3
votes
1answer
42 views

Undistinguishable elements in posets

Given a finite partially ordered set $P = (V, <)$, I say that $x$ and $y$ in $V$ are indistinguishable in $P$ if for all $z \in V \backslash \{x, y\}$, I have $z < x$ iff $z < y$, and $x < ...
1
vote
1answer
38 views

compactness and maximal elements

Let $C$ be a nonempty compact subset of $R^n$, with a certain partial order defined on it. I am trying to prove that $C$ contains a maximal element. My idea is: start with a certain element of $C$. ...
0
votes
0answers
68 views

Have you seen this property of tolerance relations before?

Let $A$ be a set equipped with a binary reflexive and symmetric relation $\uparrow$ (such relations are often called tolerances, see also "Are there real-life relations which are symmetric and ...
1
vote
2answers
63 views

Introductions to posets on algerbaic structures (Everything I need to know about them)

I need a good and complete introduction to Tree-like orders and partial orders on algebraic structures with one operations. I accept basic texts too. I'm looking for free online texts mostly because ...
2
votes
1answer
29 views

Properties of the Majorization Order on $\mathbb{Z}^n$

I'm looking for background material on the majorization (aka dominance) order over $\mathbb{Z}^n$ (rather than over partitions). Let $v=(\psi_0,\dots,\psi_{n-1})$ and $u=(\phi_0,\dots,\phi_{n-1})$ ...
3
votes
1answer
74 views

What are some examples of “homogeneous” linear orders, other than $\mathbb{R}$ and $\mathbb{Q}$?

Let $L$ denote a linear order that is unbounded. Then it may or may not satisfy: Globally homogeneous. For all $x,y \in L \cup \{-\infty,\infty\}$, if $x < y$ then the interval $(x,y)$ is ...
1
vote
0answers
17 views

Recommendation for Order Theory texts

Order Theory, Lattice Theory, or any directly relatable subject is not taught at my university, and quick searches don't give much clue into good textbooks for the subjects. My question is: what are ...
0
votes
0answers
124 views

A function that is both log-supermodular and submodular?

Functions that are log-supermodular and submodular do they have some special name? Or any special properties? Any references? Is the function below log-super-modular and sub-modular? Consider ...
4
votes
0answers
93 views

adjoints and duality

Let $X$ and $Y$ be posets seen as a category. Let $F:X \to Y$ and $G: Y \to X$ be an adjunction. That is, $(G \circ F) (x) \ge_X x$ and $(F \circ G) (y) \le_Y y$. Now, $F$ is additive and $G$ is ...
1
vote
1answer
106 views

Which mathematician has argued that we should move from “set” to “order”?

I recall reading (on this very site, in fact) that there is a mathematician whom argues that we ought to switch from "set" to "order" in the foundations, so as to "recover duality" or some such. Does ...
2
votes
0answers
70 views

Do subhomomorphisms / subfunctors have a standard name, and where can I learn more?

Sorry for all the mistakes in the original! I think they're mostly fixed now. Thank you for your patience. Part 1. If $A$ and $B$ are models in $\mathrm{Pos}$ of an algebraic signature $\sigma$, ...
6
votes
1answer
129 views

Reference Request on Order Theory topics

I am looking for some references (especially a good recent book) that covers important topics involving partial orders such as: order polytopes, sorting/selection in partially ordered sets, upper and ...
3
votes
0answers
129 views

Power-set in Hypercube: historical background of indexing each term like Hasse Diagram?

My instructor wants references about the indexation over the hypercube, related question here, he does not believe that I was the first who used it -- [update] thanks to a comment, the name is Hasse ...
1
vote
1answer
44 views

Counting Self-dual posets

An exercise in Stanley's Enumerative Combinatorics (Chapter 3, ex. 8) asked for an example of a finite self-dual poset, (i.e. there is a bijection $f: P\to P$ such that $s\le t \Longleftrightarrow ...
7
votes
0answers
256 views

Proving equivalence of a tree-based version of Countable Choice for families of finite sets.

In this paper by Good and Tree, the following result is mentioned without proof as part of Proposition 6.5: Each of the following statements imply those beneath it. The countable union of ...
3
votes
1answer
82 views

Where can I learn more about order-reflecting functions? And is the following result well-known?

I stumbled across a cute result about order-reflecting functions. A couple of questions: Is the following result well-known? I know lots of place to learn about order-preserving functions, but is ...
5
votes
0answers
106 views

Reference request - the topology generated by upward and downward closures of antichains.

By definition, the order topology on a totally ordered set $T$ is the coarsest topology such that every subset of $T$ that can be expressed as the upward or downward closure of a singleton is closed. ...
5
votes
2answers
250 views

The order type of the rationals.

Herewith another mind-numbingly naive question from a reader of philosophy. My question concerns the order type of the rational numbers. Omega squared seems a natural first choice, but obviously ...
2
votes
1answer
82 views

Are the closed ordinals (apart from $0$ and $1$) precisely the regular cardinals?

Given a partially ordered set $P$, a collection of partially ordered sets, call it $\mathcal{Q}$, and an arbitrary function $f : P \rightarrow \mathcal{Q},$ we can form a new partially ordered set ...
3
votes
2answers
140 views

Good resource to learn transfinite induction and/or recursion?

I'm currently reading John H. Conway's On Numbers and Games, but without a good understanding of transfinite induction and/or recursion, progress is very slow. What's a good resource for learning ...
3
votes
0answers
76 views

Are there any texts on order theory that treat it as decategorified category theory?

I read this post on nLab and now I want to learn some order theory. What are good texts that contain the above referenced topics, and are there any that are explicitly about order theory as ...
1
vote
1answer
105 views

Ask book to deeply understand partially ordered sets

I learn little about ordering and poset before, but I think it's not enough and want to learn more about ordering and Poset. Can anyone please recommend some best books to learn about this topic. I ...
5
votes
1answer
207 views

Theory of structures with infinite Partial orders. $\langle H, \{\sqsubset_i \}_{i\in I} \rangle$

My recent interests have led me to have to deal with particular structures I have never seen before. Sets equipped with an infinite numbers of partial orders $\{\sqsubset_i:i\in I\}$. I'm a bit ...
6
votes
1answer
96 views

A new(?) partial order on the set of continuous maps

Let $X,Y$ be topological spaces. Define a partial order on $\hom(Y,X)$ as follows: $f \leq g$ if $f^{-1}(U) \subseteq g^{-1}(U)$ for all open subsets $U \subseteq X$. Equivalently, $f(y)$ is a ...
3
votes
1answer
106 views

Robertson-Seymour fails for topological minor ordering? (I.e., subgraphs and subdivision)

The Robertson-Seymour theorem says that the set of isomorphism classes of finite graphs, with the minor ordering ($G \le H$ if $G$ is a minor of $H$) is a well partial ordering (it is well-founded and ...
7
votes
3answers
250 views

Help me get hyped about lattices

I own a small book on Lattice theory published by Dover. Unfortunately, since I bought it almost a year ago, I have gotten nowhere in its study. What I am asking for are freely available papers that ...
11
votes
2answers
820 views

Every partial order can be extended to a linear ordering

How do I show that every partial order can be extended to a linear ordering? I think that I manage to prove that claim for finite set, how can I prove it for infinite set? Thank you.
5
votes
1answer
172 views

Extending a partial order to antichains

Let $(S, \leq)$ be a partial order. Let $T$ be the set of antichains of $S$ (i.e., subsets of $S$ whose elements are pairwise incomparable). Define a relation $\leq'$ on $T$ as follows: for all $A$, ...
3
votes
3answers
228 views

Counting directed acyclic graphs with the same partial ordering

We are given a partially ordered set $P$. Let $L$ denote the set of all linear extensions of $P$ (or equivalently, the set of all topological sortings of the nodes). We want to count the number of ...
4
votes
0answers
100 views

Monoids with positive and negative elements

By a pointed monoid I mean a monoid $M$ together with an absorbing element $0 \in M$ (i.e. $0x=x0=0$). Equivalently, this is a monoid in the monoidal category $(\mathsf{Set}_*,\wedge)$. By a ...
2
votes
1answer
213 views

properties of the poset of self adjoint elements in a $C^*$ algebra

It is well known that the set of self-adjoint elements in a $C^*$ algebra naturally forms a poset. I assume that the general properties of this poset are known. Can anybody point me to a reference ...
6
votes
1answer
156 views

Every poset is embedded into a meet-semilattice

I "discovered" a few minutes ago that every poset can be embedded into a meet-semilattice. Let $\mathfrak{A}$ be a poset. Then it is embedded into the meet-semilattice generated by sets $\{ x \in ...
0
votes
1answer
148 views

What is a valuation associated to an ordering on a field?

If $(K,\leq)$ is a totally ordered field with $P\!=\!\{\alpha\!\in\!K;\, 0\!\leq\!\alpha\}$, how is the valuation associated to $P$ defined? I was searching through Prestel & Delzell's Positive ...
4
votes
1answer
143 views

number of linear orders

It is well known that for every infinite cardinal $\kappa$ the number of non-isomorphic total orders of cardinality $\kappa$ is $2^\kappa$. Who first proved this, and in what context? Was it proved ...