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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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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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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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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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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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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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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42 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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32 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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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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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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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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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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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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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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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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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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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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28 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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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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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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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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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 ...
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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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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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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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47 views

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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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 ...
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Why is this lattice modular?

I'm reading Stanley's Enumerative Combinatorics, and he says every lattice with at most six elements is modular. But what about the lattice below on the right? If $x,y$ are the two elements at the ...