Questions tagged [order-theory]
Order theory deals with properties of orders, usually partial orders or quasi orders but not only those. Questions about properties of orders, general or particular, may fit into this category, as well as questions about properties of subsets and elements of an ordered set. Order theory is not about the order of a group nor the order of an element of a group or other algebraic structures.
4,283
questions
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vote
1
answer
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Partial order on power set & set of partial orders
Consider a set $X$ and a partial order $\preceq$ on the power set $2^X$ of $X$. We assume that $\preceq$ extends the usual subset relation $\subseteq$, i.e. whenever $A\subseteq B\subseteq X$ then $A\...
- 438
3
votes
0
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Show the following forcing poset is $\sigma$-centered
In Kunen, there's the following exercise:
Assume MA(κ)
and (X,<)
be a total order with |X|≤κ
, then there are $a_x$⊂$ω$
such that if x<y
then $a_x$⊂∗$a_y$
. (x⊂∗y
if |x−y|<ω
and |y−x|=ω
.)
...
- 31
2
votes
1
answer
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Convexity structures and partial orders
Can any convexity structure be defined by a partial order $\preceq$ in the sense of the order topology: a given set $A$ is convex if for any $a,b \in A$ and any other element $c$ for which $a\preceq c ...
- 438
0
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0
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42
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Is $\mbox{row}_i(A) \cup \mbox{col}_j(A)$ for a matrix $A$ a thing?
In working on a research problem in order theory, I have encountered a symmetric rank-1 matrix that can be expressed as
$$
A = 30 \begin{pmatrix}
1 \\\\ 1/5 \\\\ 1/2 \\\\ 1/6 \\\\ 1/15 \\\\ 1/30
\...
- 2,170
3
votes
1
answer
17
views
Permutations maximally matching given pairwise order relations
Given $n \in \mathbb{N}$ and a sequence of $T$ pairwise orders $(i_t, j_t)$'s for $1 \leq t \leq T$.
Q: Are there any existing algorithms to find permutations of $[n]$ ($\sigma \in S_n$) such that as ...
- 2,150
1
vote
3
answers
120
views
Symbolic notation for "$A_1\subseteq A_2\subseteq\cdots$"
Background
Definition: A ring $R$ is said to satisfy the ascending chain condition (ACC) for left (right) ideals if for each sequence of left (right) ideals $A_1,A_2,\ldots$ of $R$ with $A_1\subseteq ...
- 3,683
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0
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35
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Why Is the Following Proof of a Finite Nonempty Totally Ordered Set Containing Its Maximum Wrong?
I wish to prove the result suggested in the title without induction on the cardinality of set. Here is my approach:
Let $S$ be a finite nonempty totally ordered set, i.e. $S=\lbrace x_{1},x_{2},\ldots,...
- 1
0
votes
1
answer
50
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Congruences on the pentagon lattice $\mathcal{N}_5$
Let $\mathcal{N}_5$ refer to the Pentagon lattice, or the lattice generated by the set $\{0, a, b, c, 1\}$ subject to $1 > a$, $1 > c$, $a > b$, $b > 0$ and $c > 0$.
My aim is to find ...
- 497
1
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0
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List of all posets of size $n$ for small $n$? [duplicate]
Is there a good reference for, or an easy way of generating, all Hasse diagrams of partially ordered sets of small size (say $n\leq 6$)? I am familiar with the OEIS entry A000112 listing the number of ...
- 1,307
2
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0
answers
50
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Converting "improper" partial order to total order
I suspect that if I knew what to search for, this would be easy to find an answer to, but I don't know what the proper name is for the input portion of the problem statement.
I have a set and a ...
- 663
0
votes
0
answers
20
views
Optimization of totally ordered set valued function.
I am familiar with the meaning of optimizing a function $f : \Omega \to \mathbb{R+}$. However I was just wondering if there's some theory of math explaining how to optimize mapping from $f : \Omega \...
- 5,317
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0
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Necessary and sufficient conditions for finding graphs based on posets
Let $\Gamma$ be any graph (say finite, simple, undirected), then denote by $P(\Gamma)$ the set of all non-isomorphic subgraphs of $\Gamma$. Let $\gamma$ be another graph, then denote $\gamma \subseteq ...
- 967
1
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1
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Lattice with supermodular height function is lower semimodular
Question
Let $(L,\leq)$ be a lattice of finite length and let its height function $h$ be supermodular, meaning that
$$h(x \wedge y) + h(x \vee y) \geq h(x) + h(y) \quad \forall x,y\in L.$$
Does it ...
- 23.1k
0
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1
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Understanding ordered fields and the subset $P \subseteq \mathbb{F}$ of positive elements.
I'm following Real Analysis: A Long-Form Textbook (Jay Cummings) and there is a part about defining the positive set $P \subseteq \mathbb{F}$.
The following definition is given:
An ordered field is a ...
1
vote
0
answers
63
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Is it possible to order proper classes?
Let's assume that we have NBG/MK, with its global choice.
Assume a relation F, a family of classes, is given. (a class-function, such that $F(x)=\bigcup\{s|(x,s)\in F\}$ is considered to be "in&...
- 842