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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Find the Fixed points

Let $L=\mathcal P(\mathbb N)$ be a complete lattice of subsets of $\mathbb N$. Find the smallest and the greatest Fixed Point: 1) $F(X)=\left\{ x \mid x+1\in X \right\}$ 2) $F(X)=X \setminus \left\{ ...
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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. ...
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Crazy Set Theory Analogies

I think the following analogies are too interesting to be ignored: Union = Least Common Multiple If $G_1,...,G_n$ denote a number of sets of points (either linear or in any number of dimensions), ...
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The lattice of closed subsets of an algebraic closure operator is an algebraic lattice

Let $A$ be a set. Let $C:Su(A)\longrightarrow Su(A)$ be a function, where $Su(A)$ denotes the set of all subsets of $A$. Suppose that 1) $X\subseteq C(X)$ 2) $X\subseteq Y\rightarrow C(X)\subseteq ...
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Lattice Path Spaces.

It is well known that the number of paths from $(0,0)$ to $(n,k)$ in $\mathbb{N^2}$ with the set of steps $\{(1,0),(0,1)\}$ is ${n+k \choose k}$. This is the minimum number of steps needed to get to ...
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In a lattice, does $x \vee y \leq x\vee z$ and $x \wedge y \leq x\wedge z$ imply $y\leq z$?

Let $L$ be a lattice and $x,y,z\in L$. If $y \leq z$, then clearly $x\vee y \leq x\vee z$ and $x\wedge y \leq x\wedge z$. Now I wonder about the reverse direction. In general, $x\vee y \leq x\vee z$ ...
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Distributive lattices and Birkhoff theorem

I am trying to prove the teorem (Birkhoff) $L$ is a nondistributive lattice iff $M_5$ or $N_5$ can be embedded into $L$ The only part of the proof which I can't understand is this (I am copying from ...
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number of Lattice paths from origin to diagonal after removing vertices

i am stuck with the following problem: consider the quarter plane $\mathbb{N}_0^2$ with vertices $(i,j)\in\mathbb{N}_0^d$ and edges from each vertex $(i,j)$ to $(i+1,j)$ and $(i,j+1)$, i.e. one ...
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Lattice-Theoretic Interpretation of the Fundamental Theorem of Arithmetic

When equipping $\mathbb{N}^\ast=\mathbb{N}\setminus \{0\}$ with the divisibility relation, it forms a lattice with minimum 1, supremum given by the least common multiple, and infimum given by the ...
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Is there such a thing as 'overtification' (dual to compactification)?

The dual to the notion of compactness for spaces is overtness. This duality is not manifest in the category of spaces but rather in the quantifiers used to define these notions. Is there a process ...
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54 views

Lower bounded lattice to complete lattice

My problem is to show that any lower-bounded lattice satisfying the maximal condition is a complete lattice. Let's call the lattice $L$. I'm having some trouble with this. I have tried to look at it ...
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uniqueness of order-isomorphisms for a finite lattice

Let $(X,\le)$ be a finite lattice and $f,g:(X,\le)\to (X,\le^{-1})$ be order-isomorphisms. Is $f$ the same as $g$‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌?
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Help to conceive a name

Filter $F$ is defined by the formula $$A\cap B\in F \Leftrightarrow A\in F\wedge B\in F.$$ Ideal $F$ is defined by the formula $$A\cup B\in F \Leftrightarrow A\in F\wedge B\in F.$$ In my book I ...
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Defining principal elements of every poset. Is this a new idea?

Fix an arbitrary complete lattice $\mathfrak{A}$ with order $\sqsubseteq$. I call elements $a,b\in\mathfrak{A}$ intersecting and denote $a\not\asymp b$ iff there is a non-least element ...
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35 views

increasing subset of a partial order and characteristic function

Can someone help me understand this? Suppose that $\preceq$ is a partial order on a set $S$ and that $A\subseteq S$. If $\mathbf{1}_A$ is the indicator function then $A$ is increasing if ...
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the indicator function of a increasing poset [duplicate]

Can someone help me understand this If $X$ is a poset, $X'$ is a subset of $X$, and $X\cap[x,\infty)$ is a subset of $X'$, then $X'$ is an increasing set. Equivalently, a subset $X'$ of a poset $X$ ...
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22 views

Are surjective order-homomorphisms necessarily complete lattice homomorphisms?

Let $X$ and $Y$ denote complete lattice, and suppose $f : X \rightarrow Y$ is a surjective order-homomorphism. Does $f$ necessarily preserve arbitrary suprema, therefore being a complete lattice ...
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31 views

Show that B has infimum

Let $(A,\leq)$ is a lattice. if $B\subseteq A$ and $|B|=3$ then show that there is $\inf(B)$ If A is a lattice we know for every $x,y\in A$ there are $\inf\{x,y\}=x\wedge y $ and ...
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Dense Boolean subalgebras

I was reading this page and, in the third part of the first remark I found the definition of dense sub-algebra of a Boolean algebra. It is stated that there are various equivalent definitions of this ...
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Monotone log-supermodular function is supermodular.

Let $X$ and $Y$ be lattices. Let $f: X \times Y \rightarrow \Re$. Function $f$ is log-supermodular if for all $x'>x$ and $y'> y$ \begin{equation} f\left(x', y'\right)f\left(x, y\right) \geq ...
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Closure operators and complete lattices.

A closure operator on a set $A$ is a function $C: \mathcal{P}(A) \to \mathcal{P}(A)$ satisfying following axioms: $X ⊆ Y \implies C(X) ⊆ C(Y)$ $X ⊆ C(X)$ It may also satisfy some additional ...
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Lattice inside a finite dimensional vector space

I have an integral domain $R$ and its field of fractions $K$. Let $V$ be a finite dimensional $K$ vector space. Let $M$ be a finitely generated $R$-module contained in $V$. Why is $K\cdot M=V$ ...
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All tree orders are lattice orders?

Say that a set is tree ordered if the downset $\downarrow a =\{b:b\leq a\}$ is linearly ordered for each $a$. In a comment, Keinstein says that such sets are also semi-lattices, provided they are ...
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Number of join-irreducible elements of a lattice: is it monotonic?

Let $\mathcal L$ be a sub-lattice of $\mathcal P(X)$, where $X$ is a finite set. Denote by $\mathcal I(\mathcal L)$ the set of union-irriducible elements of $\mathcal L$ (i.e. $A\in \mathcal ...
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What is $|Aut(D_n,|)|$?

Let $n=p_1^{\alpha_1}\dots p_k^{\alpha_k}$ with the $p_i$ distinct primes and $ \alpha_i\in \Bbb N$. Just to check if I'm correct, is it true that $k!$ is the number of order-isomorphisms of the form ...
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Example of a bounded lattice that is NOT complete

I know that every complete lattice is bounded. Is there a simple example for a bounded lattice that is not complete? Thank you
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Is every sub-lattice of $\mathcal P(X)$ isomorphic to a sub-lattice of $\mathcal P(X')$ containing singleton sets?

Let $X$ be a finite set. $\mathcal{P}(X)$ denotes the set of all subsets of $X$. Let $\Gamma$ be a sub-lattice of $\mathcal{P}(X)$, i.e. $\Gamma$ is a collection of subsets of $X$ closed under union ...
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Proving isomorphism in lattices

The question is as follows: Let f be a monomorphism from a lattice $L$ to a lattice $M$.Show that $L$ is isomorphic to a sublattice of $M$. My attempt: Since $f$ is a monomorphism from a lattice ...
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extension of an increasing function over a lattice

Let $X$ be a finite set. $\mathcal{P}(X)$ denotes the set of all subsets of $X$. Let $\Gamma$ be a sub-lattice of $\mathcal{P}(X)$, i.e. $\Gamma$ is a collection of subsets of $X$ closed under union ...
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Lattice with $3$ operations.

If $R$ is a commutative ring and $\mathcal I(R)$ denotes its set of ideals then I know that $\mathcal I(R)$ can be looked at as a complete lattice with intersection $I\cap J$ and addition $I+J$ as ...
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Exercise 1.17 from Bell & Slomson's Models and Ultraproducts

I'm attempting to prove the following theorem left as an exercise from Bell & Slomson's Models & Ultraproducts (1969). I'd like to know whether my attempted proof is correct, and if not, I'd ...
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Examples of Stone algebras which are not Boolean algebras

Grätzer, in his Lattice Theory: Foundation, describes a Stone algebra as a distributive lattice with pseudocomplementation $L$ which satisfies the Stone identity: for every $a \in L$, $\neg a \vee ...
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When do DeMorgan's laws hold in a Heyting algebra

I'm working a bit with Heyting algebras (which are pseudocomplemented distributive lattives, right?) and I have a question about DeMorgan's laws. I know that, in general, it's not the case that $-(X ...
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Showing another result on distributive, complemented lattice

Currently reading Bell & Slomson's Models & Ultraproducts. Looking for clues, tips hints for showing the following: Let $L$ be a distributive, complemented lattice. Then for any $x$ $\in$ ...
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What is so amazing about having least upper bound (and dually, the greatest lower bound)?

Why are these properties so important, that they spawned lattice theory? Or, why were posets having these properties (lattices) identified as being of special interest compared to other posets?
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Theorem on complemented lattices

I'm looking through the book Models and Ultraproducts by Bell & Slomson, and I'm checking that my proof is correct (exercise 1.13). I'm pretty sure it is, but I thought I'd post it anyway. It is ...
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Operation table of Hasse diagram

Consider the following Hasse diagram: My book gives the following join and meet operation tables for this diagram: $$\begin{array}{|c || c | c|} \hline Subset & x \wedge y & x \vee y \\ ...
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Properties of the Majorization Order on $\mathbb{Z}^n$

I'm looking for background material on the majorization (aka dominance) order over $\mathbb{Z}^n$ (rather than over partitions). Let $v=(\psi_0,\dots,\psi_{n-1})$ and $u=(\phi_0,\dots,\phi_{n-1})$ ...
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Constructive fixed-point theorems where finite iteration yields the fixed point

I would like to show that $p$, a fixed point of some effective map $f : S\rightarrow S$, can be constructed effectively. Ideally, I would like there to exist a finite $n$ such that $p = f^n(p_0)$, ...
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Good introduction to number theory that develops and/or makes heavy use of commutative ring theory and lattice theory?

I'd like to learn some number theory, since it provides a lot of motivation for commutative ring theory and even some motivation for lattice theory (at least, that's the impression I'm under). ...
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Distributive lattices (interpretation of distributivity)

Simple counter example Ok, there is a very simple counter-example ^^ : This lattice isn't distributive, because $M=x\wedge(a\vee b)=x>0=(x\wedge a)\vee(x\wedge b)=m$, but for all $n<N$ the ...
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Lattice Definition

I see two Lattice definitions in Mathematics. Partial order set with each pair of elements have a least least upper bound and greatest lower bound. Integer linear combinations of vectors. Is ...
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Congruence lattice of N5

I calculated the the congruence lattice of $N_5$ using hit and trial and then verified it with Universal Algebra calculator. But I need to prove that it is the congruence lattice of $N_5$ How should I ...
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Isomorphism of intervals of a distributive lattice

Let $P$ be a distributive lattice and let for $a,b \in P$ by $a \land b$ and $a \lor b$ denote the usual notion of infimum and supremum. How can I show the following isomorphism? $$[a \land b , a ...
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Going down from filters to sets for specific relations

Let $\mathfrak{A}$ be a bounded lattice. I call a $2$-staroid a relation $f\in\mathscr{P}(\mathfrak{A}\times\mathfrak{A})$ such that $i\sqcup j \mathrel{f} b\Leftrightarrow i\mathrel{f} b\vee ...
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how to get a lattice morphism from a lattice join-morphism of complete lattices?

Let $L$ and $K$ be complete lattices, and $F:L\rightarrow K$ an isotone (ie order-preserving) poset map that preserves arbitrary joins. Assume that $L$ and $K$ have minimums, so we can define binary ...
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Distributive lattice and prime ideal

How to prove that a distributive lattice D with 0 and 1 is complemented if and only if every prime ideal of D is maximal?
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Is there a name for a 'weak' sublattice?

For a subset of a lattice to be a sublattice, we need that it is closed under the meet and the join operations. However there are other subsets which are not sublattices, but such that the poset on ...
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Semilattices are congruence-semi-distributive

A semilattice $(S,\cdot)$ is a commutative idempotent semigroup. A congruence on a semilattice is an equivalence relation that preserves multiplication, i.e. $x_1\mathrel{\theta} y_1$ and ...