Tagged Questions

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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Bounded and Complete Lattices [duplicate]

Prove or disprove: Every bounded lattice is complete. It can be easily proved that every complete lattice is bounded. But is the converse true?
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Books on pseudocomplemented lattices and Heyting algebras

I was wondering if anyone knows a good reference for pseudocomplemented lattices and/or Heyting algebras. Ideally, it should be something like Givant & Halmos's Introduction to Boolean Algebras, ...
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Is the set of pseudo-complements of the elements of an ideal in a pseudocomplemented lattice a filter?

Let $L$ be a pseudocomplemented distributive lattice with $0$ and $1$, $I \subseteq L$ an ideal and set $F = \{\neg x \; | \; x \in I\}$, where $\neg x$ is the pseudocomplement of $x$. My question is: ...
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Show that a interval from a boolean algebra is also a boolean algebra and that a function is surjective

We have an boolean algebra $(B,\lor, \land, ', 0, 1)$ and $b \in B - \{0\}$. We consider $[0,b] = \{x \in B | 0\le x\le b \} \subset B$, where $\le$ means an order relationship introduced in the ...
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1answer
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Lattice of a POSET Realtion

Given a set $S=\{1,2,3,4,5,6,7,8\}$, defined by a partial order relation Divisibility. Now consider all 4 elements containing sub-graphs, out of which $\{1,2,4,8\}$ is a Lattice obviously . Is ...
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About Knaster-Tarski theorem

The Knaster-Tarski theorem states the following: Let $L$ be a complete lattice and let $f : L → L$ be an order-preserving ...
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Kleen fixed-point theorem and complete lattice

The Kleene fixed-point theorem states the following: Let $(L, \sqsubseteq)$ be a CPO (complete partial order), and let $f : L → L$ be a Scott-continuous (and therefore monotone) function. Then $f$ ...
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Definition of complete partial order and difference with complete lattice

According to this wiki page, a complete lattice is a partially ordered set in which all subsets have both a supremum (join) and an infimum (meet). Whereas, the ...
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What kind of Choice am I making in this argument?

I have an argument that's supposed to imply Choice, but I'm afraid it may be using some choice. If it does, how much choice? This is the part of the argument that might use some Choice. I marked the ...
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28 views

What is a Lattice?

To be up front, I was helping my friend with his programming assignment and I stumbled upon the following sentence List all the lattices (subsets of S) on S with size K it later says that the ...
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1answer
20 views

Question about function on a lattice.

Let $X$ be a complete lattice, and $g$ a function from $X$ to $X$ s.t. $x_1\le x_2$ $\implies g(x_1)\le g(x_2)$. Show that there must be some element in $X$ that maps to itself. Here is what I am ...
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Compact frames, an equivalent reformulation

$\top$ denotes the greatest element of a poset. Adapted from nLab: Definition 1. A frame is compact is and only if for every collection of opens whose union is $\top$ (which covers $\top$), there is ...
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Is the locale of filters on an arbitrary lattice compact?

A mathematician has claimed in a private email to me, that the lattice of filters on every lattice is compact. I have proved it only for distributive lattices. I need help for non-distributive case. ...
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1answer
74 views

Is every chain a lattice?

I am asked to prove that every chain is a distributive lattice. Is it true that every chain is a lattice? I am told that a chain is a poset where we can compare any two elements. A lattice is a ...
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1answer
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Showing finite lattices are isomorphic to their sets of ideals.

I'm working out a problem from Gratzer '71. We are to show that any finite lattice $L$ $\simeq$ $I(L)$. My attempt was as follow: prove the contrapositive. So assume $L$ is not isomorphic to $I(L)$. ...
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a question about distributive lattice

$L$ is a distributive lattice with top and bottom element($1$ and $0$ respectively). Show that if an element has a complement, the complement must be unique. This is what I have so far, but am ...
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Lists of small lattices and posets

Does any one know where I can find a table that lists, up to isomorphism, all the lattices for a set with small order? and the same thing for how many posets can be formed from a set with small order. ...
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Exercise on posets and antichains in Steven Roman's Lattices and Ordered Sets

I have just began reading through Steven Roman's "Lattices and ordered sets", and I came across an exercise in Chapter 1 that I can't seem to find a good answer to. All the others are fairly easy, so ...
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1answer
52 views

Lattice homomorphism

I have an order on $\mathbb R ^ \mathbb R: f \le g $ iff $\forall x \in \mathbb R: f(x) \le g(x)$. Now I have a function $\mathbb R ^ \mathbb R \to \mathbb R ^ \mathbb R: F (f)(x)= f(x)+f(-x)$. ...
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The size of largest antichain to the total number of incomparable elements

Given a poset $>$ over a set $A$, every two elements $x,y\in A$ stands in exactly one of three cases: either $x>y$ or $y>x$ or $x\bowtie y$. The last case says $x$ and $y$ are incomparable. ...
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cardinality of maximum antichains in power set posets

Let $\mathcal{P}(S)$ be the power set of a non empty set $S$. Consider the poset $\succ$ for the inclusion relation over the elements of $\mathcal{P}(S)$ (which is equivalently represented by a single ...
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2answers
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How is a complete lattice defined solely by a least-upper bound?

I'm pretty far into "Concrete Semantics With Isabelle HOL" and I've come to a section on complete lattices. Their definition of "complete lattice" goes like this: A type $'a$ with a partial order ...
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Pointfree topology: can frames be characterized in terms of forbidden substructures?

Does there exist a class of complete lattices $\mathcal{C}$ such that for all complete lattice $L$, the following are equivalent? In $L$, finite meets distribute over arbitrary joins. i.e. $L$ is a ...
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Extra information needed to distinguish combinatorially isomorphic polytopes

The title pretty much sums up my question: what extra information do we need (or what is an example of sufficient information) on top of the face lattice in order to completely characterize a convex ...
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1answer
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Order preserving functions that do not preserve binary operations

According to Tarski's Fixpoint Theorem for lattices, if I have a complete lattice, $L$, and an order-preserving function, $f:L \to L$, then the set of all fixpoints of $L$ is also a complete lattice. ...
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How to find all maximal chains and antichains in a finite bounded lattice

Is there a (possibly efficient) algorithm to find all maximal chains and antichains in a finite bounded lattice?
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88 views

How does the index of this subgroup is a power of 2?

I am reading an article about coset codes (for answering this question having knowledge about these codes and lattice theory is not necessary) which are defined by $(\Lambda ,\Lambda ',C)$ in which ...
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26 views

Lattice which is not bounded lattice

I want to find an example of a lattice which is not a bounded lattice . Diagrams would be good with an explanation .
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43 views

Equality of two expressions describing a filter

Let $U$, $W$ be boolean lattices with order $\sqsupseteq$, and $U \supseteq W$. The top element of $U$ is the same as the top element of $W$. The bottom element of $U$ is the same as the bottom ...
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On modules over simple rings

We suppose that all rings are left Artinian simple rings and all modules over a ring are of finite length. Let $M \neq 0$ be a left module over a ring $R$. By Wedderburn theorem, $R$ is a matrix ring ...
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19 views

Order on the set of partitions (terminology)

Let $S$ and $T$ be partitions of some set $U$. What is the name for the partition $\{ X\cap Y \mid X\in S, Y\in T, X\cap Y\ne\emptyset \}$? Should it be called the infimum of $S$ and $T$? meet of ...
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1answer
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What does a lattice of the direct power of the two-element chain look like?

In universal algebra, it is known that every finite Boolean lattice is isomorphic to a direct power of the two-element chain. I am having hard time figuring out what a lattice of the direct power of ...
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1answer
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Probability of picking from a sublattice

Short version The set of partitions of a four-element set forms a lattice. Suppose that I pick $n$ times from the set of tri- and bipartitions (i.e., the top element = quadripartition and the ...
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Compositions of filters on finite unions of Cartesian products

Let $\Gamma$ be the lattice of all finite unions of Cartesian products $A\times B$ of two arbitrary sets $A,B\subseteq U$ for some set $U$. See this note for other equivalent ways to describe the set ...
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Proof on Lattices

Prove that a partially ordered set is a lattice if every two elements in the set have a unique least upper bound and a unique greatest lower bound. I was unable to find a way to prove this.
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Alternative definition for order-embedding

I realize that the traditional definition for an order-embedding f is that $a_1 \sqsubseteq a_2 \iff f(a_1) \sqsubseteq f(a_2)$ However, is it also fair to say that if $(A, \sqsubseteq)$ is a ...
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A lattice generated by two particular sublattices of the lattice of binary relations

Let $U$ be some set. Let $\Gamma$ be the set of all finite unions $\bigcup_{X\in S}(X\times Y_X)$ where $S$ is a finite partition of $U$ and $Y_X\in\mathscr{P}U$ for every $X\in S$ (that is $Y$ is a ...
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Another conjecture about filters and cartesian products

Now I present a similar but different (weaker) question (the difference is a different definition of the set $\Gamma$): Let $U$ be some set. Let $\Gamma$ be the set of all finite unions ...
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A criterion for compactness in complete lattices [closed]

Let $(X,\le)$ be a complete lattice, $a\in X$ and for each chain $C\subseteq X$, $$a\le\sup C\to (\exists c\in C)(a\le c)$$ Is $a$ compact?
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1answer
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Lattice theory question

I am having trouble with the following question Show that a lattice is distributive iff for any element $a,b,c$ in the lattice $$(a\lor b)\land c \leq a \lor(b\lor c)$$ My attempt: Let the ...
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1answer
61 views

Every subgroup of a cyclic group is characteristic (using lattice theory).

I want to show for $n\in\Bbb N$, which is not square-free, that every subgroup of $Z_n=\langle x\;|\;x^n\rangle$ is characteristic. But I want to show it in a convoluted way. Every automorphism ...
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2answers
140 views

Equivalence between middle excluded law and double negation elimination in Heyting algebra

It's well-know that in intuitionistic logic, middle excluded law and double negation elimination are equivalent. For example, in Johnstone - Topo theory, I read that, in a Heyting algebra, $p\vee\neg ...
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1answer
46 views

Irreducible elements in a Lattice.

Let $L$ be a lattice. We say that $a\in L$ is irreducible if for every $b,c\in L$ such that $a=b\vee c$ we can conclude that $a=b$ or $a=c$. If $L$ is a finite lattice prove that every element ...
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1answer
130 views

A conjecture about filters and finite unions of cartesian products

Let $U$ be some set. Let $\Gamma$ be the set of all finite unions of cartesian products ($X_1 \times Y_1 \cup \dots \cup X_n \times Y_n$) of sets on $U$. Obviously, $\Gamma$ is a a distributive ...
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1answer
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closedness of a sublattice under complements

Let $A$ be a bounded sublattice of the bounded lattice $(X,\le)$ with $$\max A=\max X, ~~\min A=\min X$$ Let $a,b\in X$ be complements and $a\in A$. Is $b\in A$?
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Finite Linearly Ordered Abelian Monoids

This question concerns the proper definition of the phrase "finite linearly ordered abelian monoids". The sequence A030453 of OEIS counts the number of "finite linearly ordered abelian monoids". The ...
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A standard terminology for different definitions of complete sublattice

Let $(X,\le)$ be a complete lattice and $A\subseteq X$. I'm trying to find a standard terminology for special types of sublattice. What is $A$ called if $(A,\le_A)$ is a complete lattice. ...
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113 views

Who first studied semilattices?

Historically, who first studied semilattices, as opposed to lattices or Boolean algebras? (With or without identity, I do not mind.)
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32 views

traingular matrix

I am not much in to the matrices. so I am sorry if I can not put forward the question properly. Assume $G_{n\times n}$ is a rank $n$ matrix (Indeed $G$ is the generator matrix of a lattice). I need ...
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1answer
30 views

Lattice homomorphism induced by topology homeomorphism?

My book mentions this. However, I am not seeing why 1. and 2. are true. I guess part of the problem is that the lattice (homomorphic) map induced by the continuous topology map isn't even defined on ...