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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What is a “lattice” in set theory??? [on hold]

NOTE: There is another question asking "What is a Lattice?" but when reading the question, it has to do with programming, and that is not what my question has to do with. The answer provided to that ...
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1answer
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Finding a poset with an antichain and failing pairwise join

In an introductory exercise on posets and lattices, I am asked to give an example of a poset $(P,\leq)$ in which there are three elements $x,y,z$ s.t. $\{x,y,z\}$ is an antichain (EDIT: $A\subseteq ...
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1answer
51 views

Finding the congruences of a lattice

Part of the excercise I am currently doing is finding the congruences of the following lattice: The problem I struggle with the most is what happends when $1 \sim d$ - how to find what is the ...
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22 views

Interpreting N vector function

For $\mathcal{N} = \mathcal{N} \bigcup\{ \bot_{\mathcal{N}} , \top_{\mathcal{N}} \} $ and $\mathcal{N}^\vec{} = \mathcal{N} \vec{} \mathcal{N} $ (monotonic function over $ \mathcal{N} $), how ...
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1answer
29 views

Can every countable Boolean algebra be embedded into $\mathcal{P}(\mathbb{N})$?

Can every countable Boolean algebra be embedded into $\mathcal{P}(\mathbb{N})$? And if so, is the same true for countable semi-lattices?
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1answer
52 views

Interpreting functions for poset

I'm stuck at a question with the following expression of poset $(\mathcal{N}, \sqsubseteq)$: $$\forall x, y \in \mathcal{N}: x \sqsubseteq y \mbox{ iff } (x = y \vee x = \bot_{\mathcal{N}} \vee y = ...
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1answer
25 views

Common fixed point of commuting monotonic functions

Let $P$ be a chain-complete poset with a least element, and let $f_1,f_2,\ldots,f_n$ be order-preserving maps $P\to P$ such that $\forall i,j: f_i \circ f_j = f_j\circ f_i$. Claim. The functions ...
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1answer
43 views

Poset where every monotonic function has a least fixed point

Let $P$ be a poset such that every order-preserving map $f:P\to P$ has a least fixed point. Must $P$ be chain-complete?
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1answer
33 views

Enough homomorphisms to separate its elements

For a lattice $L$ what does the statement mean that there are enough homomorphisms $L\to \{0,1\}$ to separate its elements? What exactly is meant by "separating"?
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1answer
52 views

Chain-complete and least element iff every order-preserving map has least fixed point

Let $P$ be a poset. I want to show the following are equivalent. $P$ is chain-complete and it has a least element. For every order-preserving map $f:P\to P$, the set $P_f$ of fixed points of $f$ has ...
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Do the subspaces of a vector space form a distributive lattice?

Ordered by inclusion. There is a least element $\{0\}$ and a greatest element $V$. Also for two subspaces $V_1,V_2$ we have $V_1\land V_2 = V_1 \cap V_2$. But what is $V_1\lor V_2$? The union of two ...
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1answer
43 views

Different Zorn's lemma statements

Given a chain-complete poset $P$, every $x\in P$ lies below some maximal element. Every inductive poset has enough maximal elements a maximal element. Chain-complete means every chain has a least ...
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1answer
49 views

Least fixed point of restricted function

Let $P$ be a poset with the property that every order-preserving map $f:P\to P$ has a least fixed point $\mu(f)$. Now for any $p\in P$, the poset $\downarrow(p)=\{x\in P|x\leq p\}$ must also have ...
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1answer
53 views

Algebraic lattices and distributivity over joins of upward directed sets

I am reading Burris & Sankappanavar, Chapter 1 on lattices, and I am doing Exercise 6 in Section §4: If $L$ is an algebraic lattice and $D$ a subset of $L$ such that for each $d_1$, $d_2 \in D$ ...
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Elements of bounded distributive lattice belonging to same prime ideals are equal?

I have read in a paper that by an easy application of Zorn's lemma one may show that two elements of a bounded distributive lattice are equal iff they are contained in exactly the same prime ideals of ...
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1answer
25 views

How many sets are created by repeatedly intersecting a family of sets?

I have a finite set $X$ and a finite family of subsets $X_i \subset X$, $0 <= i < n$, $n \in \mathbb{N}$. What can we say about the size of the transitive hull of this family with regards to ...
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1answer
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Lattice breadth $k$ same as having Boolean sublattice of $2^k$ elements?

The breadth of a lattice is the largest integer $n$ such that any join of elements $X=\{x_1,x_2,\ldots,x_{n+1}\}$ is join of a proper subset of $X$. Birkhoff's classical book has an exercise: "Show ...
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1answer
25 views

Complete Lattice and fixed point

I am wondering how to show: An order-preserving map $f$ of a complete lattice $A$ into itself has at least one fixed element.
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15 views

Is there a common notation for $x \sqcap Yy\neq \bot$

Is there a commonly used shorthand to express the following relation: $x R y \iff X \sqcap Y \neq \bot$? That is, the greatest lower bound of the two elements is not bottom. In terms of sets, the ...
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1answer
13 views

Distributivity of lattice $\left(N,\:\le \right)$

The exercises asks me to prove/verify the distributivity of the lattice $\left(N,\:\le \right)$ I've no clue on how to approach this problem, because at the seminar we didn't really study lattices as ...
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1answer
29 views

Prime ideal theorem for modular lattices?

There's a well-known theorem for distributive lattices commonly referred to as the "prime ideal theorem:" Let $L$ be a distributive lattice, $I$ an ideal of $L$, and $F$ a filter of $L$ such that ...
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1answer
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What are the endomorphisms of $(X \sqcup \{\infty\},\wedge,\infty),$ where $X$ is the set of finitely-supported $\mathbb{N} \leftarrow \mathbb{N}$?

Write $X$ for the set of finitely-supported functions $\mathbb{N} \leftarrow \mathbb{N}$. Then $(X,+,0)$ is the commutative monoid freely generated by $\mathbb{N}$-many generators. So the ...
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2answers
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How to prove that $\frac{1}{n}L/L\simeq (\mathbb{Z}/n\mathbb{Z})^2$?

Let $L$ be any lattice in $\mathbb{C},$ and $L'$ a lattice containing $L$ with index $n$ (i.e $n=\sharp L'/L$) I found this statement "The lattice $L'$ must be contained in $\frac{1}{n}L = ...
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Proof: Every lattice has a maximal filter iff AC

I'm working through a proof of Herrlich's book Axiom of Choice, p.58 (Google books): Equivalent are Every lattice has a maximal filter. Axiom of Choice. In this book, a lattice is ...
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31 views

About Kurosh-Ore theorem

Where can I find the proof of Kurosh-Ore theorem in lattice theory? The statement of this theorem is: Let $L$ be a modular lattice with $0$ and $1$ that satisfies both chain conditions. Then for ...
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Let (S,≤) be a partial order with two minimal elements a and b, and a maximum element c. Let P: S → {True, False} be a predicate defined on S.

Suppose that P(a) = True, P(b) = False and P(x) ⟹ P(y) for all x,y∈S satisfying x≤y, where ⟹ stands for logical implication. Which of the following statements CANNOT be true? (A) P(x) = True for all ...
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1answer
27 views

ordering of intervals

Suppose I have a set of N objects, {a,b,c,d,e,...}, and an NxN matrix whose values are the overlap (in length, area, volume, etc) of each pair of objects. With this matrix, can I recover the ordering ...
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28 views

Does GLB imply LUB and conversely?

Let $(X,\le)$ be a totally ordered set such that it satisfies the Least Upper Bound property (LUB). Does it necessarily satisfy the Greatest Lower Bound Property (GLB) and vice versa? In ...
4
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1answer
73 views

What does it mean if a free algebra has an unsolvable word problem?

I wonder how hard identity testing (similar to polynomial identity testing) can be for a free algebra. I thought that in a certain sense, the problem should always be semi-decidable, because the free ...
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31 views

Power set of A is a complete lattice

I am currently trying to proof that the power set of A is a complete lattice. Since $\mathcal{P}(A),\subset$ is a partially ordered set, we still have to proof that $\sup(X)$ and $\inf(X)$ exist, ...
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1answer
29 views

Completeness of the lattice of projectors of a von Neumann algebra

Consider a von Neumann algebra of operators $R$ in a complex generally non-separable Hilbert space $H$ and let $L\subset R$ be the lattice of orthogonal projectors included in $R$. Is $L$ complete? ...
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1answer
19 views

On non-modular lattices and orto-modularity

I would like to have a definition for non-modular lattices which clearly sets them appart from their modular counterparts, thereby focusing on their main distinctive feature. Besides, I would be very ...
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What are universal abstract $\sigma$-algebras on $\sigma$-frames?

In this paper, the authors make the following definitions: An (abstract) $\sigma$-algebra is a boolean algebra with countable joins. A $\sigma$-frame is a bounded lattice with countable joins, where ...
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1answer
31 views

Describe which partial orderings yield boolean algebras

I thougt about propositional logic and boolean algebras and how propositional logic is (at least from one point of view) not really about $\land,\lor,\neg,...$ but about boolean operators, i.e. n-ary ...
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1answer
32 views

All non-distributive lattices of 6 elements

I'm struggling to find all non-distributive lattices of 6 elements. I looked for those, that have at least 1 sub-lattice isomorphic to M3 or N5 and found some, but I don't know how to guarantee that i ...
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1answer
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Relative chinese remainder theorem and the lattice of ideals

Let $R$ be a ring with two-sided ideals $I,J$. The proof of the "absolute" chinese remainder theorem revolves around the fact that if $I,J$ cover $R$ in the lattice of ideals, i.e $I+J=R$, then the ...
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1answer
24 views

Question related to Boolean Algebra.

I am asked to prove that order of a Boolean Algebra cannot be prime greater than 2. I have a dificulty to show this in an appriopriate way. I know the definition of Boolean Algebra. The definition I ...
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1answer
12 views

Sublattices , lattice of subgroups of a group is complete?

It is asked to show that sublattices of a lattice form a complete lattice under subset relation( I am not sure if the symbol is of just a relation R) and also to show that the lattice of subgroups of ...
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23 views

A question related to distributive bounded lattice.

If L is a distributive bounded lattice then show that the complemented elements of L form a sublattice of L. The question is very simple to understand but I am not confirm about my answer. My question ...
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1answer
45 views

How many chains are there in a finite power set?

Let $A$ be a finite set with $n$ elements. How many chains are there in $\mathcal P(A)$ -- that is, how many different subsets of $\mathcal P(A)$ are totally ordered by inclusion? It's easy enough to ...
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Is this a sufficient condition for distributivity of the lattice?

If a lattice $L$ is distributive then it can be shown that for $a,b,c\in L$: $$[a\wedge b=a\wedge c\text{ and }a\vee b=a\vee c]\implies b=c$$ So for fixed $a,u,v\in L$ there is at most one $b$ such ...
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1answer
25 views

A question related to lattice theory.

I am asked to show that in a Boolean Algebra $$(a' \lor b') \lor (a \land b \land c') \;=\; (b \land c') \lor (a' \lor b')$$ My question is - Is it absolutely okay to show this using truth table ...
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1answer
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How to identify lattice in given hasse diagrams?

Consider the following Hasse diagrams. and given here , Counter example on wiki : Says " Non-lattice poset: b and c have common upper bounds d, e, and f, but none of them are the least ...
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The $n$-cells of a modular lattice are antichains

The terminology "$n$-cell" is made up by me, and I'd love to hear if this has an official name. Given a poset $(X,\le)$, define the set $A_n$ of $n$-cells of $X$ recursively as follows: An ...
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1answer
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name or characterization for the “partition” lattice of integer partitions of some n? (Young lattice row partial order)

Is there a name or characterization for the "partition" lattice of integer partitions of some n? Young's Lattice depicts the integer partitions of numbers. Often Young diagrams are used in displaying ...
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61 views

Equivalence of two definitions of complete distributivity

I would like to know if the following alternative definitions of complete distributivity are equivalent. Let's begin with defining choice functions: For any set $S$ and $U\in ...
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1answer
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Is the width of a poset well-defined?

According to the Wikipedia definition (current revision), the width of a poset is the cardinality of any maximum antichain, where "maximum antichain" here means an antichain of maximal cardinality. ...
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2answers
47 views

Compatible Hilbert space subspaces - need help understanding a statement made in a book

A book I'm reading has the following in a section on lattices formed by subspaces of a Hilbert space : Two subspaces $M$ and $N$ are compatible if there exist three mutually disjoint subspaces ...
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31 views

Is this a lattice? X = {1,3,4,12}, Divisilibity relation.

I am just starting to learn about lattices. I am trying to see what examples I can come up with. It is helpful if I can have some outside confirmation about my thinking. Let $X = \{1,3,4,12\}$ ...
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1answer
24 views

For a Poset $(P,\sqsubseteq)$ and given is $A \subseteq P$ need to prove is $s=m$

There is a Poset $(P,\sqsubseteq)$ and given is $A \subseteq P$ It has both supremum $s$ and maximum $m$ Need to prove is $s=m$ My Work:- Let $x \in A$ then $x \sqsubseteq m$ $\qquad$ { m is ...