# 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.

3,783 questions
Filter by
Sorted by
Tagged with
1 vote
49 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 ...
41 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, ...
85 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
13 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 ...
• 13.2k
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 ...
• 4,869
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 ...
• 13.2k
56 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
34 views

### 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?
92 views

### A property of partitions of the real numbers

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 $x < y$, ...
• 1,363
1 vote
61 views

### What does it mean that "All diagrams commute in a posetal category"?

I am quite familiar with posetal categories, however, I just randomly came accross the claim that "all diagrams commute in a posetal category" on Wikipedia. I am confused, what does it even ...
• 1,922
20 views

### On extending a monotone map

I often find myself needing to invoke a result like the one given below, but I have not been able to find it in any textbook on set theory or order theory. I can think of two possibilities for this: (...
• 13.2k
78 views

### Unique Complemented lattice which is not distributive lattice!

I know that : "if a distributive lattice is also complemented lattice then lattice should only have unique complement" but i am not able to find a case where A lattice is uniquely ...
• 29
### Longest antichain among $k$ posets whose union is the total order
I'm wondering if the following question (or something close) has already been considered in the literature. Consider $k$ posets $P_1, P_2, \ldots P_k$ on the same set of $n$ base elements, whose pairs ...