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.
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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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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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Are all complete lattices a pointed complete partial order, and vice versa?
A friend of mine asked for my help in drawing a venn diagram that includes the notions of partial orders (PO) in general, complete partial orders (CPO), pointed complete partial orders (CPPO), total ...
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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 ...
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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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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) ...
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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 ...
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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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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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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 ...
