# Tagged Questions

1answer
44 views

0answers
259 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 ...
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 ...
0answers
107 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. ...
2answers
254 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 ...
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 ...
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 ...
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 ...
1answer
107 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 ...
1answer
208 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 ...
1answer
97 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 ...
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 ...
3answers
252 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 ...
2answers
837 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.
1answer
174 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 $A$, ...
3answers
231 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 ...
0answers
101 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 ...
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 ...
1answer
159 views