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 $x$) and transitive ($x\leq y$ and $y\leq z$ implies $x \leq z$).

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234 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 $H$. ...
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352 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 ...
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143 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: ...
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409 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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86 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 ...
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138 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 $g$ ...
4
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159 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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127 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 ...
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354 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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48 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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61 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, ...
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122 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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66 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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105 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 ...
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173 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 ...