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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When a vector space will be a complete lattice?

Let $E$ a vector space, and let $P$ a strict cone in $E$ (i.e) $P\subset E$ verify: $$ \mathbb{R}^+ P\subset P \\ P+P\subset P\\ P\cap (-P)=\{0\} $$ So we can easily construct a partial order on $E$ ...
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Total Turing reducibility

For $x, y\in 2^\omega$, say $x$ is totally reducible to $y$ - and write "$x\le_{Tot}y$" - if there is some Turing machine $\Phi_e$ which is total on every oracle (that is, $\Phi_e^z$ is total for all ...
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Free lattice in three generators

By general results for every set $X$ there is a free bounded lattice $L(X)$ on $X$. I would like to understand the element structure of this lattice. The cases $X=\emptyset$, $X=\{x\}$ and $X=\{x,y\}$ ...
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Creating n-dimensional lattices from lower dimensional parts

Suppose I have n orthogonal unit vectors (in Euclidean space). These unit vectors may be used to describe an n-dimensional unit hypercube. A subset of these unit vectors may be combined to make lower ...
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25 views

Lattices and Hasse diagrams

I would like to know which is the exact difference between lattices ans Hasse diagrams. In some cases, such as when lattices are used to represent maximal and minimal ideals, it seems to me that both ...
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Draw figures for the 5 different lattices with 5 elements.

I can only think of 4 lattices. Those are: a 1-1-1-1-1 (a chain), a 1-3-1, a 1-1-2-1 and a 1-2-1-1 (if this notation isn't clear, I'll provide images). I really can't figure out what the 5th lattice ...
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Proof that any finite subset of a lattice has a supremum.

Let $C\subseteq D$ where $C = \{c_{1}, c_{2},\ldots, c_{n}\}$ and $D$ is a lattice. I believe that this supremum is $((c_1 \vee c_2)\vee c_3)\vee \ldots \vee c_n)$ (sorry about the notation). However ...
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n-dimensional lattice as a collection of lower dimensional spaces.

Suppose I have n orthogonal unit vectors (in Euclidean space). These unit vectors may be used to describe an n-dimensional unit hypercube. A subset of these unit vectors may be combined to make lower ...
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1answer
28 views

Are lattice operations in set of orthogonal projections in Hilbert space continous?

Let $H$ be Hilbert space and denote set of all orthogonal projections in $H$ by $\Pi$. Then $\Pi$ can be given structure of a lattice. We partially order it by declaring $P \leq Q$ if $Q-P$ is ...
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An order isomorphism between lattices is a lattice isomorphism

Suppose that $(X, \leq_X)$ and $(Y, \leq_Y)$ are ordered sets. Let $T:(X, \leq_X) \rightarrow (Y, \leq_Y)$ be an order isomorphism. Is it true that $T$ is a lattice isomorphism? I have this question ...
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How do we identify $\mathfrak{R}$-automorphisms of a group?

If $G$ is a finite group, a bijection $f\colon G\to G$ is called a (normed) $\boldsymbol{\mathfrak{R}}$-automorphism if $f$ maps subgroups of $G$ to subgroups of $G$, and $f(gH) = f(g) f(H)$ for any ...
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Is the Fibonacci lattice the very best way to evenly distribute N points on a sphere? So far it seems that it is the best?

Over in the thread "Evenly distributing n points on a sphere" this topic is touched upon: http://stackoverflow.com/questions/9600801/evenly-distributing-n-points-on-a-sphere. But what I would like to ...
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Framing a lattice problem from information available on multiple runs of GLV decomposition

I have posted a similar question here. The GLV method [ref] is used to speed up ECDSA signature generation. In this method, an input scalar $k$ is decomposed into two scalars, $k_1$ and $k_2$. Then ...
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Is there a connection between lattices in the sense of orders and lattices in the sense of groups?

I'm wondering about this for some time now - is there a intuitive connection between those concepts or have they been named the same by chance? In particular, my interest in this question was ...
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Decomposition of a Set System into Distributive Lattices

I would like to decompose an arbitrary set system $S$ over a universe $U$ into a number of distributive lattices such that these lattices partition $S$. Now, I am interested in the least number of ...
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Properties of infimum and supremum w.r.t. conic orders, in particular positive semidefinite matrices

Given a convex cone $\mathcal{K}\subseteq \mathbb{R}^n$, we can define a partial order $\leq_\mathcal{K}$on $ \mathbb{R}^n$ by setting $$x\leq_\mathcal{K} y \Leftrightarrow y-x\in \mathcal{K}.$$ For ...
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coproduct of lattices preserving filtered property of positive elements

Let $L$ be a complete and completely distributive lattice. An element $x\in L$ is well above $0$, denoted by $x\succ 0$, if for all $S\subseteq L$ with $\bigwedge S =0 $, there exists $s\in S$ with ...
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Partial order relation on the group of integers

We know that the usual $\leq$ is a partial order relation on the group of integers $\mathbb Z$ and $\mathbb Z$ is a totally ordered with this partial order relation. Is there any other partially order ...
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What is subgroup lattice of GL$(n,\mathbb F_q)$?

I am trying search for subgroup lattice diagrame for the general linear group GL$_n(\mathbb F_q)$ but could not find any thing in the net. Can some one help me by providng some link on it ? thank ...
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Is this thing a value quantale?

I am currently trying to understand R. C. Flagg's "Quantales and continuity spaces". However I am struggling a bit with his definitions and would like to have a good simple (but not too simple) ...
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Finding subgraph in DAG in which all nodes are smaller than all others

I think my problem might be one of terminology, so let me explain what I am trying to do. Given a direct acyclic graph $G$ interpreted as a partial order, I want to find a subgraph $S$, so that ...
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Generalizing a theorem about filters on a boolean lattice

Let $\mathfrak{A}$ be a bounded distributive lattice with binary meet and join $\sqcap$ and $\sqcup$. I will denote $\partial F = \{ X\in\mathfrak{A} \mid \forall Y\in F: X\sqcap Y\ne \bot \}$ where ...
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How to create lattice diagrame in maple 14?

I am studying lattice diagrame of subgroups of groups and I have already posted one query over here. Now my present query is: I am using MAPLE 14. Can anyone suggest me how to create lattice ...
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What does $C_2^8, C_2^4$ etc in lattice diagrame of subgroup represent?

I am studying lattice diagrame of subgroups of groups. and I came to know about the lattices of $C_4\times C_2$ and $C_8\times C_2$ over here and here. But the problem is: I am unable to understand ...
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Intersection of two filters on a poset

Fix a poset. A filter on the poset is its nonempty subset which is both a down-directed set and an upper set. Conjecture Intersection of two filters is also a filter. I have proved this conjecture ...
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Does the subnormal subgroups always form a lattice?

Let $G$ be a group and $W(G)$ the set of all subnormal subgroups of $G$, partially ordered by inclusion. My question is if $W(G)$ always forms a lattice (but not necessarily a sublattice of $L(G)$). ...
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Is $(\mathbb{Z} \cup \{-\infty, \infty\},\leq)$ a complete lattice?

I read that $\mathbb{Z}$ is not a complete lattice because it has no greatest lower bound (and lub). If we define $-\infty$, $\infty$ to be integers, i.e., include them in $\mathbb{Z}$, does ...
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The groups with symmetric subgroups lattice

Let $G$ be a group and $\frak L (G)$ be set of all subgroups of $G$. Clearly, $\frak L (G)$ is a lattice. If we know that $\frak L (G)$ is symmetric then what can be said about the group $G$ ? Any ...
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Quotients vs equivalence relations

In a recent preprint I defined the category of quasi-frames (qframes for short) as follows: a qframe is a modular and upper continuous complete lattice; a morphism of qframes $f:L_1\to L_2$ is a map ...
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Ternary algebra satisfying some identities is a join-semilattice

A join-semilattice with greatest element is an algebra $(S,\vee, 1)$ of type $(2,0)$ such that $\vee$ is idempotent, commutative, and associative, and $a\vee 1=1$ for all $a\in A$. Now, let ...
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A complemented lattice satisfying de Morgan's laws is an ortholattice?

Suppose you have a bounded, complemented lattice $\mathfrak{L} = \left<L, \vee, \wedge, \neg, 1, 0\right>$ that satisfies De Morgan's laws. I want to prove that this is an ortholattice. The ...
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Is the congruence relations lattice of a lattice a sublattice of all equivalence relations on it?

By this Wikipedia link, it seem the set of all congruence relaions on a lattice $(X,\le)$ is a complete lattice with inclusion. Is this lattice a (complete) sublattice of the lattice of all ...
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Characterizing affine subspaces order-theoretically

Let $V$ denote a real vectorspace and $\mathrm{Con}(V)$ denote the poset of convex subsets of $V$. The goal is to identify those elements of $\mathrm{Con}(V)$ that happen to be affine subspaces of $V$ ...
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A distributive lattice with finite length must have a finite number of join-irreducible elements.

Let $L$ be a distributive lattice. Let $\mathcal{J}(L)$ be the set of all join-irreducible elements. Recall that $x$ is join-irreducible if $x \neq 0$ and $x = a \vee b$ implies that $x = a$ or $x = ...
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A question on Join homomorphism and Ideals

On page 287 of the book Mathematical Methods in Linguistcs, by Barbara Partee, Alice Ter Meulen and Robert E. Wall (Dordrecht, Kluwer Academic Press, 1993), I find the following theorem, which they ...
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Is there a cyclic subgroup C of S8 such that the interval lattice [C,S8] is distributive?

I've checked by hand that for any $n \le 7$, there is a cyclic subgroup $C$ of $S_n$ such that the intermediate subgroups lattice $\mathcal{L}(C \subset S_n)$ is distributive. Question: Is it the ...
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proving properties of (graph) dominance defined via a system of equations

Some notions on graphs can be defined via a system of equations with values in a lattice. For example, dominance $d(v_1, v_0)$ ($v_1$ dominates $v_0$) in a graph $g$ is defined by a system $\forall ...
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Representing a Concept

From Formal Concept Analysis: If I have a set of objects: G={Shark, Penguin, Bat} and a set of attributes: M={Breathe Underwater, Can Fly, has Vertebrae} and I make a chart to represent my ...
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Does disjunction of two Boolean algebra cuts always produce their ideal sum?

Let $(B, 0, 1, \leq, \wedge, \vee, \neg)$ be a Boolean algebra. For a subset $A \subseteq B,$ denote by $L(A) = \{l \in B \mid (\forall a\in A) \, l \leq a\}$ the set of all lower bounds of $A,$ and ...
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Is orthocomplementation unique in a weakly modular lattice?

I've read in [1] that an equivalent definition of weak modularity in a lattice is that the orthocomplementation is unique in that lattice. However this is just stated without proof and I can't manage ...
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Is Least Upper Bound of Empty Set equal to Greatest Lower Bound of a another Set?

We had this discussion today that $\operatorname{LUB}(\varnothing) = \operatorname{GLB}(L)$ in a complete lattice $(L,\leq)$. I'm not still not getting it why that is the case.
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from linear extension to partial order

Going from partial order to a linear one is easy, we just do topological ordering. I am wondering about the other way around. In particular: let $>$ be a linear order over a set $A$. Let $\Gamma$ ...
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What are conditions for Tarski's fixed point theorem so that greatest and least fixed points are not the same?

Tarski's fixed point theorem claims that for any monotone function on a complete lattice there are least and greatest fixed points (or more generally, that the set of fixed points is a complete ...
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Lattice orders and number of elements in a set

My discrete mathematics lecture notes give the following definition of a lattice order: A 'Partial order R is a lattice order if the set of lower bounds for any two elements $x, y ∈ X$ has the ...
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Three theorems for the price of one? (like duality)

Why the notion of "duality" (when we get two theorems for the price of one) are ubiquitous in mathematics (order and lattice theory, category theory, group theory), but "triality" (three theorems for ...
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A chain with more than two elements

A chain with two elements $0$ & $1$ is complemented as complement of $0$ is $1$ and that of $1$ is $0$.How to show that every chain with more than two elements is not complemented?
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some distributive laws in the Bousfield lattice

It is know that for any α-well generated tensor triangulated category T the collection of Bousfield classes forms a set, furthermore this set is a complete lattice, denoted by B(T), and this lattice ...
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Restriction of a monotone self-map (“endomorphism of $\mathbf{Pos}$”) to a wide subposet

Here's a poset, call it $P.$ $\hspace{1cm}$ Define a function $S : P \rightarrow P$ as follows. $$S(0) = 1, \qquad S(1) = 2, \qquad S(2) = 2$$ Clearly, $S$ is monotone. Now consider the following ...
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Proving Crapo's Lemma

Let $L$ be a finite lattice with least and greatest elements $0, 1$, respectively, and let $X\subseteq L$. Let $n_k$ be the number of $k$-element subsets of $X$ with join $1$ and meet $0$. I want to ...
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Sets of non-complements elements in a lattice.

Let $L$ be a finite lattice with a least element $0$ and a greatest element $1$, where $0\neq 1$. Fix a $t\in L$, and let $X$ be the set of non-complements of $t$, i.e., the set of all $x$ such that ...