# 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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### Existence of a utility function on the reals

Suppose I have $\preceq$, a total order on $\mathbb R^n$. I wish to show that there is a utility function $u:\mathbb R^n\to\mathbb R$ such that $x\preceq y \leftrightarrow u(x)\leq u(y)$. I came up ...
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### Is there a relation that is irreflexive, anti-symmetric and not transitive?

from the set $\{a, b, c, d\}$? Of the one's I have tried, it at best is two of the three, but never all.
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### How many ways are there to represent a monomial order by term order via matrices?

During the lecture, my professor brought up a list of project ideas to work on. One of the ideas I am interested and currently working on is term order via matrices. That is: I need to find the number ...
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### Zorn's Lemma related statement

Consider the following statement: If $X$ is partially ordered set such that every chain in $X$ has un upper bound, then for every $x \in X$ there is a maximal element $m$ in $X$ such that $x \le m$. ...
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### Not well-ordered sets satisfying the least upper bound property

Before proving that every closed interval in $\mathbb{R}$ is compact, James R. Munkres remarks that, in Section 27 entitled "Compact Subspaces of the Real Line" of Topology (2nd edition), Remark: ...
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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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### In the transition from set theory to order theory, what is the appropriate generalization of “group”?

Given a set $X$, the collection of all bijective endofunctions on $X$ forms a group, called the symmetric group on $X.$ Furthermore, Cayley's theorems says that every group embeds into a subgroup of ...
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### Injectivity of rank function

Let $R$ be well founded and set like on $A$. I want to show that $R^{*}$ is a total order iff $rank_{A,R}(y)=\sup\{rank(x)+1|$ $xRy$ $\wedge{x\in{A}}\}$ is injective. Here $R^{*}$ is the transitive ...
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### Completeness of posets

I think this is rather a simple question in order theory, but if someone could explain it step by step that would be really useful. If we have an arbitrary set, then let's denote the set of all finite ...
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### Which of these constructions are left adjoints?

A groupoid can be regarded as a small category in which every arrow is an isomorphism A monoid can be regarded as a small category with only a single object A preorder can be regarded as a small ...
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### Likeness of strictly well-ordered sets

The question itself: Given a strictly ordered set $\langle A, < \rangle$ (< merely symbolises some kind of an order in this case) which is dense, with a least and greatest elemnts that are not ...
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### 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 ...
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### Elementary Properties of Order

Need help with this question: Prove: Let $x = a_1, a_2...a_n$, and let $y$ be a product of the same factors, permuted cyclically. (That is, $y = a_k, a_{k+1}...a_na_1...a_{k-1}$. Then $ord(x)=ord(y)$...
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### The null set as the underlying set of a relation structure

Can the null-set underly a relation structure? Can it underly a well-order? If so, would such a well-order have order-type $0$? (Can one even have an order-type of $0$?) My motivation for this ...
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### Question about sets of well-orderings

I recently posted a question on here about sets of well-orders and isomorphisms between well-orders. I have a related question about order-type-comparison: if $W$ is any set of well-orders, doesn't ...
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### Question about sets of well-orders and isomorphisms between well-orders

I was thinking about this problem in trying to better get a feel for and understand well-orders (I'm trying to learn some set theory) - right now, I'm not really hoping for anybody to supply me with a ...
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### Could someone check this proof for me?

The problem (and a definition first): The following definition explained Let $U \subseteq \mathbb{N}$. We say that a function $x: \{0,1\}^U\to \{0,1\}$ is a $xor$ on the set $U$ if the following ...
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### How many pairwise nonisomorphic dense subsets exist

in the completion of a linear order of cardinality $\kappa$? Can there be more than $\kappa$? To be more precise, suppose $L$ is a linear order of cardinality $\kappa$. Let $\overline L$ denote ...
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### Interesting question with ordering integers

The numbers $1,2,3,...,101$ are written down in a row in some order. Is it always possible to cross out $90$ numbers in a way such that all $11$ numbers left will stay either in increasing or in ...
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### Proof by contradiction: $c=\max (A) \wedge d=\max (B) \to \max (A \cup B ) \in \{c,d\}$

Let be $A,B \subseteq \Bbb{R}$, with $A \neq \emptyset$ and $B \neq \emptyset$ and $\Bbb{R}$ totally ordered by $\leq$, and $c,d \in \Bbb{R}$, with $c=\max (A)$ and $d=\max (B)$. I must proof by ...
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### Does the Riemann zeta function tell us about the order theoretic properties of the natural numbers?

The classical Möbius 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 ...
Consider a group $G$ acting on a set $\Omega$. For example, let $G=\{g\in A(\mathbb R):(\alpha +1)g=\alpha g+1\}$ for all $\alpha\in\mathbb R$, where $A(\mathbb R)$ are the order-preserving ...