# 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.

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### Finding the first, last, minimal and maximal elements in these relations.

I'm now learning about relations of order. This is what I gather: First element: Precedes all elements Minimal elements: Have no predecessors. The first element is always a minimal. Last element: ...
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### Direct products in a partially ordered category

Consider a category, whose set of objects is a poset. Let $f=(f_0,f_1)$ is an indexed family of objects of this category. (Thus $f$ is a 2-indexed family, we can extend it to any index set in an ...
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### Interval in Product Space

I am reading a paper now that refers to an "interval" in $\mathbb{R}^{[0,1]}$. But what does interval here refer to? Does this mean there is some ordering of the elements in $\mathbb{R}^{[0,1]}$? I am ...
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### Finite ordered sets and their isomorphism

This is a slightly strange question perhaps. How many ways are there to prove that if X and Y are two finite orders (total orders) on n elements, then X and Y are isomorphic? There is a direct proof ...
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### On the alphabetical order of monomial

I found this definition of alphabetical order for monomials in $k[x_1,\ldots,x_n]$. We say that $x_1^{a_1}\cdots x_n^{a_n}>x_1^{b_1}\cdots x_n^{b_n}$ if for the least $i$ such that $a_i\neq b_i$ we ...
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### Partially ordering finite graphs

What are some interesting partial orders on the set of all finite graphs (identified up to isomorphism), apart from the usual (induced) subgraph relation and the (topological) minor relation, and why ...
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### Distance of binary strings to produce the lexicographical order

Indexing objects like elements of a Cantor Set or nodes of a Binary Tree can result in a enconding system of binary strings like illustrated bellow: The illustrated indexes form a finite set, C_3=\...
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### Partially Ordered Set and Equivalence Relationship

Is it fine if one could add an equivalence relationship for the following case, to make a partially ordered set become totally ordered? Suppose (X, d) is metric space, and X has infinite elements. ...
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### Order relation proof …

Consider the set $A = \{1, 2, 3\}$, and set $B = A × A$. In the set $B$, consider then the relationship $C$ defined by placing $(a, b) C (c, d)\iff a ≤ c$ and $b | d$ where $≤$ and $|$ denote ...
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### X is totally ordered under ≤ if and only if X follows the law of trichotomy?

I'm having trouble with this proof. I know that this proof requires two parts. My attempt so far: Proof: ($\Rightarrow$) Assume $X$ is a totally ordered set under $\le$. Let $m,n$ be two ...
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### Strengthening of a theorem about filters vs a counter-example

Let $S$ be a non-empty set of filters on a meet-semilattice. If our semilattice is a distributive lattice, then the supremum (on the poset of filters ordered by set-theoretic inclusion) of $S$ is the ...
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### Does consistency of a test imply continuity of a preorder?

Let $F$ and $G$ be two real-valued probability distributions, considered as cadlag functions with the uniform norm [not the Skorohod metric], and let $\mathbb{F}_n$ and $\mathbb{G}_m$ be empirical ...
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### How to understand the proof of the below statement similar to Zorn's lemma?

Proposition: Let $A$ be a partially ordered set such that every chain (total ordered subset) of A has a supremum in A; assume that A has a least element p. Show that there exists an element $m ∈ A$ ...
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### Prove that the partial order ((D * D), ⊆) is a complete partial order with bottom.

Let D be a non-empty set and (D-->D) be the set of all partial functions from D to D. Prove that the partial order ((D--> D), ⊆) (i.e., the set of partial functions ordered by set inclusion) is a ...
Let $\mathfrak{A}$ is a poset. For $a, b \in \mathfrak{A}$ we will denote $a \curlyvee b$ if only if there is a non-least element $c$ such that $c \leqslant a \wedge c \leqslant b$. Let $\mathfrak{A}$...
### Let $≺$ be a relation on a set $P$, reflexive and transitive. Def. $∼$ on $P$ by $p ∼ q \iff p ≺ q ∧ q ≺ p$. Show $∼$ is an equiv. relation.
Can I get some help on this one? Any solutions? Let $≺$ be a relation on a set $P$ that is reflexive and transitive. Define the relation $∼$ on $P$ by $p ∼ q$ iff $p ≺ q ∧ q ≺ p$. (a) Show $∼$ is ...