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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questions about well-order

In Wikipedia, well-order is defined as a strict total order on a set S with the property that every non-empty subset of S has a least element in this ordering. But then later, well-order is defined ...
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order and binary relation

Are order and binary relation the same thing, or order is just some special binary relation? If it is the latter, what kinds of properties distinguish an order from a general binary relation? Thanks ...
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In a lattice, ( x ∧ z ) ∧ ( y ∧ z ) = z ∧ ( y ∧ x )

Let ( L, ≤ ) be a lattice, x, y, z ∈ L I am unable to understand why ( x ∧ z ) ∧ ( y ∧ z ) = z ∧ ( y ∧ x ) is mentioned as fact in Y.N. Singh's "Mathematical Foundation of Computer Science" - Pg ...
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In a lattice, $x \leq y$ and $x \leq z \iff x \leq y ∧ z$

Added later: Proof now in progress. Let $(L, \leq)$ be a lattice, $x, y, z \in L$. Prove that $x ≤ y$ and $x ≤ z \iff x ≤ y ∧ z$. I proceed to prove as follows: The statement can be split into ...
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In a Lattice, x ∧ y ≤ x and x ∧ y ≤ y

Let ( L, ≤ ) be a lattice, x, y, z ∈ L. Prove that : $x ∧ y ≤ x$ and $x ∧ y ≤ y$ Here is how I proceeded to solve it: Approach 1 : Proceeding by definition: $x ∧ y ∈$ { x, y } Therefore, x ∧ y ...
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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 ...
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Showing that well-ordered subsets of $P(\omega)$ are countable

I have the following problem: Show that no uncountable subset of $P(\omega)$ is well-ordered by the inclusion relation. I think they want me to do by embedding it in a separable complete dense ...
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Separative Quotients, and the Induced Order

Let $P$ be some partial order. We say that $x$ and $y$ are compatible if $\exists r\in P (r\le x \wedge r\le y)$, we denote this by $x \perp y$. Otherwise, we say that $x$ and $y$ are incompatible. ...
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Showing the Inclusion is sup-continuous

I fear I over simplified the following problem: For any partially ordered set $(A,\leq)$, let $A^* = A- \{\max A,\min A\}$ if $\max A$ and $\min A$ exist. Show the inclusion ...
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For some sets $S\subseteq T\subseteq U$, when is $\inf_T S=\inf_U S$?

I've been struggling with the following problem I found for a while now: Suppose $(T,\preceq)$ is a partially ordered subset of $(U,\preceq)$ and $S\subseteq T$. If $\inf_T S$ and $u=\inf_U S$ both ...
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when inf and sup of a subset achievable in itself?

I was wondering if the following is true: In a topological space with partial order, the inf and sup for a closed subset are achievable inside the subset so that they become minimum and maximum. Is ...