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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Eulerian posets and order complexes
To every poset $P$ it is possible to associate its order complex $\Delta(P)$. The faces of $\Delta(P)$ correspond to chains of elements in $P$. An Eulerian poset is a graded poset such that all of its ...
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Action of $V$ on the homology of a subposet of the poset of affine subspaces of $V$
Let $(V,Q)$ be a pair, with $V=\mathbb{F}_2^{2n}$ ($n\geq 2$) and $Q$ a nondegenerate quadratic form on $V.$ We consider the poset $\mathcal{P}_n$ of affine totally isotropic (with respect to $Q$) ...
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Which posets arise from closed, transitive relations?
This a follow up question of Chain components and posets.
Let $X$ be a compact metric space and $R\subset X^2$ a closed, transitive relation. Denote by $|R|=\{x\in X: xR x\}$ the diagonal of $R$.
The ...
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Approximation of Poset
Let $(X,\leq)$ be a poset, $X=\{x_1,x_2,...,x_n\}$. Preference matrix $\textbf{P}=[p_{ij}]$ (which is known and fixed), satisfies $p_{ij}=1-p_{ji},p_{ii}=\frac{1}{2}$, and
$$\forall i \neq j, x_i \leq ...
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Is a finite lattice determined by its Hasse diagram (as a graph)?
If finite lattices $L_1,L_2$ have Hasse diagrams that are isomorphic as undirected graphs, must $L_1$ and $L_2$ be isomorphic?
NOTE: Sam Hopkins points out that the answer is “no” because there are ...
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Size of antichains in powerset of $\mathbb N$
Take a countably infinite set $S$, say $\mathbb N$. Is it possible for there to be an antichain in $\mathcal P(S)$ (with the inclusion ordering) of continuum cardinality?
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Probability distribution of total time for a job, given a workflow graph
$$
\begin{array}{cccccccccccc}
& & \text{A} \\
& \swarrow & & \searrow \\
\text{B} & & & & \text{C} \\
& \searrow & & \swarrow \\
\downarrow & &...
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Non-isomorphic $T_0$-spaces with order-isomorphic topologies
Are there non-isomorphic $T_0$-spaces $(X_i, \tau_i)$ for $i = 1,2$ such that $\tau_1 \cong \tau_2$ when considered as partially ordered sets?
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When are increasing functions on posets (specifically, lattices) the CDF of a probability measure?
This is perhaps a basic question, but I couldn't find a reference. Let $P = (X,\leq)$ be a poset. Given a probability measure $\mu$ on $P$ (with respect to the Borel $\sigma$-algebra generated by sets ...
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Set-theoretic trees with ordering between siblings
In set-theoretic trees, we have the binary relation $<_T$ (which defines the "ancestor-descendant" ordering). This relation is partial, as children are incomparable in that ordering.
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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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