# Questions tagged [posets]

A poset or partially ordered set is a set endowed with a partial order, meaning a binary relation $\leq$ which is reflexive ($x \leq x$ for all $x$), antisymmetric ($x\leq y$ and $y\leq x$ implies $x=y$), and transitive ($x\leq y$ and $y\leq z$ implies $x \leq z$).

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### Reeb Chords as Posets

I have been thinking recently about Legendrian knots (specifically, the paper by Shende, Treumann, and Zaslow). Basically, a Legendrian knot $\Lambda$ is derivable by permuting the Reidemeister moves [...
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### Find a finite semimodular poset such that

For definitions, see Section 1 of Chapter 3 of Richard Stanley, Enumerative Combinatorics, Volume I (second edition). Also see Section 8 of Chapter II of Garrett Birkhoff, Lattice Theory (third ...
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### Order on Euclidean space in which a finite poset embeds

Fix positive integers $k$ and $n$. For which finite posets $(X,\lesssim)$ with $\#X=k$ does there exist an order embedding $\phi\colon(X,\lesssim)\to (\mathbb{R}^n,\le)$, where $\le$ is the standard ...
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### Are there more than 2 types of posets $P\cong\mathcal O_{\rm fin}(P)\setminus\{\emptyset\}$?

We use notation derived from Davey and Priestley, Introduction to Lattices and Order. Let $\mathcal O_{\rm fin}(P)$ be the poset of finite down-sets of the poset $P$. A finite poset is ranked if all ...
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### Can we say a partial order set is 2-dimensional if its comparability graph does not contain an asteroidal triple?

I think it is true that if the comparability graph of a poset contains an asteroidal triple then it is at-least 3 dimensional. I want to know if the converse is true, i.e. if there exists no ...
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### Embedding of a poset with "desirable" characteristics

Let $(P, \succeq)$ be a poset (you can assume $\succeq$ is not empty). I am interested in finding a poset $(X, \succeq^*)$ that embeds $(P, \succeq)$, and $\forall x,y \in X$ both of the following ...
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### Multidimensional power series with coefficients equal to an order of stabilizer of a set of powers

I have encountered a necessity to work with a series of the following form. There are $N$ variables $x_1,\ldots x_N$. It is convenient to introduce monomial symmetric polynomials $m_{\lambda}$. They ...
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### Are the Boolean algebras ${\cal P}(\omega)/(\text{fin})$ and ${\cal P}(\omega)/(\text{thin})$ isomorphic?

A set $A\subseteq \omega$ is said to be thin if $$\lim\sup_{n\to\infty}\frac{|A\cap \{0,\ldots, n\}|}{n+1} = 0.$$ We say for $A, B\subseteq \omega$ that $A\simeq_\text{fin} B$ if the symmetric ...
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### Reference request: number of antichains of a partially ordered set

Let $\mathbb{N}$ denote the set of all positive integers. For each $n \in \mathbb{N}$, define the set $$P_n = \{ (a,b) \in \mathbb{N} \times \mathbb{N} : 1 \leq a \leq b \leq n \}$$ and consider the ...
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### Reference Request for "Finite Semilattice with Top and Bottom is a Lattice"

Let $\mathcal{O}(P)$ be a finite, completely distributive lattice of all lower sets ordered by set inclusion. Moreover, let $K =\; \mathrel{\{} h(x) \mathrel{|} x \in \mathcal{O}(P) \mathrel{\}}$ be ...
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### Sperner property of a distributive lattice associated to a divisor poset and the free distributive lattice

Let $P_n$ denote the poset with elements $P_n=\{1,...,n\}$ ordered by divisibility and let $L_n$ denote the distributive lattice of order ideals of $P_n$, whose elements should correspond to primitive ...
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### If $P\times{\bf2}$ order-embeds in $Q\times{\bf2}$, does the poset $P$ embed in the poset $Q$?

A function $f:P\to Q$ from a poset $(P,\le_P)$ to a poset $(Q,\le_Q)$ is an order-embedding if, for all $p,p'\in P$, $p\le_P p'$ if and only if $f(p)\le_Q f(p')$. We partially order the Cartesian ...
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### Find an order-embedding of $S_3\times{\bf2}\times{\bf k}$ into a product of $3$ chains, one of size at most $k$

A function $f:P\to Q$ from a poset $(P,\le_P)$ to a poset $(Q,\le_Q)$ is an order-embedding if, for all $p,p'\in P$, $p\le_P p'$ if and only if $f(p)\le_Q f(p')$. We partially order the Cartesian ...
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### Find an order-embedding of $S_3\times{\bf2}\times{\bf2}$ into ${\mathbb Z}^4$

A function $f:P\to Q$ from a poset $(P,\le_P)$ to a poset $(Q,\le_Q)$ is an order-embedding if, for all $p,p'\in P$, $p\le_P p'$ if and only if $f(p)\le_Q f(p')$. We partially order the Cartesian ...
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### Universal poset for cardinals $\kappa \geq \aleph_0$

Given a cardinal $\kappa\geq \aleph_0$, is there a poset $(P,\leq)$ with $|P| = \kappa$ such that every poset of cardinality $\kappa$ can be order-embedded into $(P,\leq)$?
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### Let $X$ be a finite set of $n$ ($>1$) elements and $\tau$ be a topology on $X$ having exactly $m$ elements. Can we give any description of $m$?

Let $X$ be a finite set of $n$ ($>1$) elements and $\tau$ be a topology on $X$ having exactly $m$ elements. Can we give any description of $m$ as it relates to $n$? Obviously $2\le m\le 2^n$ and ...
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### Is every homogeneous poset a lattice?

A poset $(P,\leq)$ is homogeneous if $P\cong [a,b]$ for all $a,b\in P$ with $a<b$ (where $[a,b] := \{x\in P: a\leq x\leq b\}$). Examples of homogeneous posets include $[0,1]$, $[0,1]\cap \mathbb{Q}$...
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### Causal-net category and poset category

Order is a fundamental mathematical structure. There are two natural ways to represent order structures, by posets and by causal-nets (acyclic directed graph). How can we compare these two ways, and ...
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### Characterization of edge posets

Given an acyclic directed graph $G$, the set $E(G)$ of edges of $G$ equipped with the reachable order $\to$ is called the edge poset of $G$, where for two edges $e_1\to e_2$ means that there is a ...
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### Request for literature recommendations on isotonic mappings

An isotonic mapping is a function between two partially ordered sets that preserves the ordering between the elements. Specifically, given two partially ordered sets $(X,\le)$ and $(Y,\le)$, a ...
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### What is the cofinality of the co-infinite subsets of ${\bf N}$?

Let ${\mathcal A}$ be the family of subsets $A$ of the natural numbers ${\mathbf N}$ which are co-infinite (i.e., their complement is infinite). We partially order this family by set inclusion. A ...
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### Is the face poset of a compact intersection of cylinders and half-spaces shellable?

Let the $n$-disk $D^n$ be stratified hemispherically (so there are two 0-dimensional strata at the poles, two 1-dimensional strata for the prime meridian and the international date line, two 2-...
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### Is the set of sub-dcpos a dcpo (directed-complete partial order)?

$\newcommand{\sub}{\mathrm{sub}}$Given a dcpo (directed-complete partial order) $\mathcal{X} = (\le, X)$, consider the set $\mathcal{X}^{\sub}$ of all sub-dcpos of $\mathcal{X}$. Can one define a ...
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### Zorn's lemma: old friend or historical relic?

It is often said that instead of proving a great theorem a mathematician's fondest dream is to prove a great lemma. Something like Kőnig's tree lemma, or Yoneda's lemma, or really anything from this ...
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### Cofinal well-founded subset in mod finite order

The mod finite order on ${}^\omega \omega$ is defined as $f \leq^\ast g$ if and only if $f(n) \leq g(n)$ except for finitely many $n \in \omega$. My question is: can we always extract a cofinal well-...
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The starting point of this question is the observation that if $\lambda$ is a countable ordinal, then there is an order-embedding $e:\lambda \hookrightarrow \mathbb{Q}$. Given an infinite cardinal $\... 8 votes 1 answer 313 views ### First inaccessible Suslin trees in L, an interesting detail It's known (but quite nontrivial) that$V=L$implies that if$\kappa$is the 1st inaccessible cardinal then there are$\kappa$-Suslin trees$T$. Such a tree$T$can be considered as a forcing notion ... • 2,321 2 votes 0 answers 113 views ### Product-decomposition of${\cal P}(\omega)/\rm{fin}$[closed] For$A,B\in {\cal P}(\omega)$let us say that$A\simeq_{\rm{fin}} B$if both$A\setminus B$and$B\setminus A$are finite. It is easy to see that this establishes an equivalence relation on${\cal P}(\...
Let $(P,\leq)$ be a directed set with uncountable cofinality. For every element $p\in P$, we are given a finite set $c_p\subset P\smallsetminus \{p\}$ of "incompatible elements". We say that ...
Suppose $\langle L,\leq\rangle$ is a lattice with join $\sqcup$. Let $F_1$ and $F_2$ be principal filters on $L$. Thus, for $i\in I=\{1,2\}$ there are $x_i\in L$ so that $F_i=\{y\in L:x_i\leq y\}$. In ...