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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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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23 views

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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lattice theory in algebraic number theory

.Let V be a finite dimensitional vector space .A lattice in V is complete if and only if there exist a bounded subset whose translation covers the whole space V.
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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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67 views

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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107 views

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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1answer
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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 ...
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What are some illustrative examples that demonstrate how $\succ$ can differ in behavior from $>$ and/or $\geq$?

I really, really want to understand the generalization of metric spaces known as continuity spaces. Unfortunately, I always get tripped up right at the beginning. The problem is that I have little or ...
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Correct definition of subframe

Given a frame $L$, is the following the correct definition of sub-frame? $S\subseteq L$ is a subframe if $0,1\in S$; $x,y \in S$ implies $x\land y \in S$; $S'\subseteq S$ \implies $\bigvee_L S'\in ...
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Obout the poset of subspaces of the vector space $\mathbb{R}$ over $\mathbb{Q}$.

Let $L$ the set of all subspaces of the vector space $\mathbb{R}$ over $\mathbb{Q}$, ordered by the set strict inclusion: $V_1<V_2$ iff $\{x\in V_1 \Rightarrow x \in V_2$ and there exists $y \in ...
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does a lattice have a minimal item

A lattice (L, ≤) is a partially ordered set, where each two elements of the set have a least upper bound and a greatest lower bound. Let's consider the situation where L is finite. I think that this ...
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Does the “equality semigroup” have an accepted name?

Given a set $G$, we get a semigroup on $G \cup \{0\}$ as follows: Define $x^2 = x$ for all $x \in G \cup \{0\}$. Define $xy = 0$ for all distinct $x,y \in G \cup \{0\}$. Question 0. Does this ...
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When is the direct product of two cores itself a core?

Given graphs $X$ and $Y$, a graph homomorphism $f : X \to Y$ is a function $f : V(X) \to V(Y)$ such that if $uv \in X$, then $f(u)f(v) \in E(Y)$—that is, it is a function of the vertices mapping ...
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join-semilattice vs Upper-semilattice ?! definition problem ?! [closed]

In mathematics, a join-semilattice (or upper semilattice) is a partially ordered set that has a join (a least upper bound) for any nonempty finite subset. I ran into some definition challenge. I ...
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81 views

is differ between distributive lattice vs semi-lattice on Turing Degrees

We know a Posed Closed under suprema but not necessarily under infima is an upper semi-lattice. We now r.e set forms a distributive lattice. But my question is why following statement is hold? I ...
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lattices made by $a\wedge b = ab$ and $a\vee b=a+b-ab$

We define the binary operations $\vee$ and $\wedge$ on $\Bbb R$ by $a\wedge b = ab$ and $a\vee b=a+b-ab$. Then for $A=\{0\}$ and $A=\{1\}$ and $A=\{0,1\}$, the set $(A,\wedge,\vee)$ is a lattice. Is ...
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Least upper bound and greatest lower bound of the void set.

Let $(L,\ge)$ a partially ordered set. Suppose that for avery $S \subset L$, there exists an element $LUB(S)= a$ such that $x \ge a \iff x \ge u \quad \forall u \in S$ and, with the obvious meaning of ...
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Birkhoff's definition of semilattice?

could anyone provide me with the original definition of semilattice by Garrett Birkhoff in his book on lattices? If you could also provide, page number and edition, it would be great (as well as some ...
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Function on a Power Set

Let $f\colon \mathcal{P}(A)\mapsto \mathcal{P}(A)$ be a function such that $U \subseteq V$ implies $f(U) \subseteq f(V)$ for every $U, V \in \mathcal{P}(A)$. Show there exists a $W \in \mathcal{P}(A)$ ...
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Showing that poset of set of supports of a vector space is semimodular

Let $W$ be a subspace of the vector space $\mathbb{K}^n$, where $\mathbb{K}$ is a field of characteristic $0$. The support of a vector $v = (v_1,\ldots, v_n) \in \mathbb{K}^n$ is given by ...
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What would be the join and meet of this lattice?

I'm working on the following problem: Let $W$ be a subspace of the vector space $\mathbb{K}^n$, where $\mathbb{K}$ is a field of characteristic $0$. Let $L$ denote the set of supports of all vectors ...
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What is reverse inclusion?

I'm learning about posets for the first time. What does it mean for a collection of sets to be "ordered by reverse inclusion"? Thank you.
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Problem on distributive lattices

I'm trying to prove the following: Show that a lattice is distributive if and only if it does not contain a sublattice isomorphic to either of the two lattices below. I was able to prove that ...
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Which of the following are Hasse diagrams of lattices?

I'm trying to figure out why each of the following figures are not Hasse diagrams of lattices. Could someone, for example, explain why (A) is not a lattice? Thanks!
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inverse relation in a vector lattice

Let $X$ be a vector lattice endowed with an ordering $\leq$. If for some operator $\phi:X_+\rightarrow X$ and $m,M\in X$, we have $$m\leq \phi(x)\leq M,\quad x\in X_+. $$ What would be the bound for ...
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Conditions for monotone function to take maximal chains to maximal chains srujectively

Suppose that $P$ and $Q$ are graded (with rank function $r$) connected posets with least elements and suppose that all maximal chains of $P$ and $Q$ have length $n$. Let $f:P \to Q$ be a surjective ...
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how to find $\alpha(a\wedge b)=\alpha(a)\wedge\alpha(b)$

for $\alpha(a\wedge b)=\alpha(a)\wedge\alpha(b)$ how to do? answer: $\alpha $ and $ \alpha^{-1}$ are order-preserving and $a \leqslant b$ and $a=a\wedge b$ so $\alpha(a) = \alpha(a\wedge b)$ so ...
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show that N(G) is a lattice?

if $G$ is a group let N(G) be the set of normal subgroups of $G$ define $\wedge$ and $\vee$ on$ N(G)$ by $N_{1}\wedge N_{2}=N_{1}\cap N_{2}$ and $N_{1}\vee N_{2}=N_{1} N_{2}=\{n_{1}n_{2}:n_{1}\in ...
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Dual atoms of the lattice of varieties

I'm reading Jaroslav Ježek's "Universal algebra". There is a Theorem. For a signature containing at least one symbol of positive arity, the lattice of varieties of that signature has no coatoms. ...
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Coproduct of bounded distributive lattices given as lattices of subsets

Let $X$ be a set. A lattice of subsets of $X$ is a subset of $\mathcal{P}(X)$ containing $\emptyset$ and $X$ and closed under finite intersection and finite union. Such a lattice is therefore a ...
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Are all finitely distributive and join-complete lattices infinitely distributive?

The infinite distributive law on a join-complete lattice $L$ is as follows: $\displaystyle a \wedge\left( \bigvee_{b \in B} b \right) = \bigvee_{b \in B}(a \wedge b) $ for all $a \in L$ and $B ...
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What does the Dedekind Rule `say'?

In Relation Algebra, the modal law or dedekind rule $$R;S \,\cap\, T \;\subseteq\; (R \cap T;S^\circ);S$$ appears often and I wonder what is the motivation behind it. Moreoever, what does it "say". I ...
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Volume of a compact set, not necessarily convex

Looking through my lecture notes, I came across the notion that if a set $X\subset \mathbb{R}^n$ is compact and convex and $vol(X)=2^n$, then by choosing an $0<\epsilon <1$, then $X\subsetneq ...
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What does a distributed lattice have to do with GCD and LCM?

$\newcommand{\lcm}{\operatorname{lcm}}$I am lost while following this explanation: Let $$A(g, i) = \gcd(F_{g}, \lcm(F_{a_1}, F_{a_2}, \ldots , F_{a_i}))$$ and $$X = \lcm(F_{a_1}, F_{a_2}, \ldots , ...
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Definition of algebraic structure

Is there a definition of algebraic structure? Wikipedia says: a set (called carrier set or underlying set) with one or more finitary operations defined on it. In particular, what is the ...
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Stone-Weierstrass: Lattice

This is just a prework. Given a compact domain. Regard the function space: $$\mathcal{C}(\Omega,\mathbb{R}):=\{f:\Omega\to\mathbb{R}:f\text{ continuous}\}$$ Clearly it is an algebra: ...
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An example of continuos functions on lattices

The question is as following: Design a finite complete lattice $A$, with at least 15 elements. Provide four non-trivial examples of $f:A -> A$ such that $f$ is: a) non monotone b) monotone but ...