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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298 views

Is the Poset of Graphs Automorphism-free?

For $n\geq 5$, let $\mathcal {P}_n$ be the set of all isomorphism classes of graphs with n vertices. Give this set the poset structure given by $G \le H$ if and only if $G$ is a subgraph of $H$. ...
11
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391 views

Asymptotic growth of antichains in divisibility posets

The following question is inspired by a problem that Erdős used to ask epsilons. It asks to prove that if one chooses a subset of $\lbrace 1,\dots,n\rbrace$ with more than ...
9
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109 views

A family of posets

Consider the family of all (finite) posets that can be obtained by repeatedly applying one of the following three operations (starting e.g. with the empty poset): (O1) Disjoint union of one or more ...
8
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71 views

Ideals in strong Bruhat order

Recall that a (lower) ideal in a poset $(P, \le)$ is a subset $I\subset P$ such that $x\in I \Rightarrow y\in I$ for all $y\le x$. In am interested in ideals in finite Coxeter groups $W$ equipped ...
8
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148 views

Fixed points and universal maps for posets

In a recent post about f.p.p. for poset products, @M.Mirabi brought back an old-standing problem about the fixed point property of the product of two arbitrary posets which already enjoy the fixed ...
7
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222 views

Pattern Avoidance in Poset Permutations

I am not sure if it is appropriate to use MathOverflow to publicize a conjecture, but I think this is an interesting question and I have no real ideas of how to solve it. A permutation on a ...
7
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161 views

Strength of claims about extensions of partial preorders and orders to linear ones

Consider these two axioms: Every partial order extends to a linear order. Every partial preorder (reflexive and transitive relation) extends to a linear preorder while preserving strict orderings: ...
7
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469 views

The name for a partial order

In a recent paper of mine, my co-authors found that a partial order that we were using was contained in a paper by Kundgen. In it, he called it "right-shifted partial order". I was curious, and found ...
6
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159 views

“Double convolution” with the Mobius function on a poset

Let $f$ and $g$ be arbitrary (say integer-valued) functions on some poset $P$, and say $\mu$ is the Mobius function of $P$. I'm studying a quantity that's a sort of "double convolution" of $f$ and $g$ ...
5
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71 views

Minimal Hausdorff topologies compatible with a bunch of functions

Let $X$ be an infinite set, let ${\cal F}$ be a set of functions $f: X\to X$. We say that a topology $\tau$ is compatible with ${\cal F}$ if every $f\in {\cal F}$ is a continuous function $f:(X, ...
4
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75 views

Face structures of chain polytopes

For a finite poset $P$ the chain polytope $\mathscr C(P)\subset\mathbb{R}^P$ consists of such $g$ that $g(p)\ge 0$ for all $p\in P$ and $$g(p_1)+\ldots+g(p_n)\le 1$$ for any chain ...
4
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172 views

When can we determine an $f$-vector or rank-generating function from its unordered list of coefficients?

Let $f_i$ be the number of $i$-dimensional faces in a $d$-dimensional simplicial complex $\Delta $. Recall that the $f$-vector of $\Delta $ is the vector $(f_{-1},f_0,f_1,\dots ,f_d)$ where ...
3
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81 views

Weighted maximal number of disjoint singly-generated ideals in the divisibility poset for $\{1,2,\ldots,n\}$

In the mathoverflow question here the asymptotic growth of antichains in the divisibility poset ${\cal P}_n$ of the set of natural numbers $\{1,\ldots,n\}$ is considered. I have a somewhat dual ...
3
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140 views

references for properties/examples of breadth in (semi)lattices

This is in some sense following up on my earlier question and the answer given by NN. I am currently revising the paper which used the condition mentioned in my question. It was pointed out in NN's ...
3
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387 views

a poset with small “cycles”

(a followup to this recent question) I noticed the following curious property of a poset (which I strongly believe to be a lattice, I'm still trying to prove that...): Suppose that $z$ is covered by ...
2
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43 views

Decidability of first order theory of subclasses of posets

Is the first order theory of finite posets known to be undecidable? Does anyone know a survey about such results?
2
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55 views

Computing basis of a lower set given basis of complementary upper set

In a poset $P$, $U\subseteq P$ is an upper set when for all $x\in U$, we have $y\ge x$ implies $y\in U$. Any subset of $P$ generates an upper set, and the basis of an upper set $U$ is the smallest ...
2
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125 views

non-degenarete tools to calculate a derived functor on a model category which is a poset?

Are there theorems (esp. computational tools) on model categories which survive and do not trivialise when its underlying category is a (quasi-)poset ? Are there tools that may help to calculate ...
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27 views

Form of binary function over poset that is monotone over first and antitone over second argument

if I have a partially ordered set $P$, and I have a function $f: P \times P \to \mathbb{R}$ that is monotone over the first and antitone over the second argument, i.e. for any $a,b,c \in P$ $a ≤ b ...
1
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71 views

reconstructing a linear order corrupted by noise

Suppose we have a partial order (efficiently computable), $\leq$, on $\mathbb{R}^n$, a set $S \subset \mathbb{R}^n$, and let $\rho$ be the standard Euclidean metric. We want to find a set $S^\prime = ...
1
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84 views

Suprema and infima in spaces ordered by non-normal cones

Background We shall call a subset $V_+ \subseteq V$ of a Banach space $V$ a cone if $V_+$ is closed, $\alpha V_+ \subseteq V_+$ for all $\alpha \geqslant 0$, and $V_+ \cap (-V_+) = \{0\}$. Cones ...
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98 views

Rank generating functions on a graded poset and a linear extension of it

Let $A$ be a possibly infinite ensemble and let $\leq$ be a partial order between elements of $A$. We denote $P_{\leq}=(A,\leq)$ the corresponding poset. Furthermore, we suppose that $P$ is graded, ...
0
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65 views

An equality for the trace of the join of a non-degenerate indecomposable system of projections in a finite factor

Let $M \subset B(H)$ be a finite factor (see for example here p2, or there) with a trace $tr$. The subset of projections of $M$ is naturally a lattice, noted $(\mathcal{P}(M), \wedge, \vee)$. A ...
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66 views

Is the order convergence topology on a poset always Hausdorff?

In this post two topologies on a poset $(P,\leq)$ were defined: the interval topology $\tau_i(P)$ and the order convergence topology $\tau_o(P)$. It turns out that $\tau_i(P)$ is always $T_1$ and ...
0
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107 views

A search for optimal order ideals

At the behest of Gerhard Paseman I'll describe the problem that I alluded to in name for a partial order. Let $M = M(\infty)$ denote the set of all finite subsets of the positive integers ...
0
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176 views

Ordering labellings of a fixed poset.

Let $\{A_1,\ldots, A_m\}$ be a family of sets and $I=\{1, \ldots, m\}$. Assume for any $J\subset I$, $B_J=\bigcap_{i\in J}A_j$ satisfies $1\leq |B_J| \leq m-1$ as long as $|J|>1$. We define a ...