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 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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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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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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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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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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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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44 views

Lattice Properties Challenge [closed]

i see in one note that the following is true and false, anyone could help me why? if + be a minimum upper bound and * be a maximum lower bound why this properties be true for Lattice? $ a + (b*c) ...
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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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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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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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25 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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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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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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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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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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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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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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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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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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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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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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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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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 ...
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51 views

Definition of Jordan–Dedekind chain condition

Let $(X,\le)$ be a lattice. Which is the correct definition for Jordan-Dedekind condition: 1) all maximal chains in $X$ have the same cardinality. 2) for any interval $I$ in $X$, all maximal chains ...
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A non-algebraic complete lattice

Do you have an example of a complete lattice which is not algebraic‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌?
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The Mini-Max theorem for lattices

I'm asking for help on an exercise in Davey and Priestleys's Introduction to Lattices and Orders. For those with the book, the exercise is specifically 2.9. Let $A=(a_{ij})$ be an $m\times n$ ...
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Self-duality in a lattice

Is there any finite self-dual lattice $(X,\le)$ such that there is not any self-duality $f:X\to X$ such that $f\circ f = 1_X$? Let $f,g:X\to X$ be a self-dualities. Then $f^{-1}\circ g$ is an ...
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How many Homomorphisms are there from one bounded lattice to another?

for a project that I work on, I need to know how many homomorphisms there are from one finite lattice with 0 and 1 to another. I remember that I already worked it out if one of them is the trivial ...
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Proving that a convex subset of a chain is the intersection of a lower segment and an upper segment of a chain.

The book I am using is very naive so AC is assumed. Call a subset $S$ of a chain $L$ a lower segment if: $x \in S$ and $a < x$ implies $a \in S$. An upper segment is defined equivalently. Call a ...
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Find the Fixed points (Knaster-Tarski Theorem)

Let $L=\mathcal P(\mathbb N)$ be a complete lattice of subsets of $\mathbb N$. a) Justify that the function $F(X)=\mathbb N \setminus X$ does not have a Fixed Point. I don't know how to solve this. ...