All Questions
Tagged with order-theory relations
351
questions
0
votes
1
answer
30
views
Reconciling Continuity of Binary Relations with Continuity of Functions/Correspondences
I asked this question in the Economics StackExchange as well, but figured it may be better-suited here.
There are various ways to express the concept of continuity of a binary relation, but one I've ...
- 91
0
votes
0
answers
34
views
Definition for orders corresponding to directed acyclic graphs (DAG)
My question
What is the name for a binary relation $R$ on $V$ that corresponds to a graph $G = (V,E)$ that is a directed acyclic (simple) graph?
Background
There is a bijection between simple directed ...
- 414
0
votes
2
answers
35
views
Relations Symmetry and Transitivity
Given the following Relations over the set $M := \{α, β, γ\}$
$R1 := \{(α, α), (α, β), (β, α), (β, β), (γ, γ)\}$
How is $R1$ transitive?
The condition for transitivity is
$(a,y)\in R1 \text{ and }(...
2
votes
0
answers
35
views
Number of partial orders such that a given function is monotone (order homomorphism)?
Given a set $X$, a poset $(Y, \preceq)$, and an arbitrary function $f: X \to Y$, how many partial orders $\le$ can one construct on $X$ such that $f$ becomes an order homomorphism $(X, \le) \to (Y, \...
- 1,458
0
votes
0
answers
68
views
How to prove "A finite saturated chain is maximal if and only if it contains both a minimal and a maximal element of the poset"
The wikipedia says:
Maximal chain. A chain in a poset to which no element can be added without losing the property of being totally ordered. This is stronger than being a saturated chain, as it also ...
- 416
0
votes
0
answers
69
views
One elegant proof of the transitivity part in "Show that lexicographic order is a partial ordering on the set of strings from a poset."
Recently, when I self-learnt Discrete Mathematics and Its Applications 8th by Kenneth Rosen, I did only the even-numbered exercises which the author offers the detailed description instead of the odd ...
- 416
0
votes
3
answers
30
views
Proving if a defined relation satisfies the properties of an order
I am trying to solve the below problem from an MIT OCW course on real analysis.
Suppose that $S$ is a set and $\preceq$ is a relation on $S$ with the following properties:
For all $x \in S$, $x \...
- 189
0
votes
1
answer
135
views
How is the order "less than" on Natural number a total order?
First of all, I'm confused between the terms Order and Relation; I know what a relation on a set means, but don't know exactly a what order is. I may have used these terms as if they almost mean the ...
- 181
3
votes
0
answers
57
views
general notion for 'smallest' binary relation
Let $R\subseteq X\times X$ denote a relation on the set $X$.
There are various names for properties that $R$ can satisfy: $R$ is
reflexive if $\forall x.xRx$,
symmetric if $\forall x,y.(xRy\implies ...
- 807
0
votes
0
answers
38
views
Is the defintion given of a Hasse diagram correct?
I am currently reading Proofs: A Long Form Mathematics Textbook written by Jay Cummings, and am on the section about Partial Orders. When discussing how to visualize a POSET he defines a Hasse diagram ...
- 195
1
vote
1
answer
58
views
Counting possible number of relations that have certain properties
I was wondering whether it would be possible to count number of binary relations possible from a set of $n$ elements (I am mainly interested in the cases where $n=3,4,5,6,7$ but a general formula if ...
- 89
3
votes
1
answer
461
views
Difference between partially ordered, totally ordered, and well ordered sets.
I just started studying set theory and I'm a bit confused with some of these relation properties.
Given a set A = {8,4,2}, and a relation of order R such that aRb means "a is a multiple of b"...
- 41
-1
votes
1
answer
24
views
Prove that if A is a finite non-empty partial order set and it has a unique maximal element, then it is the greatest element (proof by contradiction) [closed]
I was given this question as an extra question on the lecture notes. I had been thinking about this question for the past 2 weeks but to no avail. Before making this post, I have scoured through ...
- 1
0
votes
0
answers
23
views
Showing that a union of subset partition spans a whole set
Again, a question based on "An Invitation to Applied Category Theory". This time, exercise 1.15 on page 10.
Here we have an equivalence relation $\sim $.
Under $ \sim $ we form a partition ...
- 418
2
votes
0
answers
40
views
The set of predecessors of a node in a tree is finite?
I came across the following definitions of tree and transitive tree, respectively, in the book Modal Logic by P. Blackburn:
Definition 1. A tree is a relational structure $(T, R)$, where:
$T$, the ...
