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 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 ...
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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 ?!

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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70 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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23 views

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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Is an open map with open relative diagonal necessarily a local homeomorphism? [migrated]

Let $f : X \to Y$ be an open (and continuous) map of locales. Suppose the relative diagonal $\Delta_f : X \to X \times_Y X$ is an open embedding of locales. Does it follow that $f : X \to Y$ is a ...
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40 views

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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ideal of ring of integer modulo z12 and lattice

my question is that how to find ideals of ring of integers modulo Z12. Is that a lattice ? if yes then how to draw its hasse daigram?
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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 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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28 views

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 ...
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Coproducts and pushouts of Boolean algebras and Heyting algebras

I am having trouble of find a reference explaining how to compute coproduts and pushouts in the category of Boolean algebras and in the category of Heyting algebras. To be precise I am looking for ...
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Can this lattice be realized as an intermediate subgroups lattice?

Let $G$ be a finite group and $H$ a subgroup. Let $\mathcal{L}(H \subset G )$ be the lattice of all the intermediate subgroups between $H$ and $G$. Let the lattice $\mathcal{L}$ as follows: ...
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Free ordered vector space over an ordered abelian group

Let $G$ be a partially ordered abelian group (written additively). I want to add $\mathbb{R}$-multiples to $G$ in a "free" way ,thus extending $G$ to an ordered vector space. Construction: To this ...
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A normal intermediate subgroup in $B_3$ lattice with an additional index condition?

This post is a sequel of: A normal intermediate subgroup in $B_3$ 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 $B_3$ 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 $B_{3}$?

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 $B_{n} = \mathcal{L}(\{ e\} ...
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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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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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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 ...