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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Lattice of subgroups

I'm trying to find the lattice of subgroups of the symmetric group $\mathfrak S_3$ and of the diedral group $\mathcal D_8$ (the group of order 8). I searched on google, but I didn't find anything.
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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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How to transform a semilattice into lattice [closed]

I need to transform a complete semilattice according to intersection and with a unity member into a lattice. I have been researching this problem and can't understand where to start and how to resolve ...
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+50

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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28 views

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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22 views

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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27 views

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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29 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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41 views

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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42 views

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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48 views

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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33 views

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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40 views

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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24 views

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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31 views

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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28 views

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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47 views

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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31 views

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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22 views

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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89 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 ...
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57 views

Showing finite lattices are isomorphic to their sets of ideals.

I'm working out a problem from Gratzer '71. We are to show that any finite lattice $L$ $\simeq$ $I(L)$. My attempt was as follow: prove the contrapositive. So assume $L$ is not isomorphic to $I(L)$. ...
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a question about distributive lattice

$L$ is a distributive lattice with top and bottom element($1$ and $0$ respectively). Show that if an element has a complement, the complement must be unique. This is what I have so far, but am ...
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40 views

Lists of small lattices and posets

Does any one know where I can find a table that lists, up to isomorphism, all the lattices for a set with small order? and the same thing for how many posets can be formed from a set with small order. ...
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Exercise on posets and antichains in Steven Roman's Lattices and Ordered Sets

I have just began reading through Steven Roman's "Lattices and ordered sets", and I came across an exercise in Chapter 1 that I can't seem to find a good answer to. All the others are fairly easy, so ...
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1answer
55 views

Lattice homomorphism

I have an order on $\mathbb R ^ \mathbb R: f \le g $ iff $\forall x \in \mathbb R: f(x) \le g(x)$. Now I have a function $\mathbb R ^ \mathbb R \to \mathbb R ^ \mathbb R: F (f)(x)= f(x)+f(-x)$. ...
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The size of largest antichain to the total number of incomparable elements

Given a poset $>$ over a set $A$, every two elements $x,y\in A$ stands in exactly one of three cases: either $x>y$ or $y>x$ or $x\bowtie y$. The last case says $x$ and $y$ are incomparable. ...
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cardinality of maximum antichains in power set posets

Let $\mathcal{P}(S)$ be the power set of a non empty set $S$. Consider the poset $\succ$ for the inclusion relation over the elements of $\mathcal{P}(S)$ (which is equivalently represented by a single ...
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How is a complete lattice defined solely by a least-upper bound?

I'm pretty far into "Concrete Semantics With Isabelle HOL" and I've come to a section on complete lattices. Their definition of "complete lattice" goes like this: A type $'a$ with a partial order ...
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Pointfree topology: can frames be characterized in terms of forbidden substructures?

Does there exist a class of complete lattices $\mathcal{C}$ such that for all complete lattice $L$, the following are equivalent? In $L$, finite meets distribute over arbitrary joins. i.e. $L$ is a ...
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Extra information needed to distinguish combinatorially isomorphic polytopes

The title pretty much sums up my question: what extra information do we need (or what is an example of sufficient information) on top of the face lattice in order to completely characterize a convex ...
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1answer
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Order preserving functions that do not preserve binary operations

According to Tarski's Fixpoint Theorem for lattices, if I have a complete lattice, $L$, and an order-preserving function, $f:L \to L$, then the set of all fixpoints of $L$ is also a complete lattice. ...
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How to find all maximal chains and antichains in a finite bounded lattice

Is there a (possibly efficient) algorithm to find all maximal chains and antichains in a finite bounded lattice?
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89 views

How does the index of this subgroup is a power of 2?

I am reading an article about coset codes (for answering this question having knowledge about these codes and lattice theory is not necessary) which are defined by $(\Lambda ,\Lambda ',C)$ in which ...
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32 views

Lattice which is not bounded lattice

I want to find an example of a lattice which is not a bounded lattice . Diagrams would be good with an explanation .
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Equality of two expressions describing a filter

Let $U$, $W$ be boolean lattices with order $\sqsupseteq$, and $U \supseteq W$. The top element of $U$ is the same as the top element of $W$. The bottom element of $U$ is the same as the bottom ...
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On modules over simple rings

We suppose that all rings are left Artinian simple rings and all modules over a ring are of finite length. Let $M \neq 0$ be a left module over a ring $R$. By Wedderburn theorem, $R$ is a matrix ring ...
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Order on the set of partitions (terminology)

Let $S$ and $T$ be partitions of some set $U$. What is the name for the partition $\{ X\cap Y \mid X\in S, Y\in T, X\cap Y\ne\emptyset \}$? Should it be called the infimum of $S$ and $T$? meet of ...
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What does a lattice of the direct power of the two-element chain look like?

In universal algebra, it is known that every finite Boolean lattice is isomorphic to a direct power of the two-element chain. I am having hard time figuring out what a lattice of the direct power of ...
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Probability of picking from a sublattice

Short version The set of partitions of a four-element set forms a lattice. Suppose that I pick $n$ times from the set of tri- and bipartitions (i.e., the top element = quadripartition and the ...