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
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Galois connections give rise to complete lattices
I am reading Introduction to Lattices and Order, second edition, by Davey and Priestly. On page 161, it says
Every Galois connection $(^\rhd,^\lhd)$ gives rise to a pair of closure operators, $^{\rhd\...
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The supermodularity of probability of intersection [closed]
Given a finite sample space $E$, let $E=\{A_1,A_2,\dots,A_n\}$ be a collection of random events.Then, is $f(S)=\mathbb{P}\{\cap_{A_i\in S}A_i\}$ a supermodular function for $S\subseteq E$?
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How to phrase the proof of $m \lt n$ if and only if $m \le n-1$
I have been reading Knuth's "The Art of Computer Programming" and in the mathematical preliminaries chapter of volume 1 there is on page 476 the answer to an exercise where he states
... ...
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If sub-universe $S$ of lattice has congruence $\theta$, does the lattice have a congruence $\lambda = \theta \cap S^2$? [duplicate]
Let $(L, \lor , \land )$ be a lattice and $S$ a sub-universe of the lattice. A sub-universe of a lattice will be any subset of the lattice set that is non-empty and closed under $\land$ and $\lor$. ...
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Understanding the definition of congruences over a lattice
Let $(L, \land, \lor)$ a lattice and $\theta$ a binary relation over $L$. We say $\theta$ is a congruence iff
$$
x_0\theta x_1, y_0 \theta y_1 \Rightarrow (x_0 \lor y_0)\theta(x_1 \lor y_1)
$$
(and ...
- 3,468
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Motivation of inventing concept of well-ordered set?
I've started studying set theory for a while. I understand what is an ordered sets but i still fail to see what motivated mathematicians to invent these concept.
Could you please enlightment me ?
...
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Number of lattices over a finite set
I'm interested in finding a general formula for the number of lattices possible over a finite set $S$ as a function of the set's cardinality.
For instance, how many lattices over $\{1, 2, 3\}$ are ...
- 3,468
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Do the monotone maps from a poset into a Heyting algebra form a Heyting algebra?
I am interested in generalizing the fact that the up-sets of a poset always form a Heyting algebra.
Let $P$ be a poset and $H$ a Heyting algebra. $\operatorname{Hom}(P,H)$ can be made a bound lattice ...
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Every countably infinite linear order has a copy of $\omega$ or $\omega^{op}$
Every countably infinite linear order $L$ has a copy of $\omega$ or $\omega^{op}$. I'm interested in different kinds of proofs of this fact.
One I came up with is: pick $x_0 \in L$. Wlog $[x_0, +\...
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Is the grading of a poset unique?
A graded poset is a poset $(P,\leq)$ with a map $\rho:P\rightarrow\mathbb{N}$ where $\rho$ is strictly monotone, and if $x,y\in P$ where $y$ covers $x$, (i.e. $x\lessdot y$), then $\rho(y)=\rho(x)+1$. ...
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Problem with dense set
On ' Set theory with an introduction to real point sets'(Dasgupta, Abhijit ,2014) i found this exercise:
This is interesting because compare the topological (left,1) and order (right,2) definition of ...
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Extending the $M_3,N_5$ theorem from distributive lattices to frames
It is known that a lattice $L$ is distributive if and only if it does not contain the diamond $M_3$ or the pentagon $N_5$ as sublattices.
A complete lattice is one in which every subset has an infimum ...
- 53
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Would the category of directed sets be better behaved with the empty set included, or excluded?
In a topology book of mine, a directed set is defined as a nonempty set $D$ equipped with a relation $R$ that is transitive, reflexive, and for all elements $x$ and $y$ of $D$, there exists a $z$ in $...
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terminology related to inducing a total order
Imagine I have 10 students in an elementary school. I believe it is proper to say the following: The students' age induces a total order on the students (assuming no 2 students have the exact same age)...
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Proof of variant Sperner's Theorem for divisibility posets
I'm trying to determine the size of the maximal antichain in the poset of divisors of $N$ where the partial order is divisibility. Looking at the prime factorization of $N=p_1^{e_1}\cdots p_d^{e_d}$ ...
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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\...
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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
.)
...
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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 ...
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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
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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 ...
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3
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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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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,...
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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 ...
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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
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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 ...
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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 \...
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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 ...
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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 ...
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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 ...
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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&...
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