# 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.

2,595 questions
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### Does the frame of open sets in a topological space or locale really have all meets?

According to the nLab article on locales, a frame has all meets by the adjoint functor theorem: This seems a bit strange to me, since it's well-known that an infinite intersection of open subsets is ...
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### Is a topological sorting of a poset a total ordering?

I have been taught by my professor that the topological sort gives a total ordering of a partial order. However, I do not see how this is the case. You are simply rearranging the elements in the ...
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### Finite chain and finite antichain implies that the poset is finite [duplicate]

The problem with which I am struggling is the following, Let $(P,\le)$ be a poset. If every chain and antichain of $P$ are of finite then $P$ is also finite. My Attempt Notice that by Hausdorff ...
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### Can we expand “induction principle” to a partial order $(X, \leq)$?

We know that every infinite can be made well-ordered with an unknown order. Also we can expand the induction principle on any infinite set in the sense that it can made well ordered. Now partially ...
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### Necessary and sufficient condition for existence of a partial order

I'm trying to find a necessary and sufficient condition for the existence of a partial order such that an arbitrary relation on a set X is a subset of the partial order. So far all I have is that ...
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### Name for a poset where incomparability is an equivalence relation

Say I have a partial order $\leq$ on a set $S$. Let me write $a \sim b$ if $a$ and $b$ are incomparable under this order. Is there a name for the following restriction on the partial order? $\sim$ ...
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### Lattice definition and example

Guys I am struggling to understand the lattice concept: Could you help me with this silly example? Take the collection $\{\emptyset, \{0\}, \{1\}\}$ ordered by inclusion. This is a poset, but not a ...
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### Ordered Set example. Why is partially ordered?

I am studying these concepts of order for the first time, and I am having a certain difficulty: I define an Order relation in $A=\mathbb{R_{+}^{2}}$ as : $x,y \in A$, $x\geq y \iff x_{1} \geq y_{1}$ ...
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### Is it possible to have a single axiom that subsumes axioms 8-10 in this list?

Think of a totally ordered set as an “order-theoretic line”. Similarly, cyclic orders are “order-theoretic circles”. I want to find the right axioms for an “order-theoretic plane”. My ultimate goal ...
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### Examples of non-orderable fields.

I wish to find examples of non-orderable fields. We know that fields with finite characteristics cannot be ordered, especially finite fields. Also $\mathbb{C}$ - the field of complex numbers cannot be ...
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### Cayley digraphs of a group

I could not find an answer to the questions online. How many loopless Cayley digraphs of a group G there are? How many loopless Cayley graphs of a group G there are if |G| = n and G has i self-...
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### Relationship between Archimedean and Divisible ordered groups

Let $(G,+,\leq)$ be a linearly ordered abelian group (i.e. the order is total and compatible with the sum) and $n\cdot x$ denote the classical action of $\mathbb{Z}$ over $G$ (i.e. $0$ for $n=0$, sum ...
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### Showing equality of 2 suprema in complete lattice

Let $(M,+,0)$ be a naturally ordered commutative monoid (i.e. such that the natural preorder is antisymmetric) such that $(M,\sqsubseteq)$ is a complete lattice. Then $(M,\sum^*)$ is a summation ...
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### Whether partially ordered sets $(Z,\subseteq )$ and $(Q ,\subseteq )$ are isomorphic?

Let $Z= \left\{[k,l]:k,l \in \mathbb Z \wedge k \le l\right\}$ and $Q= \left\{[p,q]:p,q \in \mathbb Q \wedge p \le q\right\}$. Whether partially ordered sets $(Z,\subseteq )$ and $(Q ,\subseteq )$...
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### Each ordered semigroup is cancellative: reference?

It is easy enough to show that $a+b < a+c\Rightarrow b < c$ holds in totally ordered semigroups. Indeed this must be very well known. Can anyone please provide a reference for this result? A ...
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### Extend order on $\mathbb{Q}$ to $\mathbb Q(X)$

I struggle to show that there exists a unique order on $\mathbb Q(X)$ that extends the order on $\mathbb Q$ such that $\forall q \in \mathbb Q, ~X>q$.
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### On chain conditions and Zorn's Lemma, again

I'm doing a introductory course on commutative algebra, and have just been introduced to the chain conditions. I know there's a lot of questions probably similar, but I haven't really understood this. ...
Starting from the cardinal $|\Bbb N| = \aleph_0 = \beth_0$, we can generate a larger cardinal in two ways: Take the set of all subsets, generating the cardinal $\beth_1$ Take the set of all well-...