# Tagged Questions

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.

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### How to find a total order with constrained comparisons

There are 25 horses with different speeds. My goal is to rank all of them, by using only runs with 5 horses, and taking partial rankings. How many runs do I need, at minumum, to complete my task? As ...
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### What do linearly ordered abelian groups look like?

Recently a post on MathOverflow connected a theorem about ordered abelian groups to the axiom of choice, and it made me want to try and play with these objects a bit. But it appears to me that I don'...
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### 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 ...
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### 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. ...
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### 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 $\pm$-...
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### Why is the symbol for the exterior product a meet rather than a join?

I've moved this over to HSM. It seems odd that something that looks so much like a join [see below] would get given "the wrong symbol". It's even worse when you dualise it and get something called (...
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### Does order isomorphism of linear extensions of two partially ordered sets imply order isomorphism of themselves?

This question was firstly posted on Mathoverflow. Two answers are pretty interesting. The potential counter-example given in the second answer is really interesting, but it is not surely a counter-...
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### Group orderable iff all its finitely-generated subgroups are orderable

I want to proof this specifically using the Compactness Theorem from propositional logic (this is an exercise from Model Theory, Hodges). $G$ orderable means there is a total ordering s. t. $g\leq h$ ...
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### Squares of adjunctions / Galois correspondences

$\newcommand{\Spec}{\operatorname{Spec}} \newcommand{\PS}{\mathcal P} \newcommand{\Idl}{\operatorname{Idl}}$ There are many situations where one encounters a square of things which are related by ...
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### (Almost) co-free objects?

another naming question from me, which comes up because I try to use category theory as a compass in developing some (new or not?) order-theoretic notions. According to my book (The Joy of Cats, [...
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### Prove that for every 2 elements in the set F of all functions from N to N, there's an element in F that's bigger than both

let there be $\ F$ the set of all functions from $\ N \rightarrow N$. K is a relation on F, for every f,g$\in$F , (f,g)$\in$K $\leftrightarrow$ for all $\ n\in N$, $\ f(n)\leq g(n)$ Prove that for ...
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### 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 ...
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### Expect size of $k^{th}$ layer of a POSET

Is this known? What is the expected width of the $k^{th}$ layer (anti-chain layer) of a $d$-dimensional partially ordered set of $n$ elements formed by product of $d$ random liner orders chosen from ...
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### Every subset of a poset has an inf implies every subset has a sup

P is a partial order with $\leq$, and L and U are the sets of all lower and upper bounds of A I already know that if every nonempty subset with a lower bound has an infimum, then every nonempty ...
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### How can we define “trivially orthogonal” groups?

In any lattice-ordered group, we say that two elements are orthogonal if their meet is 1. I've been thinking of groups who have only "trivial" orthogonal relations, i.e. $x\perp y\implies x=1$ or $y=1$...
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### 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 ...
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### Non-isomorphic posets

Is there any formula or counting algorithm for the number of non-isomorphic posets (defined on finite n-element set)? I'm interested how to solve task *5, p.4 in Birkhoff's "Lattice Theory" the book ...
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### Can the actual scope of “lattice theory” be summarized as “algebraic order theory”?

Lattice and order theory are often mentioned together. So I wonder how lattice theory is "intended" to differ from order theory. Lattice theory became important in the context of universal algebra. ...
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### Back-and-Forth Argument vs. “One-Way” Argument

The wikipedia article on the Back and Forth Argument claims at the end: If we iterated only step $(1)$, rather than going back and forth, then in some cases the resulting function from A to B ...
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### Cones and partial ordering

A positive cone always induces a partial ordering on a vector space. I wonder if the converse holds, so is every partial ordering in a vector space induced by a positive cone, or in other words, is ...
The classical MÃ¶bius function $\mu(n)$ fulfills the multiplicative inversion formula, e.g. see this thread. Now I see in the theory of posets, they generalize the concept of that function, see ...
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 $\sigma$, ...