# 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 algebraic structures.

3,786 questions
Filter by
Sorted by
Tagged with
1 vote
41 views

### Show that a circular relation is not a partial ordering

I'm learning about partial orderings and need some help understanding the following statement. Say we have a relation, R, between the objects of finite set $S$: If every element in R is preceded by ...
• 517
52 views

### Show that the definition of Scott topology is a topology

To verify the definition of a Scott topology is a topology, I still need to show that it's closed under intersection. Can someone help? Definition 1 (Scott topology). Let $(D,\leq)$ be a complete ...
• 29
71 views

### A complete partially ordered set.

Please help me to understand the topic of order relations on the set. I can't understand what a complete partially ordered set is. I want to summarize how I understand the topic of order relations on ...
13 views

### Are there any theorems about the restriction of a partial order? [closed]

That is, by restricting the partial order on a subset, the order becomes complete.
• 175
57 views

### The proof of Lemma 10.1 in Nik’s book about forcing

I’m an undergraduate student trying to teach myself set-theory. And I have some trouble understanding the density of a constructed set. In Lemma 10.1 of Nik’s book, it states: Let $G$ be a generic ...
1 vote
50 views

### What is an example of a nonempty subset of $\mathbb{R}$ that is bounded above that does not contain its least upper bound?

What is an example of a nonempty subset of $\mathbb{R}$ that is bounded above that does not contain its least upper bound? This is an on-a-review sheet for my final. I thought the completeness axiom ...
45 views

27 views

### A mapping of two sequences with no overlaps and partial assignments

I want to characterize a correspondence mapping of two sequences $\psi : A \rightarrow B$ for an article that I am writing. I need help describing the function class. I think this is an injective, ...
87 views

### Is the theory of linear dense orders with distinct endpoints complete?

I'm trying to solve some problems about elementarily equivalent structures. For example, I know that the structures $\langle\mathbb{Q},<\rangle$ and $\langle\mathbb{R},<\rangle$ are elementarily ...
23 views

• 17
14 views

### Construction of order-preserving map on Bourbakian poset

Given an order preserving map $$f:\alpha\to P$$ with $\alpha$ an ordinal and $P$ a Bourbakian poset, I'm trying to construct an order preserving map $$g: P\to P$$ whose fixed points are precisely the ...
59 views

35 views

### Is there a general poset representation that specializes to power set lattices in case of finite boolean algebras?

I read here that every finite, complemented, distributive lattice is isomorphic to a power set lattice. Is there a general order preserving mapping from a poset $P$ to a set inclusion poset $S$, such ...
• 123
27 views

### Is the equality-free theory of linear orders the same as the equality free-theory of linear preorders?

This is a natural follow-up to my question, here:In first-order logic without equality, is the theory of partial orders the same as preorders?. My current question is, consider first-order logic ...
• 13.3k
114 views

### Ordinals without set theory?

I'm interested in whether ordinal numbers can be described by a first-order theory without presupposing ZFC or any particular set theory. Such a theory might look like Peano arithmetic, but ...
• 4,878
91 views

### Order theory from categorical point of view

On p. 12 of Introduction to Lattices and Order by Davey and Priestley, the authors give a 1-paragraph description of Category Theory, and then write: We do not have sufficient need to call on the ...
• 13.3k
61 views

### Quick way of drawing Hasse diagrams of posets

When drawing a Hasse diagram, I have seen that you can draw a bigraph for the poset and remove the reflexive and transitive edges of the poset. However, doing this for a poset with many elements can ...
• 465
44 views

### Finding restriction of the ultraproduct that behaves like $\mathbb{Z}$

Let $\mathcal{A}= \prod_{n \in \mathbb{N}} \mathcal{A}_n /\mathcal{U}$, where $\mathcal{A}_n=(\{0, 1, \dots, n\},<)$ and $\mathcal{U}$ is a non-principal ultrafilter of $\mathbb{N}$. Can we find ...
• 1,699
### Name of property: $\phi (x)\geq x$ [duplicate]
Let $X$ be a preordered set and $\varphi : X\to X$ a function (can assume monotone if useful for the answer). Does the property of $\forall x\in X: \varphi (x) \geq x$ have a standard name?
Let a strict linear order $C = (V, <)$, be an irreflexive and transitive relation < defined on $V$, and call a section of $C$ a partition of $V$ into two sets $A, B$, such that $x < y$, ...