1
vote
1answer
50 views

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 $I(L)$. ...
3
votes
2answers
32 views

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 ...
0
votes
0answers
18 views

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. ...
0
votes
0answers
31 views

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 ...
2
votes
1answer
33 views

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. ...
1
vote
1answer
21 views

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 ...
1
vote
1answer
54 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 ...
0
votes
1answer
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 ...
1
vote
3answers
52 views

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 $m\times n$ ...
4
votes
0answers
47 views

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 ...
1
vote
2answers
48 views

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 ...
1
vote
2answers
53 views

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. ...
5
votes
1answer
46 views

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, $x\vee y \leq x\vee z$ ...
0
votes
1answer
61 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 ...
0
votes
1answer
51 views

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 ...
0
votes
1answer
43 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 ...
1
vote
1answer
30 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 ...
3
votes
2answers
39 views

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 ...
4
votes
4answers
68 views

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 ...
1
vote
1answer
56 views

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 ...
0
votes
1answer
128 views

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 ...
1
vote
0answers
20 views

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 ...
1
vote
1answer
13 views

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 ...
0
votes
0answers
51 views

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 ...
1
vote
1answer
30 views

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 $x$ $\in$ ...
0
votes
0answers
22 views

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 ...
0
votes
0answers
27 views

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 \\ ...
2
votes
1answer
38 views

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 $u=(\phi_0,\dots,\phi_{n-1})$ ...
0
votes
1answer
48 views

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 $p = f^n(p_0)$, ...
3
votes
2answers
38 views

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 ...
0
votes
1answer
26 views

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 ...
1
vote
1answer
53 views

Factor congruences of non trivial Lattices

A pair of congruences $\theta$ and $\theta^*$ are called factor congruences if $\theta \vee \theta^*$ = full congruence. $\nabla$ $\theta \wedge \theta^*$ = trivial congruence. $\triangle$ I need ...
5
votes
2answers
130 views

If a finite poset has greatest and least elements, is it a lattice?

Let $P$ be a finite partially ordered set with elements $0$ and $1$ such that $0 \le x \le 1$ for any $x \in P$. Does it follow that $P$ is a lattice? If not, what is a counterexample? I believe this ...
0
votes
1answer
60 views

Definition of finite direct decomposition of elements and indecomposable elements at arbitrary lattice

How can i define finite direct decomposition of elements and indecomposable elements at arbitrary lattice . I think i can say an element of lattice is finite direct decomposition of elements if it be ...
5
votes
1answer
128 views

Two topologies over the same set which are carried to each other by a lattice isomorphism induce homeomorphic spaces?

This is a question that I've been turning in my head and haven't been able to come up with a way to proceed to either prove it or construct a counter-example: Let $X$ be a set and $\mathfrak{T}_X$ ...
0
votes
0answers
163 views

A function that is both log-supermodular and submodular?

Functions that are log-supermodular and submodular do they have some special name? Or any special properties? Any references? Is the function below log-super-modular and sub-modular? Consider ...
2
votes
0answers
41 views

Can the actual scope of “lattice theory” be summarized as “algebraic order theory”?

Lattice and order theory are often mentioned together. So I wonder how lattice theory is "intended" to differ from order theory. Lattice theory became important in the context of universal algebra. ...
2
votes
2answers
108 views

Collection of independent sets equal to downset?

I have been struggling with the following problem. Every set here is supposed to be finite. If we have a closure $\lambda$ on $X$, we define the collection of independent sets of $X$ as $$ I_\lambda ...
0
votes
0answers
32 views

Order Embeddings

Let (S,<=) be an (partially) ordered set and A a subset of S with the inherited order, namely, <= /\ AxA Thus, A is order embedded in S. When S isn't a lattice and A is a lattice would would ...
0
votes
1answer
26 views

Order isomorphic but not Boolean isomorphic

Is there an example of Boolean lattices $(X,\le)$ and $(Y,\le)$ and a function $$f:X\to Y$$ such that for all $x,z\in X$ $$x\le z ~~~~ \leftrightarrow ~~~~f(x)\le f(z) $$ so that for some $a\in X$, ...
2
votes
1answer
112 views

non-prime maximal ideal in a complemented lattice

Is there a complemented lattice, which has a non-prime maximal ideal?
0
votes
0answers
43 views

Prove after seeing $k$ comparisons, there is no total order with $k+1$

Let $X=\{x_1,x_2,\dots,x_m\}$ be a set of $m$ values. Let $Z=\{\succ_1,\succ_2,\dots,\succ_{m!}\}$ be the set of all possible strict total orders over $X$. A comparison $c_{(i,j)}$ is a test between ...
0
votes
0answers
40 views

Recovering Ordered Monoid Operation from the Order

I have a partially ordered set $(X, \preceq)$ with the following properties: $X$ has a minimum. I'll name it $1$. For every $x \in X$, the principal filter ${\uparrow} x$ is order isomorphic to $X$. ...
4
votes
2answers
39 views

disconnected ordered set

Is there a totaly ordered infinite set $A$ with the least element $a$ and the greatest element $b$ such that for any sequence $\{\alpha_n\}$ and $\{\beta_n\}$ in $A$ which satisfies ...
0
votes
1answer
99 views

Relation between concept lattice and Dedekind–MacNeille completion

While trying to unify the (mostly linearly) ordered attributes from Statistical classification with the binary attributes from formal concept analysis, I came across an interesting section in the ...
0
votes
1answer
33 views

Ordering tuples over (0,1)

To understand the natural ordering of $\{0,1\}^N$ I first thought about $\{0,1\}^3$ which has this Hasse diagram: There's something interesting going on in the second and third rows. In 3-D it ...
2
votes
1answer
83 views

Help to understand a proof (about filters)

Niels Diepeveen claims that he has proved the following theorem: Theorem Consider the cofinite filter $F$ on an infinite set. Let $K$ be a collection of ultrafilters such that $\bigcap K=F$. Then for ...
0
votes
2answers
72 views

Is a Lattice just a linearly ordered set?

The definition of Lattice I was introduced to recently was a poset for which every two elements had a supremum and infimum. Doesn't this just mean that the poset is a chain and tus a linearly ordered ...
3
votes
1answer
50 views

Does every orthocomplemented lattice satisfy the shuffle laws?

Let $L=(X,\wedge,\vee,^\bot)$ denote a non-empty lattice having a unary operation $x \mapsto x^\bot$ that satisfies both "shuffle" laws: $$x \wedge y \leq z \iff x \leq y^\bot \vee z.$$ $$x \leq y ...
1
vote
1answer
70 views

Adjoint of forgetful functor

I want to solve an exercise about adjoints (but I am not sure about the steps). Let $U:\textbf{CL}\rightarrow\textbf{Posets}$ be the forgetful functor. Here we have a functor from the complete ...