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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.

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Math theory that deals with ordered attr-value items?

There is partially ordered sets and lattices. Is there a branch of math that deals with ORDERED Attribute-Value items/objects. F.e. av-items /see that attrs also can be missing i.e. doors&roof/ : ...
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Generalising idea of projection of continuous laticces into any category

Definition: A continuous lattice $D$ is said to be a projection of a continuous lattice $D'$ if and only if there are a pair of continuous maps $$i:D\rightarrow D'$$ and $$j:D'\rightarrow D$$ ...
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properties of identically self-dual matroids

I'm dealing with an identically self-dual matroid M on the vertices E=[2N], that is, if B is a basis of M also E-B is a basis of M itself. I need simple combinatorial properties of these, things like ...
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When a family of subspaces accept a suitable set of basis using expansion basis?

Recently, I want to prove a structure theorem of submodule of finitely generated module over PID. By several process, it reduces to the following problem in linear algebra. Given a linear space $V$...
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Are the real numbers a lattice?

A lattice is a set with a partial order, where every pair has a unique upper and lower bound. As far as I can tell, there is nothing in the definition that forces the set to be discrete. In ...
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The lattice of ideals of a distribituve lattice is itself distributive

I have found myself stuck on a problem and would appreciate a hint. The problem is to show that the lattice of ideals $I(L)$ of a distributive lattice $L$ is itself distributive. This is question 2 ...
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Maximal and minimal element in preordered set

Generally the notion of maximal and minimal element is defined in a partially ordered set (binary relation is reflexive, antisymmetric and transitive). A preorder is a binary relation that is ...
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51 views

When is the Subgroup Lattice Graded?

Let $G$ be a finite group. We say that the lattice of subgroups of $G$ is graded if it is possible to assign a non-negative integer rank $r(H)$ to each subgroup $H$ in such a way that the following ...
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Prove that $f^{-1}\mathbb{R}^+f\subsetneqq Aut(\mathbb{I})$!

Let $\mathbb{I}=([0,1],\leq)$ and suppose $Aut(\mathbb{I})=\{f|f:\mathbb{I}\longrightarrow \mathbb{I}, \text{$f$ is 1-1 and onto and $x\leq$ y iff $f(x)\leq f(y)$}\}$. For any $f\in Aut(\mathbb{I})$ ...
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47 views

Which posets arise as $P\to\Bbb B$ for $P$ a poset?

Let $\Bbb B$ be the poset $\{\top,\bot\}$ with $\bot\leq\top$. Given a poset $P$ let $P\to\Bbb B$ be the poset of order-preserving functions from $p$ to $\Bbb B$, ordered by $f\leq g$ if and only if $...
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Lattice Theory and Topkis Theorem

Topkis Theorem states that: if $f$ is supermodular in $(x,\theta)$, and $D$ is a lattice, then $x^{∗} ( θ ) = \arg\max _{x ∈ D} f ( x , θ )$ is nondecreasing in $θ$. It is not a simple concept for ...
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Proving $X_a=\{L \in I_P(L) \mid a\notin L\}$ are the only clopen downsets in the Priestley topology

Edited, have a potential solution First the setup. Let $L$ be a bounded, distributed lattice, $X=I_p(L)$ be the set of all prime ideals of $L$. Let $X_b=\{L\in X \mid b \notin X\}$. The Priestley ...
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Order preserving mapping between lattices

I would appreciate some help in solving this problem. Problem Let $F:R-\mathbf{Mod}\longrightarrow S-\mathbf{Mod}$ be an additive covariant functor. Let $M$ be an $R$-Module and for each $K\leq M$ ...
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Does the frame of open sets in a topological space or locale really have all meets?

According to the nLab article on locales, a frame has all meets by the adjoint functor theorem: This seems a bit strange to me, since it's well-known that an infinite intersection of open subsets is ...
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Lattice definition and example

Guys I am struggling to understand the lattice concept: Could you help me with this silly example? Take the collection $\{\emptyset, \{0\}, \{1\}\}$ ordered by inclusion. This is a poset, but not a ...
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46 views

If $(\alpha,\beta)$ is the factor pair congruences of algebra $\mathbb{A},$ ia $(\forall \gamma\in ConA)\alpha\circ\gamma=\gamma\circ\beta?$

Let $\mathbb{A}$ be an algebra such that $ConA$ is the distributive lattice. If $(\alpha,\beta)$ is the factor pair congruences of algebra $\mathbb{A},$ prove that $(\forall \gamma\in ConA)\alpha\circ\...
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CEP for (distributive) lattices and groups?

An algebra $A$ has the congruence extension property (CEP) if for every $B\le A$ and $\theta\in\operatorname{Con}(B)$ there is a $\varphi\in\operatorname{Con}(A)$ such that $\theta =\varphi\cap(B\...
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25 views

Representation of sigma complete Boolean algebras

In Terrence Tao's article 245B notes 4: The Stone and Loomis-Sikorski representation theorems he gives a proof that not each sigma-complete Boolean algebra can be realized as a $\sigma$-complete ...
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What are the smallest sets of $n$-permutations of $\{1, 2, \ldots, n\}$ where all n-combinations are “prefix-permutation” of one element of the set?

What are the smallest sets of $n$-permutations of $\{1, 2, \ldots, n\}$ where all combinations of $\{1, 2, \ldots, n\}$ are "prefix-permutation" of at least one element of the set? Notation ...
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Showing equality of 2 suprema in complete lattice

Let $(M,+,0)$ be a naturally ordered commutative monoid (i.e. such that the natural preorder is antisymmetric) such that $(M,\sqsubseteq)$ is a complete lattice. Then $(M,\sum^*)$ is a summation ...
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Sets equipped with sublattice of their power sets

Topology and measure theory are two examples of fields which include, in their objects of study, sets which are given structure by equipping them with a lattice of a certain type sitting in their ...
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Proof if Meet-homomorphism and Join-homomorphism satisfies, then it implies order-homomorphism

What's the proof that join and meet homomorphism lead to order homomorphism as illustrated in this diagram? https://i.imgur.com/0LslbLF.png https://i.imgur.com/yUYjo2I.jpg
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Does this equality hold for all lattices with a top?

Let $(L,\land,\lor)$ be a lattice with maximal element $1$. Let $p_1,...,p_n,q_1,...,q_m \in L: \lor_{i=1}^n p_i=\lor_{j=1}^m q_i =1$. Is it guaranteed that $\lor_{i=1}^n \lor_{j=1}^m (p_i \land ...
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90 views

Cardinality of Sub(A) is smaller than ${\aleph_0}$ or equal $2^{\aleph_0}$?

Let $\mathbb{A}=(A,\mathcal{F}^\mathbb{A})$ be a countable algebra. I need to prove that $|Sub(\mathbb{A})|\leqslant\aleph_0$ or $|Sub(\mathbb{A})|=2^{\aleph_0},$ where $Sub(\mathbb{A})$ is the ...
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28 views

Equipping finite space with metric

I'm attempting to equip this finite space with a metric. The points in this space are defined as all the intersection points, (see image). I think the metric will be approximately the Euclidean metric ...
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1answer
42 views

Describing maximal orders in quaternion algebras.

In Dorman's paper, Global orders in definite quaternion algebras as endomorphism rings for reduced CM elliptic curves, he considers the following situation: $K = \mathbb{Q}(\sqrt{d})$ where $d$ is a ...
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Equivalence of Frankl's Conjecture

Frankl’s conjecture is one of the most famous problems in combinatorics. Frankl's conjecture claims: For every finite non-empty set $A$ and for every Frankl's family $F$ on $A$ exists $a\in A$ such ...
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Show that each edge of the cyclic polytope $C_4(6)$ is contained in either three or four facets, and either three or four 2-faces.

Note: here $C_4(6)$ is the notation for the cyclic polytope of dimension 4 and of 6 vertices. By the 2-neighbourly property of $C_4(6)$ and the Dehn-Sommerville equations, I've determined that the ...
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Ordered Sets and Lattices

Recall the topic “Ordered sets and Lattices” that the set $D_m$ of divisors of $m$ is a bounded, distributive lattice with $$a+b = a\lor b =\operatorname{lcm}(a, b)$$ $$ab = a\land b =\gcd(a, b)$$ (...
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closed lattice ideal is isomorphic to $C(K)$

Let $X\in E$, where $E$ is a Banach function space on $(0,1)$. Consider the interval $[-X,X]$ and generate it to a closed lattice ideal $I$ of $E$. We may renorm this ideal $I$ such that $[-X,X]$ is ...
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Can every β∈Eq(A) be represented by some terms using α1,α2,α3,α4∈Eq(A) and ∩,∨?

$A$ set $A$ is nonempty and finite. $Eq(A)$ is the set of all equivalence relations on a set $A.$ Then Eq(A)=$(Eq(A);\subset).$ How can I prove that we can choose $\alpha_1,\alpha_2,\alpha_3,\alpha_4\...
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In a convex lattice, if $x$ is minimal, then each $y \le x$ is of the form $t x$ for some $t \in (0,1)$

Assume that $X$ is a convex lattice. We say that $x \in X$ is extremal if $x$ does not belong to any open interval, that is, if $$ x = t y + (1-t) z $$ for some $t \in (0,1)$, $y,z \in X$, then $x = y ...
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If $x$ is an element of a join lattice $L$ then what is a good algorithm for determining $\{ A \subseteq L : \bigvee A = x\}$?

For a join lattice $L$, any subset $A$ with join equal to $x$ must be a subset of $\{ y \in L : y \leq x\}$ and if the join of $A$ is equal to $x$ and $B \subseteq \{ y \in L : y \leq x\}$ then $A \...
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A property of complete lattices, and its relation to continuity

I am interested in complete lattices $L$ (with least element $0$ and greatest element 1) satisfying either of the following properties. Property 1 (stronger): Given any totally ordered subset $I \...
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How can I show every uncountable partial order is equal to the intersection of all its linear extensions?

Given an arbitrary partial order $P=(X,R)$ if for any $a,b\in X$ with $(a,b)\not\in R$ and $(b,a)\not\in R$ we define $R'=R\cup\{x\in X:(x,a)\in R\}\times \{x\in X:(b,x)\in R\}$ then I can show that $...
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Restricting a filter in a Boolean algebra to a generating set and have it generate a filter

Let $B$ be a Boolean algebra and $S \subseteq B$ be a subset that generates $B$. Is it the case that every filter $x$ of $B$ is equal to the filter generated by $x \cap S$? What if $S$ itself is a ...
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73 views

On a necessary condition for being a prime ideal

All rings below are commutative with unity. If $P$ is a prime ideal in a ring $R$, then it has the following property: (*) For every ideal $I,J$ of $R$, $I \cap J \subseteq P \implies I \subseteq P$...
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60 views

Induced total order on equivalence classes from partial order

Given a poset $\langle S, \leq \rangle$, we can define an equivalence relation on elements such that $a \sim b$ if $a \nleq b$ and $b \nleq a$, and extend via transitivity and reflexivity. Put ...
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Question on some specific property of ordered semigroup

Let $\langle S, \cdot, \leq \rangle$ be an ordered semigroup (or monoid). Suppose, we have some element $a \in S$, such that for each $b \in S$, $a \cdot b \leq a \cdot b \cdot a$ and $b \cdot a \leq ...
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Degree of supermodularity

The standard definition of supermodularity is as follows. Let $(X,\ge)$ be a complete lattice. A function $f:X\to\mathbb{R}$ is supermodular if for all $x,y\in X$, $f(x)+f(y)\le f(x\wedge y) + f(x\...
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What is a compact element?

I was reading the definition of an Algebraic Lattice: "An algebraic lattice is a complete lattice L, such that every element x of L is the supremum of the compact elements below x". Then I looked for ...
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Why wouldn't the distributive laws hold for a lattice?

My understanding is that the distributive laws $$A\cap (B\cup C) = (A\cap B) \cup (A\cap C)$$ $$A\cup (B\cap C) = (A\cup B) \cap (A\cup C)$$ hold for any set. A lattice is defined as a partially ...
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What do you call a “multi-dimensional semilattice?”

Semilattices are useful for modeling certain types of systems that describe precedence or superceding. For example, in a semilattice that models "authority" systems, we can say that the join relation ...
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Is this simple demand-based prices game a submodular game?

I have this simple market game: $I=\{1,2...,n\}$ players $S_i$ strategy space of each player $i\in I$ $u_i(s_i,s_{-i})=R_i(s_i)-C(s_i,s_{-i})$ There's only one type of resource. The resource is ...
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Establish Archimedean property of a vector-lattice

I am trying to find ways to prove the Archimedian property of a certain vector lattice and got stuck on the following type of problem. I feel the statement below (or in fact weaker versions) should ...
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A problem in proving isomorphism classes of cores forming a lattice

I am reading the book "Algebraic Graph Theory" by Chris Godsil and Gordon Royle and I got confused with Lemma 6.3.3: The set of isomorphism classes of cores, partially ordered by "$\rightarrow$", ...
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Does the dimension converge? [closed]

I'm doing an exercise in a textbook and I need to calculate the effective dimension of a possibly non-Euclidean network lattice. I have $r^d=2r^2+2r+1$ where $d$ is the dimension of the network and I ...
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101 views

Definition and example of a bounded lattice?

I understood what a lattice is and I read about some examples. I understod what a semilattice is, what a complete lattice is but I don't know why I find some difficulties to get the concept of a ...
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19 views

Primary source for definitions re partitions, coarsest common refinement, join/meet etc.?

As much as I do appreciate the contributions on m.se, I need to be able to cite a primary source for definitions related to partitions, e.g., what a refinement/coarsening is, coarsest common ...
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1answer
38 views

Equivalent properties of a proper ideal of a generalized boolean algebra

I do not understand the item c) of the following question, the exercise 9 from section 1.2 from the book "Lattice-ordered Rings and Modules" from Stuart A. Steinberg: A generalized boolean algebra is ...