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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Look for the group members $\left ( (\mathbb Z/24\mathbb Z)^*,\underset{24}{\odot} \right )$ and calculate their orders.

Look for the group members $\left ( (\mathbb Z/24\mathbb Z)^*,\underset{24}{\odot} \right )$ and calculate their orders. The elements are $\overline1,\overline2,\ldots,\overline{23}$. Have the ...
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Evidence about the group $\left ( (\mathbb{Z}/p\mathbb{Z})^*,\underset{p}{ \odot} \right )$

Be $p$ an odd prime number. Show that the group $\left ( (\mathbb{Z}/p\mathbb{Z})^*,\underset{p}{ \odot} \right )$ has a unique element of order $2$, namely $\overline{p-1}$, and show that ...
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Be $G=\{ e,g_1,g_2,\ldots, g_n \}$, $|G|=n+1$. Suppose $G$ has a unique element of order $2$, say $g_1$. Show that $eg_1g_2\ldots g_n=g_1$.

Be $G=\{ e,g_1,g_2,\ldots, g_n \}$ an abelian group of order $n+1$. Suppose $G$ has a unique element of order $2$, say $g_1$. Show that $eg_1g_2\ldots g_n=g_1$. I have serious difficulties with ...
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How to show a quasi-order ~ on a set S with relation // induces a partially ordered relation?

Let ~ be the quasi-order relation. and let // be defined as the relation s~t and t~s(s,t elements of S). Show that ~ induces a partially ordered relation on the set of equivalence classes relation //, ...
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Proving questions about posets

How do I tackle a proof about posets? I have know idea how to approach this problem. Thanks! Prove that if all subsets of a poset P have least upper bounds, then all subsets of P have greatest lower ...
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Going down from filters to sets for specific relations

Let $\mathfrak{A}$ be a bounded lattice. I call a $2$-staroid a relation $f\in\mathscr{P}(\mathfrak{A}\times\mathfrak{A})$ such that $i\sqcup j \mathrel{f} b\Leftrightarrow i\mathrel{f} b\vee ...
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What is a binary relation like whose reflexive transitive closure is a partial order?

Let $R$ be a reflexive binary relation and $R^*$ be its reflexive transitive closure. The question is what is the equivalent condition in terms of $R$ to $R^*$ being a partial order. Intuitively, a ...
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Prove that for every 2 elements in the set F of all functions from N to N, there's an element in F that's bigger than both

let there be $\ F$ the set of all functions from $\ N \rightarrow N$. K is a relation on F, for every f,g$\in$F , (f,g)$\in$K $\leftrightarrow$ for all $\ n\in N$, $\ f(n)\leq g(n)$ Prove that for ...
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Building an antichain in a finite poset

Given some finite poset $P$ we would like to find an antichain $A$ which intersects each maximal chain. How to do that? Note that each chain $C$ and each antichain $A$ intersects at one element as ...
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Order of cycles in graph

In a school assignment, I am to use contradiction to prove that if a graph is bipartite then all of its cycles have even order. In this context, what does it mean for a cycle to have even order? I ...
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How many ways can we totally order a set $X$ (but not necessarily the whole set $X$, perhaps just a subset) up to isomorphism?

Here's Attempt #2 at asking this question. Suppose $X$ is a set (not necessarily finite), and let $T$ denote the collection of all totally ordered sets $(A,\leq)$ such that $A \subseteq X.$ Now let ...
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Factor congruences of non trivial Lattices

A pair of congruences $\theta$ and $\theta^*$ are called factor congruences if $\theta \vee \theta^*$ = full congruence. $\nabla$ $\theta \wedge \theta^*$ = trivial congruence. $\triangle$ I need ...
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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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Show that a particular set is a poset

I would like to know if my understanding of the concept of a poset is correct. From what I've learnt from the class: A poset must be transitive, reflexive, and antisymmetric. Am I right? Therefore, ...
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Isomorphism of intervals of a distributive lattice

Let $P$ be a distributive lattice and let for $a,b \in P$ by $a \land b$ and $a \lor b$ denote the usual notion of infimum and supremum. How can I show the following isomorphism? $$[a \land b , a ...
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Lattice from Preorder

I have seen many examples defining a lattice over a partially ordered set $(P, \sqsubseteq)$ together with a greatest lower bound $\sqcap$, a least upper bound $\sqcup$ and a top $\top$, and bottom ...
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Chains of Well-ordered sets

Let $\{X_i\mid i\in I\}$ be a chain of sets and $\{R_i\mid i\in I\}$ be the corresponding chain of relations such, $R_i$ well orders $X_i$. Show that $\bigcup_{i\in I}R_i$ well orders $\bigcup_{i\in ...
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Does the empty set have a supremum or infimum? [duplicate]

I actually dont need this question to be answered for university or something, it was just a discussion I had with a friend. Does the empty set have an upper or lower bound? Since the supremum is ...
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Two topologies over the same set which are carried to each other by a lattice isomorphism induce homeomorphic spaces?

This is a question that I've been turning in my head and haven't been able to come up with a way to proceed to either prove it or construct a counter-example: Let $X$ be a set and $\mathfrak{T}_X$ ...
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define an order relation with respect to which …

Let $(S;<)$ and $(T;<')$ be two well-ordered sets. On $S \times T$, define an order relation with respect to which it becomes a well-ordered set. Give proofs for your answer. I was thinking ...
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Theory of structures with infinite Partial orders. $\langle H, \{\sqsubset_i \}_{i\in I} \rangle$

My recent interests have led me to have to deal with particular structures I have never seen before. Sets equipped with an infinite numbers of partial orders $\{\sqsubset_i:i\in I\}$. I'm a bit ...
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Help with Quotient Groups, Order in D4.

What group is D4/ isomorphic? |D4/| = 8/2 = 4 so it can be isomorphic to V or Z4. How can I determine which group is it isomorphic to? I know that V (Klein-4 Group) is noncyclic and abelian and Z4 ...
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Order and Quotient Group Help

How do I find the order of 2+<6> in Z15/<6>? I know that |Z15/<6>| = 3 which is isomorphic to Z3, but I do not know how to find the order of 2 + <6>.
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If a finite poset has greatest and least elements, is it a lattice?

Let $P$ be a finite partially ordered set with elements $0$ and $1$ such that $0 \le x \le 1$ for any $x \in P$. Does it follow that $P$ is a lattice? If not, what is a counterexample? I believe this ...
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What is total number of inversions in a permutation that is sorted except at indices that are a multiple of a certain number.

So say we have a permutation of integers that is indexed from 0 to n-1, and it is sorted in ascending order except for indices that are a multiple of a number call it x where x is smaller than all ...
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Finitely many minimal elements

I've been working on various exercises to get a better understanding of some topics for an upcoming course. I have a relation $R$ on $\mathbb{N} \times \mathbb{N}$ defined as follows: $(x_0, x_1) R ...
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How to show a directed set always has a “largest element” using induction

I know induction from my lower division classes, but have no idea what to do when it comes to sets. For example a directed set is such that every two elements has an upperbound. We are looking at ...
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Find a voting system - possibly Schulze Method

I remember the Schulze Method from back when I was studying Econ... that was years ago and my math isn't the greatest, go easy on me. We are piloting a project to make a to-do list app for teams, I'm ...
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A set with a supremum and an infinum inside

What is the name of a set $A$ that has its infimum $i \in A$ and a supremum $s \in A$? Examples: $[0,1]$ has a supremum $1$ and an infimum $0$ which are inside $[0,1]$. $\{0,1,2,3\}$ has a supremum ...
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Prove that this partial order relation is a chain with infinite length.

The natural numbers are denoted as a divisibility partial order prove that this relation is a chain with infinite length.
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Given the Hasse diagram tell if the structure is a lattice

Let's consider the following Hasse diagram: I need to tell whether this is a lattice. By lattice definition I can prove the above shown structure $M_5$ to be a lattice if and only if $\forall x,y ...
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Prime order of a number

Prove that if $a^p=e$ where $p$ is a prime number, then $a$ has order $p$ ($a\ne e$). Perhaps I should prove this using the contrapositive? Assume that $p$ is not the order of $a$, meaning there ...
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A function that is both log-supermodular and submodular?

Functions that are log-supermodular and submodular do they have some special name? Or any special properties? Any references? Is the function below log-super-modular and sub-modular? Consider ...
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Notation/terminology: Existence of a nonleast element which is less of any element of a set

Let $A$ is a subset of a partial order $X$. Are there any name and/or notation for the following predicate $P(A)$? $P(A)$ iff there is a non-least element $x$ of $X$ which is a subelement of each ...
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What are some examples of “homogeneous” linear orders, other than $\mathbb{R}$ and $\mathbb{Q}$?

Let $L$ denote a linear order that is unbounded. Then it may or may not satisfy: Globally homogeneous. For all $x,y \in L \cup \{-\infty,\infty\}$, if $x < y$ then the interval $(x,y)$ is ...
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Computational commutative algebra: term orders [closed]

Show that in $k[x,y]$ the monomial orders deglex and degrevlex are same. Here, deglex and degrevlex are defined as in Sage.
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For a set $\mathbb{X}$ given order relation extended from that of $\mathbb{R}$, if $\mathbb{X} \supseteq \mathbb{R}$ then $\mathbb{X} = \mathbb{R}$?

For a set $\mathbb{X}$ given order relation extended from that of $\mathbb{R}$, if $\mathbb{X} \supseteq \mathbb{R}$ then $\mathbb{X} = \mathbb{R}$ ? Motivation for this question rose from an ...
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Determining minimum and maximum elements in partial orders

Let $A$ and $B$ be two subsets of some universe $U$, the partial order is $\langle\mathcal P(U), \subseteq \rangle$. a. does a maximum exists to the group: $\{C: C\subseteq A\text{ and }C\subseteq ...
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Regularization for unordered vectors

Let suppose we have two vectors u, v $\in \mathbb{R}^n$ and we want a function that returns $0$ if the ordering of the elements of both vectors are the same or a positive number otherwise, where the ...
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Total Order Relation

Given $X = \{ a , b , c \}$ and $R = \{ (a,a) , (b,b) , (c,c) , (a,b) , (a,c)\}$ Is $R$ a total order on $X$? I know that total order requires the relation to be comparable on all elements, ...
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Recommendation for Order Theory texts

Order Theory, Lattice Theory, or any directly relatable subject is not taught at my university, and quick searches don't give much clue into good textbooks for the subjects. My question is: what are ...
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Order-preserving embeddings

(Follow-up to Existence of a utility function on the reals.) Say we have a totally ordered set $X$ which has a countable, dense subset $C$. I believe we can find an $f:C\to\mathbb R$ which is ...
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Definition of finite direct decomposition of elements and indecomposable elements at arbitrary lattice

How can i define finite direct decomposition of elements and indecomposable elements at arbitrary lattice . I think i can say an element of lattice is finite direct decomposition of elements if it be ...
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On Tarski-Knaster theorem

Let $P(X)$ denote the power set of $X$ (the set of all subsets of $X$) with a partial order given by inclusion. If $F: P(X) \to P(X)$ is monotone (order preserving), then $F$ has a fixed point. How ...
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Topology inducing Order

We know by Munkres that any (total) ordering induces a topology and (luckily) for the real line that coincides with the euclidean topology. In fact, this construction can be carried over to any ...
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isomorphism between divisible, totally ordered, abelian groups

Let $G$, $H$ be divisible, abelian, linearly ordered groups, whose cardinalities are equal and satisfy $\mu := |G|=|H|>\aleph_{0}$. These are supposed to be (order!) isomorphic. And just about ...
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Not having c.c.c. implies existence of sequence of open sets with strictly increasing closures?

Suppose a topological space $X$ does not satisfy the countable chain condition (by that I mean that there exists an uncountable family of pairwise disjoint non-empty open subsets of $X$). Does it ...
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Strictly Convex and Strictly Monotonic Preferences

On the assumptions for a well behaved preference, I read recently that a preference must be complete, transitive, continuous, strictly monotonous (if the vectors x≥y and x≠y, then x is strictly ...
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Infinite linear order with endpoints which is non-dense

In the process of answering questions about normal models, I had to prove the following: Any normal model of $\chi$ is a non-dense linear order with a least and greatest element. The next question ...
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Particular “upper bound” and “lower bound” of sequence

Is it possible to define the following? Let $a:\Bbb{N} \to \Bbb{R}$, and $M \in \Bbb{R}$, $M$ is ? if $$\exists n \in \Bbb{N}(\forall p \in \Bbb{N}(p\geq n \to a(p)\leq M))$$ Let $a:\Bbb{N} \to ...