathcal{P}(n),E)$ be a power set graph for $[n]$ elements with the inclusion relation. The width of such graph is known by Sperner's theorem $w=\binom{n}{n/2}$. By Dilowrth's theorem we can ...
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Bounded and Complete Lattices [duplicate]

Prove or disprove: Every bounded lattice is complete. It can be easily proved that every complete lattice is bounded. But is the converse true?
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Books on pseudocomplemented lattices and Heyting algebras

I was wondering if anyone knows a good reference for pseudocomplemented lattices and/or Heyting algebras. Ideally, it should be something like Givant & Halmos's Introduction to Boolean Algebras, ...
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Is the set of pseudo-complements of the elements of an ideal in a pseudocomplemented lattice a filter?

Let $L$ be a pseudocomplemented distributive lattice with $0$ and $1$, $I \subseteq L$ an ideal and set $F = \{\neg x \; | \; x \in I\}$, where $\neg x$ is the pseudocomplement of $x$. My question is: ...
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Show that a interval from a boolean algebra is also a boolean algebra and that a function is surjective

We have an boolean algebra $(B,\lor, \land, ', 0, 1)$ and $b \in B - \{0\}$. We consider $[0,b] = \{x \in B | 0\le x\le b \} \subset B$, where $\le$ means an order relationship introduced in the ...
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Lattice of a POSET Realtion

Given a set $S=\{1,2,3,4,5,6,7,8\}$, defined by a partial order relation Divisibility. Now consider all 4 elements containing sub-graphs, out of which $\{1,2,4,8\}$ is a Lattice obviously . Is ...
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About Knaster-Tarski theorem

The Knaster-Tarski theorem states the following: Let $L$ be a complete lattice and let $f : L → L$ be an order-preserving ...
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Kleen fixed-point theorem and complete lattice

The Kleene fixed-point theorem states the following: Let $(L, \sqsubseteq)$ be a CPO (complete partial order), and let $f : L → L$ be a Scott-continuous (and therefore monotone) function. Then $f$ ...
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Definition of complete partial order and difference with complete lattice

According to this wiki page, a complete lattice is a partially ordered set in which all subsets have both a supremum (join) and an infimum (meet). Whereas, the ...
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What kind of Choice am I making in this argument?

I have an argument that's supposed to imply Choice, but I'm afraid it may be using some choice. If it does, how much choice? This is the part of the argument that might use some Choice. I marked the ...
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What is a Lattice?

To be up front, I was helping my friend with his programming assignment and I stumbled upon the following sentence List all the lattices (subsets of S) on S with size K it later says that the ...
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Question about function on a lattice.

Let $X$ be a complete lattice, and $g$ a function from $X$ to $X$ s.t. $x_1\le x_2$ $\implies g(x_1)\le g(x_2)$. Show that there must be some element in $X$ that maps to itself. Here is what I am ...
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Compact frames, an equivalent reformulation

$\top$ denotes the greatest element of a poset. Adapted from nLab: Definition 1. A frame is compact is and only if for every collection of opens whose union is $\top$ (which covers $\top$), there is ...
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Is the locale of filters on an arbitrary lattice compact?

A mathematician has claimed in a private email to me, that the lattice of filters on every lattice is compact. I have proved it only for distributive lattices. I need help for non-distributive case. ...
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101 views

Is every chain a lattice?

I am asked to prove that every chain is a distributive lattice. Is it true that every chain is a lattice? I am told that a chain is a poset where we can compare any two elements. A lattice is a ...