All Questions
Tagged with lattice-orders elementary-set-theory
68
questions
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86
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Show that the set of all partitions of a set S with the relation refinement is a lattice.
This one may be one duplicate of QA_1, but its example $\{\{a,d\},\{b,c\}\}\wedge\{\{a\},\{b,c,d\}\}$ seems to not meet the definition in the book because $(\{a,d\} \not\subseteq \{a\}) \wedge (\{a,d\}...
- 416
2
votes
1
answer
65
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Dedekind MacNeille completion. Definition of supremum as $\left(\bigcup M\right)^{ul}$. Counterexample for definition as $\bigcup M$?
I didn't found a mathematical text for Dedekind MacNeille completion
, so I "defined" supremum in the completion the following way:
$$\sup M := \left(\bigcup M\right)^{ul},$$ where M is a ...
- 842
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1
answer
59
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Algebraic closure for partial functions
Here's a definition (taken from here, p.9):
The $\leq$ relation is a partial order relation that stands for 'part of'. E.g. $x\leq x\oplus y$.
In another source, the same author illustrates this ...
user1048887
1
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1
answer
73
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Symmetric relations form a CABA
Fix a set X and consider the collection of all symetric relations on it. I also assume that the empty relation is by definiyion symmetric. Well, it is true that the above collection forms a complete ...
- 5,192
2
votes
1
answer
222
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L is a complete lattice, prove that there exists $x \in L$, which is a fixed point of $f$ and is the smallest $y$ that $f(y) \le y$.
I have such task:
Let $(L, \le)$ be a complete lattice and $f: L \to L$ a monotonic function. Proof that there exists $x \in L$ which is a fixed point of $f$ and is the smallest $y$ that $f(y) \le y$.
...
- 165
1
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1
answer
75
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Isomorphism of a closure system and a topped complete lattice
It is in general true that if a closure system $\mathcal{F}$ on a given ground set $\Omega$ is order isomorphic to a complete lattice $\mathcal{G}$ on $P(\Omega)$ having $P(\Omega)$ as its top element,...
- 5,192
2
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1
answer
183
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Join of two preorders and of two equivalence relations
I'm sorry for the silly doubt. What is the join of two preorders? And of two equivalence relations? The meet is given by intersections. But in general the union of two preorders (resp. equivalence ...
- 5,192
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1
answer
328
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Order relation on the union of two ordered sets.
The following exercise comes from the book Introduction to lattice and order, Second edition. David and Priestley.
Let $P$ and $Q$ two ordered sets, such that $P\cap Q\ne \emptyset$. Give formal ...
- 1,278
3
votes
2
answers
120
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When does downward closure commute with supremum?
Let $A$ be a suplattices, and suppose we have a family $\{a_i\}_{i\in I}\subseteq A.$
Is $\bigcup_{i\in I}(\operatorname{\downarrow}a_i) = \operatorname{\downarrow} \sup_{i\in I}(a_i)$ in general?
...
- 18.4k
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2
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131
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What does being smaller that to a join means in a distributive lattice?
This is a follow up question to my previous question with more restrictions. It was answered negatively for arbitrary lattices, but mentioned that the result holds "only" in distributive ...
- 18.4k
2
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1
answer
61
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What does being smaller that to a join means in a lattice?
This sounds like a very naïve question, but I couldn't find a correct argument to prove/disprove it rigorously.
Suppose we have a a subset $A$ of a lattice (or any join-semilattice), and some $x\le\...
- 18.4k
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2
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157
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About two definitions of a complete lattice. ("Introduction to Set Theory and Topology" by Kazuo Matsuzaka)
I am reading "Introduction to Set Theory and Toplogy" by Kazuo Matsuzaka (in Japanese).
In this book, the definition of a complete lattice is the following:
Let $M$ be a partially ordered ...
- 8,740
2
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2
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136
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Proving both defining properties of a distibutive lattice are equivalent
In class, we've defined distributive lattice to be a lattice L which verifies the following properties:
$$(1) \space \space a \wedge (b \vee c) = (a \wedge b) \vee ( a \wedge c)$$
$$(2) \space \space ...
- 77
3
votes
1
answer
310
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Is it possible to intersect two lattices and that their intersection is not a lattice?
Draw the line diagram of two lattices whose intersection is not a lattice. I have tried to do it with division, with containment relation, and yet I still cannot solve this exercise, I appreciate any ...
- 87
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2
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What does a "well ordering on $\Bbb R$" mean? [duplicate]
A well order is a poset such that every non empty subset of this set has a least element. Does "well ordering on $\Bbb R$" mean that every non empty subset of $\Bbb R$ (which is a poset) has ...
user782485