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.
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Order-preserving map of regressive functions on $\omega_1$
I posted the following question in March 2014 on MO. It did receive some attention, but the answer there remains incomplete. It was motivated by some paracompactness-type properties as discussed at ...
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Summation methods ordered by strength
A summation method is a partial function from scalar sequences to scalars, i.e. an element of the set $\mathbb{C}^\mathbb{N} \rightharpoonup \mathbb{C}$. A summation method $\Sigma_1$ is weaker than a ...
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Can $(\mathcal P(\mathbb N),\subseteq)$ be partitioned into maximal antichains?
If you look at the set of finite subsets of $\mathbb N$ (side question: Is there a standard notation for that?) partially ordered by the subset relation, you see that it can be partitioned into ...
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On the relationship between Martin's Axiom, the countable chain condition and the Knaster property
We say that a poset $P$ has the Knaster property (or is Knaster) if every uncountable subset of $P$ contains an uncountable subset of pairwise compatible conditions.
Let $K$ denote the statement "...
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The Term "Cofinal"
I was reading about ordered fields $\mathbb{F}$ with countable cofinality which means that there is a countable set $S$ that is cofinal. The definition of $S$ being cofinal is that for all $a \in \...
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Infinite palindromes in number of nonisomorphic posets is independent of $\mathsf{ZF}$
The following is an "exercise" in P. Stanley's book "Enumerative Combinatorics":
Let $f(n)$ be the number of nonisomorphic $n$-element posets (...) let $\mathcal{P}$ denote the statement that ...
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Reference request - the topology generated by upward and downward closures of antichains.
By definition, the order topology on a totally ordered set $T$ is the coarsest topology such that every subset of $T$ that can be expressed as the upward or downward closure of a singleton is closed.
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Monoids with positive and negative elements
By a pointed monoid I mean a monoid $M$ together with an absorbing element $0 \in M$ (i.e. $0x=x0=0$). Equivalently, this is a monoid in the monoidal category $(\mathsf{Set}_*,\wedge)$. By a $\pm$-...
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Is every set an image of a totally ordered set?
It is known that the statement "Every set admits a total order" is independent of ZF. See here, for example. However, can it be proven in ZF that for every set $Y$, there exists a totally ...
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How efficiently can the first-order theory of posets recognize the open set poset of $\Bbb R^n$?
This question requires me to write quite a bit of background, so I apologize.
Suppose our domain of discourse is the set of open subsets of a topological space. This means that, by $\forall A$ I mean &...
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Charecterization of Topologies via Galois Connections
Let $X$ and $Y$ be two sets and let $F: \mathcal{P}(X) \to \mathcal{P}(Y)$ and $G: \mathcal{P}(Y) \to \mathcal{P}(X)$ be two set functions that satisfy
$$F(A) \subseteq B \iff A \subseteq G(B). \tag{1}...
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How many total orders consistent with a partial order?
I have a finite set of objects $X$, whose power set is partially ordered by $\subseteq$.
Consider all possible total orderings of the power set $\mathscr{P}(X)$ which are compatible with the partial ...
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Why is the symbol for the exterior product a meet rather than a join?
I've moved this over to HSM.
It seems odd that something that looks so much like a join [see below] would get given "the wrong symbol". It's even worse when you dualise it and get something called (...
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Showing that poset of set of supports of a vector space is semimodular
Let $W$ be a subspace of the vector space $\mathbb{K}^n$, where $\mathbb{K}$ is a field of characteristic $0$. The support of a vector $v = (v_1,\ldots, v_n) \in \mathbb{K}^n$ is given by $\text{supp}(...
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Proof on Dyadic Trees [Smullyan: First-Order Logic, chapter 1, section 0]
I'm having difficult with a proof from Smullyan's First-Order Logic, Chapter 1 Section 0 (Reprint, Dover 1968, p. 4):
Prove:
In a dyadic tree, define x to be to the left of y if there is a junction ...
