Tagged Questions
2
votes
2answers
36 views
Name of the order with irreflexivity, antisymmetry and transitivity?
I have an order otherwise poset aka partial order but it is irreflexive so relationships such as 1R1 and 2R2 are impossible. What is the name of this order?
1
vote
1answer
67 views
A generalization of Galois connections
Let $f(x)\ast y \Leftrightarrow x\ast g(y)$ for some binary relation $\ast$ and functions $f$ and $g$. This is a generalization of Galois connections.
Are things like this studied before?
Note that ...
2
votes
1answer
45 views
“Lexicographic order” without priority, but with ties, how to define / what's the name?
I'm searching for an ordering principle which combines two (or more) preorders, akin to the lexicographic order, but not quite: I'm looking for an ordering principle that does not priorities, but ...
0
votes
0answers
53 views
Is lexicographic order complete?
I was wondering, if the lexicographic order in $\mathbb{R}^2$ is complete or not? I guess it isn't but I dont find any counterexample.
Complete means: For every partition of $M$ into two disjoint ...
2
votes
1answer
58 views
Can a Partial Order be symmetric in addition to its properties?
For example,a relation,R defined on integers.
$R = \{(a,b)\mid a^3=b^3\}$
Could this be partial order?
2
votes
1answer
25 views
Proof for a creating a partition of a countable set using chains in partial orders.
Definition: A partition of a set $A$ is a set of nonempty subsets of $A$ called the
blocks of the partition, such that
every element of $A$ is in some block, and
if $B$ and $B'$ are different ...
0
votes
1answer
50 views
The proper subset relation and strict partial order
The proper subset relation, $\subset$, defines a strict partial order on the subsets of $[1,6]$, that is $pow[1,6]$.
(a) What is the size of a maximal chain in this partial order? Describe one.
(b) ...
0
votes
2answers
35 views
Why am I getting that minimal elements are equivalent to the minimum?
I have these two definitions, about minimals and minimums in an order relation:
$b$ is minimal in $B<=>¬\exists:(x \in B \land x \prec b)$ and also equivalent to $ \forall x,[(x \in B \land x ...
1
vote
1answer
70 views
Total Ordering and relation
Consider the relation $<$ on $\mathbb{Q}$ defined by: $(m, n) < (j, k) \iff jn-mk \in \mathbb{N}.$
Where $m, j \in \mathbb{N}$ and $n, k\in\mathbb{Z}$
I want to show that $<$ is a total ...
3
votes
2answers
78 views
If $\sim$ is an equivalence relation on $X$, and there is a strict total order on $X/\sim$, what kind of ordering does $X$ have?
I would like to know if there's a special name for this kind of ordering. When I say there is a strict total order on $X/\sim$, what I mean is that two distinct elements in the same equivalence ...
2
votes
3answers
185 views
Why is 'Antisymmetry' named so?
So when we talk about order relations for the familiar number systems, we are always introduced to the antisymmetry property which is $x \le y, x \ge y \implies x=y$.
When I think of the word ...
0
votes
0answers
83 views
where are all Hasse diagram tables of n elements posets and lattices?
n=1,2,3 there are only 1 isomorphism lattice
n=4 there are all 2 kinds of isomorphism lattice
how about n=5,6,7....
I need the Hasse diagram below n=50 ,help me!
2
votes
1answer
108 views
Pointwise order of the Cartesian product of two preordered chains
Definitions: (From Categories for Types by Roy L. Crole.)
A preorder on a set $X$ is a binary relation $\leq$ on $X$ which is reflexive and transitive.
A preordered set $(X, \leq)$ is a set equipped ...
0
votes
1answer
97 views
$<$ on a preorder is a strict partial order
Definition: Suppose $X$ is a preorder. Define $x < y$ as $x \le y$ and $y \not\le x$ for each $x, y \in X$.
Question: Show that this gives a strict partial order on $X$.
1
vote
1answer
131 views
Uniqueness of meets and joins in posets
Exercise 1.2.8 (Part 2), p.8, from Categories for Types by Roy L. Crole.
Definition: Let $X$ be a preordered set and $A \subseteq X$. A join of $A$, if such exists, is a least element in the set of ...
2
votes
1answer
131 views
Preorders, chains, cartesian products, and lexicographical order
Definitions:
A preorder on a set $X$ is a binary relation $\leq$ on $X$ which is reflexive and transitive.
A preordered set $(X, \leq)$ is a set equipped with a preorder.... Where confusion cannot ...
4
votes
2answers
111 views
$<$ in a preorder
The author of the book I am studying defines $<$ for a poset as
If $x, y \in X$, where $X$ is a poset, then we shall write $x < y$ to mean that $x \le y$ and $x \ne y$.
From this, I can ...
4
votes
1answer
174 views
Lexicographical order - posets vs preorders
I found the following definition for lexicographical ordering on Wikipedia (and similar definitions in other places):
Given two partially ordered sets $A$ and $B$, the lexicographical order on the ...
2
votes
0answers
139 views
Examples of proofs by induction with respect to relations that are not strict total orders.
I have read this Wikipedia article and found it fascinating. I came across it after I tried to prove a certain statement with a method resembling induction in the set of natural numbers but ordered by ...
2
votes
1answer
161 views
When is the composition of partial orders a partial order?
I have been doing some thinking about the compostition of relations and I've had trouble remembering a number of simple facts about what happens when we compose specific types of relations. I've had ...
1
vote
0answers
59 views
About function inj, surj and something else. Is this exercise resolved correctly?
This is my problem:
For every couple of integers
$(a,b)\in\mathbb{Z}\times(\mathbb{Z}\backslash\{0\})$ we denote with
$r(a,b)$ the remainder of the division between $a$ and $b$. Consider
the ...
1
vote
2answers
52 views
Minimum size of a subset to know a complete total order
Lets say we have a set $A$. Suppose that $A$ is ordered by $<$, $A$ is completely ordered.
$<$ can be defined as $<:=\{(a,b) \in A\times A : a<b \}$
Given that $<$ is transitive, it ...
2
votes
0answers
66 views
What is a binary relation like whose reflexive transitive closure is a partial order?
Let $R$ be a reflexive binary relation and $R^*$ be its reflexive transitive closure. The question is what is the equivalent condition in terms of $R$ to $R^*$ being a partial order.
Intuitively, a ...
2
votes
1answer
97 views
Questions about power sets and their ordering
Okay, so I'm stuck on a question and I'm not sure how to solve it, so here it is:
In the following questions, $B_n = \mathcal{P}(\{1, ... , n\})$ is ordered by containment, the set $\{0,1\}$ is ...
