# 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 alegbraic structures.

2,701 questions
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### Infinite modular lattices

A finite lattice $L$ is called modular if and only if its elements satisfy the following modular identity: For all $x,y,z\in L$ such that $x\leq z$, we have $x\vee(y\wedge z)=(x\vee y)\wedge z$. How ...
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### minimal embeddings of topological spaces into connected spaces

Defintions: Let $X$ be a topological space. 1) A connected space $Y$ is a minimal connected ambient (m.c.a for short) space for $X$ if there exists an embedding $i:X\mapsto Y$, and for every ...
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### Definition: A (linear) order type $\alpha$ being dense in a (linear) order type $\beta$ - resolving a (seeming) contradiction?

I will use $\mathbf{\eta}$ and $\mathbf{\lambda}$ to respectively denote the order types of the rationals and the reals. In the book Linear Orderings, by Joseph Rosenstein (1982), he defines: ...
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### Show that every ordered set with the well ordering has the least upper bound property

Here is a proof attempt: Let $S_a =\{x\in A:x \leq a\}$ (also known as a section of $A$). We firstly prove that $\forall a \in A$, $S_a$ has a supremum in $A$. Clearly, every $S_a$ is bounded above ...
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### How can I find the connectivity graph of critical points of a function?

I have the following question. Since it is a bit hard to explain just by words, I added a figure. Suppose I know all the critical points of a function, and I would like to reconstruct a simple graph ...
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### Number of chains in a symmetric chain decomposition

I need to show that the number of chains of length $n-2k$ in a symmetric chain decomposition of Boolean Lattice $B_n$ is $\binom{n}{k}-\binom{n}{k-1}$. But I have no idea how to do it. I also have a ...
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### Spaces of Cones

In Machine Learning applications, Grassmanian and Stiefel Manifolds have exploded in popularity for studying Subspace or Orthonormal Basis valued optimization problems, among other things. I am ...
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### Munkres Topology Chapter 1.3 Exercise 15

Does $[0,1] × [0,1]$ in the dictionary order have the least upper bound property? What about $[0,1] × [0,1)$ and $[0,1) × [0,1]$? Since both $[0,1]$ and $[0,1)$ have the least upper bound property, I ...
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### Confusion about a definition in Bourbaki Algebra

I am currently reading Bourbaki Algebra and in section 2 of chapter one they define when two ordered sequences are similar as this: Two ordered sequence $(x_i)_{i \in I}$ and $(y_k)_{k \in K}$ are ...
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### Showing equality of 2 suprema in complete lattice

Let $(M,+,0)$ be a naturally ordered commutative monoid (i.e. such that the natural preorder is antisymmetric) such that $(M,\sqsubseteq)$ is a complete lattice. Then $(M,\sum^*)$ is a summation ...
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### Do we only need Choice for higher order-statements (Zorn’s lemma as example)?

In this earlier post, I asked why the axiom of choice, which is an axiom in set theory, is used in areas that are not set theory, such as group theory. The answer was that, whenever choice is used, it ...
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### A linearly ordered set without endpoints such that every closed interval is finite is isomorphic to set of integers.

Suppose $A$ is a linearly ordered set without maximum or minimum and every closed interval is a finite set. I want to show $A$ is isomorphic to the set of integers with the usual order. I know that ...
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### Why are orders formalized by weak inequalities?

Partial orders are formalized by weak inequalities $\geq$ rather than strict ones $>$. We then add an additional axiom which says that $x\geq y\land y\geq x$ implies $x=y$. But it seems to me ...
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### Filters on preordered sets vs filters on partially ordered sets

What is the advantage on defining filters on partially ordered sets versus defining them on preordered sets? Almost everywhere where I have seen filters they have been used in the setting of partially ...
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### A partial order on $\mathbb{N}$ that is not total

In the text Modern Real Analysis by Ziemer there is a question that asks to use the "natural partial order" on $\mathcal{P}(\{1,2,3\})$ to obtain a partial order on $\mathbb{N}$. I have scratched my ...
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### Quadrants of a cyclically ordered group

I am trying to prove that if there are two different positive elements in a non-linearly cyclically ordered group, then every quadrant of the group is not empty. Is this a correct statement? ...
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### Positive and negative elements of a cyclically ordered group

I am trying to prove A property of a cyclic order on a ring. In order to do it, I need the last two properties (lemmas 1.11 and 1.12) in this question. I separated them from the original theorem ...
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### Is the set of real numbers the largest possible totally ordered set?

Because I find any totally ordered set can be "lined up" in a straight line, I'm guessing that the set of all the real numbers is the biggest totally ordered set possible. In the sense that any other ...
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### Are there subsets of $\mathbb{Q}$ with a finite number of upper-bounds?

Consider the poset $(\mathbb{Q}, \leq)$. Is there a subset of $\mathbb{Q}$ with a finite number of elements that are upper-bounds? I tried to prove this as follows: Suppose $K \subset \mathbb{Q}$, ...
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### example of non order preserving bijection

I read that "the set of integers and the set of rational numbers (with the standard ordering) do not have the same order type, because even though the sets are of the same size (they are both ...
Does it make sense to introduce the new definition? Definition 2.1. An element $\pi$ of a cyclically ordered group is an apex of the group iff $\pi = - \pi \ne 0$. Considering an element $x$ of a ...