Questions tagged [lattice-orders]

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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Boolean lattice - Interval sublattice

I cannot conclude with the demonstration of the following exercise, I have already verified that the map is a homomorphism, but I do not know how to calculate $f(L)$. For a Boolean lattice $B$ and $...
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Lattice on $(M(X,\mathcal{A},\mathbb{R}),\leq)$

Show that $(M(X,\mathcal{A},\mathbb{R}),\leq)$ defines a lattice. I know that $M(X,\mathcal{A},\mathbb{R})$ with the relation $\leq$ defines a partial order. Now, a lattice consists of a partially ...
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Ordering on set of binary matrices with equivalence relation

To determine the size of a set of solutions to a combinatorial problem I would like to put a linear order on a set of square binary matrices, where two matrices are equivalent if they can be ...
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Generating Function for sublattices of $B_n$ that contain $\emptyset$ and $[n]$

This is Chapter $3$, Problem $46$(b) from Stanley's Enumerative Combinatorics. Let $f(n)$ be the number of sublattices of rank $n$ of the Boolean algebra $B_n$... Let $g(n)$ be the number of ...
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Is there a finite lattice such that every join-irreducible element is left-modular and that is not semimodular?

Is there a finite lattice such that every join-irreducible element is left-modular and that is not semimodular? I've been able to prove there are no such bounded atomistic lattices (not necessarily ...
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Mathematical analysis for economics

Even though the relation ≤ for vectors in R 2 is not rational because it is not complete, defining [(x1, x2) ∼ (y1, y2)] ⇔ [[(x1, x2) ≤ (y1, y2)] ∧ [(y1, y2) ≤ (x1, x2)]] gives an equivalence relation....
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Quotient group is of finite order? [closed]

Let $G$ be a locally compact abelian topological group. Let $L$ be a discrete subgroup of $G$ (induced topology on $L$ coincides with discrete topology on $L$). Let $\alpha:G\to G$ be an automorphism ...
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Where does the usual topology on $R^n$ fit into the lattice of topologies?

I am trying to understand how the usual topology on $R^n$ fits into the lattice of topologies. In particular, I am wondering what is the greatest lower bound and least upper bound. I have been asked ...
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Why is a Lattice called so?

Why is a Lattice called so? In my mind, I relate a Lattice to the picture of Lattice seen usually in chemistry.. like that of a salt, in 3-d space.. or just balls stacked in a plane on top of each ...
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Smallest lattice containing a poset

Given a poset $P$(we can assume $P$ is finite if necessary), how can we construct the smallest lattice containing $P$? (Does this exist?) To make the question precise, I am looking for a lattice $L$ ...
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Is $<D_n, />$ is lattice if lcm and gcd of all divisors of n belong to $D_n$

Is $<D_n, />$ is lattice if lcm and gcd of all divisors of n belong to $D_n$, where $D_n$ is set of all divisors of n PS - lcm = least common multiple and gcd = greatest common divisor Here ...
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Every bounded lattice may or may not be finite.

I want an example of a bounded lattice which is infinite with hasse diagram. As finite lattice is a lattice which surely contains greatest and least element then what is infinite lattice. Please ...
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Having multiple instances of the same element in a partial order?

I was wondering about how to represent a partial order which includes multiple instance of the same element. For example a partial order could include these two nodes: "(a ⋁ b) ⋀ a" and "(a ⋀ b) ⋁ b", ...
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Prime ideals of a lattice

Given the lattice generated by natural numbers (all except $0$) ordered by divisibility, how can I construct its prime ideals? How does a familiy of prime ideals without any given element $a$ look ...
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Non-distributive lattices with chains of different lengths

Consider the following Hasse diagramme. We will dub such lattices $\mathbf{Mkn}$. Assume a propositional language over $\{\wedge,\vee,\neg\}$. Let $v$ be a mapping from the set of all propositional ...
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Does anyone know an example of a bounded domain, not L-domain, in which every principal ideal is a join semilattice [closed]

I'm looking for an example of a bounded domain P in which every principal ideal is a Join semilattice but P isn't a L-domain. I have doubts If this kind of poset existe.
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Characterize the free lattice-ordered-group on one generator

Characterize the free $\ell$-group on one generator where an $\ell$-group is an algebra of the form (G,∧,∨,·,$^{−1}$,1), where (G,·,$^{−1}$,1) is a group, (G,∧,∨) is a (distributive) lattice and ...
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How do I find the GLB and LUB for f-i and f-c for the given lattice?

A labelled image of a lattice. I am unable to understand how the GLB for i-f will be 'a' (or 0). Since both the points are connected to b and d (which are 2 greatest LBs), in which case it won't even ...
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Will this strategy give me the whole subgroup lattice?

I have to do the following question Consider the semidihedral group $$QD_{16} = \langle\sigma, \tau|\sigma^8=\tau^2=1, \sigma \tau = \tau \sigma^3\rangle$$ This group has three subgroups: $\...
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Sublattice VS lattice under the ordering of its superset

Yet another question coming from the book Lattice Theory : Foundation, written by George Grätzer. In the third set of exercises, he proposes the following one : Find a subset $H$ of a lattice $L$ ...
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How to determine if given lattice is distributive or not?

I am trying to understand how to determine whether given lattice is distributive. I came across following: A lattice is distributive if and only if none of its sublattices is isomorphic to $M_3$ or ...
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How may one compute the proportion of isomorphisms among the total quantity of different combinations of element sequences and associative groupings?

If I have n elements in each possible sequence with each possible Tamari lattice grouping of those sequences with respect to a non-associative commutative operation, how may one compute the proportion ...
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Equivalent(?) Order-Theoretic Definition of a Lattice

Lattice Definition: A (partially) ordered set $(A,\preceq)$ is defined to be a lattice when every two-element subset of $A$ has both a meet and a join (i.e., a greatest lower bound and a least upper ...
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Ideal generated by a subset of a lattice

In George Grätzer book, Lattice Theory : Foundation, there is some fundamental result I am not sure to interpret correctly. The owner of the book will find it page 32. It is the second statement of ...
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What is the preferred convention for denoting Tamari lattice groupings?

Is there a common method or standard for denoting Tamari lattice/associative groupings with a character length less than that of the sum of the quantity of elements and parenthesis to be denoted? I ...
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Draw a tree on top of a semi-lattice

This question is rather vague, but I hope you might have some input. I'd like to create the following structure (draw it): Imagine a semi-lattice drawn on a plane in 3d space. Now I would like to ...
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Are all lattices chains?

Are all lattices chains? I think that is true because a chain is a poset where we can compare any two elements. A lattice is a poset where every subset has a lub and a gld. So, by reducing the size of ...
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Finite lattice dimension and planarity

I have been struggling for several days on this question, now it is time for you folks to enlighten me :) In the book Lattice Theory: Foundations from George Gratzer (pdf file), it is stated, page 9, ...
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Number of join-irreducibles and rank in a finite lattice

Let $(L, r)$ be a finite, ranked lattice. Is $$r(x) = \#\{j \leq x : j \in L \text{ is join-irreducible}\}$$ for all $x \in L$? This may be a naive question, though my lattice theory is weak.
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Whitman's condition for lattice polynomials

Recall that Whitman's condition for a lattice $L$ (with join $\sqcup$ and meet $\sqcap$) says that given $a, b, c, d \in L$, it holds that $a \sqcap b \sqsubseteq c \sqcup d$ iff at least one of the ...
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How can I find the Bipartite Order associated to a Lattice?

I am interested in representing a lattice as a tree. The only paper I have found about that is Bit-vector encoding for partially ordered sets. The author does not explain thoroughly what I want. I ...
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Ordered Set (Steven Roman)

How do I show 1. and 2.? Let $P$ be a poset. Prove that the down map is an order embedding from $P$ into $\wp (P)$. Let $I_n = \{ 1,...,n \}$ and let $f: \wp (I_n) \rightarrow \{ 0,1 \} ^n$ be ...
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Ordered Sets (Bernd Schröder)

Considered the power set $\mathcal{P}(\{ 1,...,n \})$ with the following relation. The set $A$ is said to be dominated by the set $B$ iff there is a $k$ such that $|\{ 1,...,k \} \cap A| < |\{ 1,......
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Ordered set - Down-sets and Up-sets

Show that $Q$ is a down-set in $P$ iff its complement $P \setminus D$ is an up-set in $P$. The down-set property is transitive, that is, $D \in \mathcal{O}(E)$ and $E \in \mathcal{O}(F)$ $\Rightarrow$ ...
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Ordered Set - Chains

Let $P$ be a poset. Let $\mathcal{C}_{a,b}$ be a chain from $a$ to $b$ and let $\mathcal{D}_{b,c}$ be a chain from $b$ to $c$. (a) Show that $\mathcal{C}_{a,b} \cup \mathcal{D}_{b,c}$ is a chain from ...
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Ordered Sets - Down-sets

Let $A$ and $B$ be decreasing sets of $P$. Prove that $A \prec B$ in $\langle {\mathcal{O} (P); \subseteq} \rangle$ iff $B = A \cup \{b \}$ for some minimal element $b$ of $P \setminus A$. Let $P$ ...
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Not understanding this definition in Lattice Theory: First Concepts and Distributive Lattices by Gratzer

In the third section "Some Algebraic Concepts" he gives a definition for sublattices. Verbatim: let $$ A_\lambda, \lambda \in \Lambda $$ be sublattices of L. Then $$ \bigcap (A_\lambda | \lambda \in ...
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Closure operators

Can a closure operator be not isotone? The definition below is a pretty standard definition of closure operator: $A \subseteq I(A)$ ($I$ is extensive) $A \subseteq B \implies I(A)\subseteq I(B)$ $I(I(...
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How can we identify Ultrafilters that “look” or “were” principal?

Let $X$ be an infinite set, and consider the lattice $\left(\mathscr P(X), \cap, \cup, \emptyset, X\right)$. The following is well known: Lemma Every Ultrafilter is either Principal or contains ...
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LUB lattice question

Consider the set of integrity levels L = {low, admin, kernel}, where kernel > admin > low. Furthermore consider the set of categories Cat containing Cat = {HRandAdmin(H),MarketTraders(T),IT Engineers(...
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Prove that convex real functions on [0,1] form a meet-semilattice

Let C be the set of all continuous strictly convex real valued functions on the interval [0,1]. For f,g $\in$ C, define f $\leq$ g IFF f(x) $\leq$ g(x) for all x $\in$ [0,1]. Prove this is a meet-...
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Number of restricted partitions dominated by a certain one

Once established a certain partition of n, I am interested in finding out which is the exact number of partitions with same amount of parts $l(\lambda)$ but that are dominated by the first one. for a ...
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$x\wedge a \wedge b=x \Rightarrow x\wedge b=x$ in a lattice viewed as an algebra

A lattice is an algebraic structure $(L,\wedge,\vee)$ such that, $\wedge$ and $\vee$ are commutative, associative and abosrbing binary operations, i.e. $$a \wedge (b\vee a)=a,\quad a\vee(a\wedge b)=a....
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A particular filter on a Heyting algebra

Let $(H, \le)$ be a Heyting algebra and $x \in H$. Consider the subset: $$F_x = \{ y \in H \mid ((x \to y) \to x) \le x \}$$ It is easy to prove that $F_x$ is a filter. Moreover, $F_x$ is proper if ...
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Upper and lower bound in distributive lattice

Let $a\in L$, where $L$ is a graded (if needed) distributive lattice. Let $x_1, \ldots, x_k$ - the set of elements which cover $a$ ($x$ covers $a$ if $a < x$ and there is no element $t$ such that $...
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Does the order induced by a self-dual cone produce a Riesz space?

Let $X$ be a Hilbert space, $K$ a closed, convex, and self-dual cone. The latter property means $$ K = \{ y\in X : \ \langle x,y\rangle \ge0 \ \forall x\in K\}. $$ Then $K$ induces an order on $X$ by $...
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W-split coequalizers

The following snippet is from Adamek, Rosicky:Algebra and local presentability,how algebraic are. It is unclear to me the end of Example 5.1: Since $e$ is the coequalizer of $\bar{u}_1,\bar{u}_2$ in ...
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How to demonstrate the finite height of a lattice?

I would like to ask you for help with a formal demonstration concerning the finite height of a lattice. My lattice is defined like this: is a lattice of vectors, each with exactly $n$ cells. In each ...
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Constructing rings with a specific lattice of ideals.

Let $R$ be a commutative ring with 1. The ideals of $R$ form a lattice with inclusion as order relation. Let me call it the ideal lattice $L(R)$ of $R$. Given an arbitrary lattice $L$, there are some ...
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Prove that Multiset and relation $\preccurlyeq$ is a lattice ($\preccurlyeq$ is defined like $\leq$)

Multiset is a set that can have more than one of each member for example $\{1,3,3,9\}$ is a Multiset. Let $\mathbb{K}$ be the set of all multisets that has exactly $k$ members. ($k$ is a fixed ...