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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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 ...
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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 ...
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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 ...
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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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