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 $x \leq z$).
449 questions
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Non-topological argument for the non-existence of an order-embedding $\iota: [0,1]\to ([0,1]\setminus \mathbb{Q})$
Starting point. I was toying around with the following question: If $A, B\subseteq [0,1]$ with $A\cup B = [0,1]$, does $[0,1]$ order-embed into at least one of $A, B$? Quickly I focused on $A = [0,1]\...
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Find a minimal cutset in this width 2 poset
A chain $C$ in a poset $P$ with partial ordering $\le$ is a subset such that for all $c,d\in C$, either $c\le d$ or $d\le c$. A chain is maximal if it is not a proper subset of any other chain.
A ...
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Is this very short proof of Zorn’s Lemma correct? [closed]
I recently came across a recent paper presenting a proof of Zorn’s Lemma that seems very short and elementary. I found it interesting because the proof does not use transfinite induction or any set ...
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Posets of quotients in category of rings
Are there some general theorems which govern global structure of posets arising as posets of epimorphisms out of a (unital, associative) ring?
One may note that it is always a lattice — it is ...
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Additively idempotent semirings that are not lattices
I am looking for examples of additively idempotent semirings (which are always join semilattices) that do not have an underlying lattice structure, i.e. either meets do not exist or exist outside the ...
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Question on extending submodularity inequalities in lattices
On a lattice $\mathcal{L}$, I have a submodular, monotone, real-valued function $\rho$.
Submodularity means it satisfies the inequality $$\rho(x)+\rho(y)\ge \rho(x\wedge y)+\rho(x\vee y)$$ for all $x, ...
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Lattice from a commutative semigroup
Consider any finite commutative semigroup $S$.
Say that $x \leq y$ iff $x = y$ or $xy = y$. This is a partial order on $S$: the only nontrivial property to check is that $x \leq y$ and $y \leq z$ ...
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Minimal cutsets containing no maximal antichain
If $(P,\leq)$ is a poset, we say that $C\subseteq P$ is a chain if $a\leq b$ or $b\leq a$ for all $a, b\in C$. Moreover, $A \subseteq P$ is an antichain if $a\not\leq b$ and $b\not\leq a$ whenever $a,...
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Possible cardinalities of maximal chains in ${\cal P}(\omega)$
Let ${\cal P}(\omega)$ denote the power-set of $\omega$. We order it by set inclusion $\subseteq$ and say that ${\cal C}\subseteq {\cal P}(\omega)$ is a chain if for all $A, B\in {\cal C}$ we have $A\...
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Is every directed graph the quotient of poset where boundary nodes are identified?
Let $k\in \mathbb{N}_+$, let $\mathcal{P}_k$ denote the set of directed graphs obtained as Hasse diagrams of posets on $k$ vertices, and let $\mathcal{Dir}_k$ denote the set of connected directed ...
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Classification of finite ordered semigroups
Question 1: Is there a classification of finite ordered semigroups (or monoids) with $n$ elements for small $n$? Is there computer algebra software that can generate those algebraic objects?
(Here I ...
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If $\tilde P$ is the poset $P$ with every element replaced by two, and $P^P\cong Q^Q$, must ${\tilde P}^{\tilde P}\cong{\tilde Q}^{\tilde Q}$?
For a poset $P$, let $\widetilde P$ be the poset with each element replaced by two twins: to be precise, $\widetilde P:=\{p_0,p_1\mid p\in P\}$ where for $p\in P$, $p_0$ and $p_1$ are incomparable in $...
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If $A$, $B$, and $C$ are bounded posets and $C$ satisfies ACC and DCC and $A^C\cong B^C$, is $A\cong B$?
For a posets $P$ and $Q$ with partial orderings $\le_P$ and $\le_Q$, a function $f:P\to Q$ is order-preserving if for all $p,p'\in P$, $p\le_P p'$ implies $f(p)\le_Q f(p')$.
The posets are order-...
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What is the Koszul dual of the incidence algebra of a free distributive lattice?
Let $L_n$ be the free distributive lattice on $n$-elements, which can be defined for example as the distributive lattice of order ideals of the Boolean lattice an $n$-set.
The incidence algebra $A_n=...
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A rank-selected generalization of a theorem of Kulakoff on $p$-groups
It is an elementary fact, due to P. Hall in 1933, that the number of subgroups of order $k$ of
a $p$-group of order $p^n$, $0\leq k\leq n$, is congruent to $1$
modulo $p$. A. Kulakoff, Math. Ann. 104 (...
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Are all topological trees contractible?
Say a Hausdorff space $X$ is a topological tree if
$X$ is uniquely arcwise connected,
$X$ is locally arcwise connected,
$X$ does not contain a copy of the long line.
Question: Is every topological ...
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Unique factorisation property in posets
Let $L$ be a finite lattice.
Say $L$ has the unique factorisation property, if every element $x$ of $L$ is the join of a unique antichain of join-irreducible elements (or maybe one should use join-...
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Is the category of Posets Locally Cartesian Closed?
I'm wondering if $\mathcal{Pos}$, the category of posets, with arrows being monotone mappings between them, is Locally Cartesian Closed?
If so, is there a reference with a proof of this?
If not, is ...
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Does a peeling sequence always exist for noncrossing perfect matchings on 2n points?
Let ${M} = \{M_1, M_2, \dots, M_N\}$ be the set of all noncrossing perfect matchings on a circle with $2n$ labeled points arranged clockwise. Then $N = \frac{1}{n+1} \binom{2n}{n}$ is the $n$-th ...
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Gorenstein algebras arising from posets
Let $L$ be a finite distributive lattice, and let $K$ be a field. Consider the binomial ideal $I_L$ in $S_L=K[x_\alpha:\alpha \in L]$ generated by $x_{\alpha}x_{\beta} -x_{\alpha \wedge \beta}x_{\...
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Spectral sequence for cohomology of inverse limit of complexes
I have the following question.
Let $\{C_i\}_{i\in I}$ be an inverse system of complexes with members in some abelian category $\mathcal{A}$ where all small limits exist + some technical conditions. ...
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Is every multiplicative lattice isomorphic to the lattice of ideals of some ring?
A multiplicative lattice is a complete lattice
$(L, \leq)$ that is
endowed with an associative, commutative multiplication that distributes
over arbitrary joins and has $1$, the top element of $L$,
as ...
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4
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answer
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Functors from the Nerve of posets to $\infty$-categories
In the corollary 3.4 of Ariotta's paper, it's been mentioned that to define a functor from $\mathbb{Z}$ to an $\infty$-category $\mathcal{C}$, it is sufficient to specify the functor on objects and ...
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Every poset is isomorphic to a collection of sets ordered with inclusion, and related
Given a (finite) poset $(P,\leq)$, we can construct the poset $(\mathcal{D}, \subseteq)$ isomorphic to $P$ simply by letting $\mathcal{D}$ be the collection of all "descendant sets" i.e. $\...
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Example of a certain lattice with $q$ coatoms and less than $4q+5$ elements
I would like to find an example of a lattice $L$ with the following properties:
it has $q$ coatoms $x_i$, $1 \le i \le q$;
for each coatom $x_i$ let $Y_i = \{y \in L :y \le x_i\} = (x_i]$: it is ...
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Looking for a finite lattice example
Consider a finite lattice $L$ such that each atom has at least $|L|/2$ elements greater than or equal to it. It can be for example a boolean lattice or the following lattice:
In the boolean lattice ...
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A well-quasi-ordering of finite interval orders?
An interval order (always assumed to be finite) is a finite poset
$P$ that is isomorphic to a set $\{[a_1,b_1],\dots,[a_n,b_n]\}$ of
closed intervals on the real line with the ordering
$[a_i,b_i]<[...
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Explicit perfect matching on adjacent levels of the Boolean poset
The post is motivated by the following exercise in Edition 5 of Miklos Bona's "A walk through combinatorics".
Exercise 11.28
Let $n>2k$ be positive integers, and let $\binom{[n]}{k}$ ...
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Adjointness of face poset and order complex constructions?
Apologies for the somewhat vague question.
Given an abstract simplicial complex $\Delta$, I can create a poset $P$ from $\Delta$: the face poset, where we order the faces of $\Delta$ by containment.
...
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Combinatorial interpretation for coefficients in Putnam 2024 B5
Fix integers $k, m \geq 0$. For each $r=0,1,\ldots,k+m+1$, define
$$ a_{k,m,r} = \sum_{j=0}^{\min(k,m)} \binom{k}{j}\binom{m}{j}\sum_{\substack{S \subseteq \{-j,-j+1,\ldots,k+m-j\},\\\#S=k+m+1-r}}\...
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Poset terminology: heads of maximal chains containing $x$
Let $x$ be an element in a complete partial order. What's the accepted terminology, if any, for the following?
Any $y\ge x$ such there does not exist any $z\ge y$. A greatest upper bound for $x$, ...
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What is the trace of this "double simplex" category?
Lately I've been trying to gather more examples of centres and traces, hoping to write a comprehensive treatment on these on Clowder.
Another example I've been trying to understand is the following.
...
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Is the interval topology on ${\cal P}(\omega)/(\text{fin})$ connected?
If $(P,\leq)$ is a poset and $x\in X$, we let $\downarrow x = \{p\in P: p \leq x\}$, and $\uparrow x$ is defined dually. The collection $$\Big\{P\setminus (\downarrow x): x\in P\Big\} \cup \Big\{P\...
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What is the trace of the category of divisibility posets?
$\newcommand{\Tr}{\mathrm{Tr}}$Lately I've been trying to gather more examples of centres and traces, hoping to write a comprehensive treatment on those on Clowder.
One of the examples I've been ...
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Partition induced by a cover
Let $X$ be a set and let $(Y_i)_{i \in I}$ be a family of (not necessarily pairwise disjoint) subsets covering $X$,
$$ X = \bigcup_{i\in I} Y_i.$$
For any subset $J \subseteq I$, we then define
$$ Y_J ...
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Has this property of well-founded posets appeared in the literature?
Let $(P,\leq)$ be a partially ordered set. Assume that $(P,\leq)$ is well-founded. Then the levels of the poset are well-defined: $L_{0}$ is the set of minimal elements in $P$, and whenever $\mu$ is ...
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If $P$ and $Q$ are finite, non-empty posets and the poset of order-preserving maps $P^P$ is isomorphic to $Q^Q$, must $P$ be isomorphic to $Q$?
Let $(P,\le_P)$ and $(Q,\le_Q)$ be posets. A function $f:Q\to P$ is order-preserving if whenever $q,q'\in Q$ and $q\le_Q q'$, we have $f(q)\le_P f(q')$. If there is a bijective order-preserving map ...
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Remove an edge from the Hasse diagram of a finite lattice
If we remove an edge from the Hasse diagram of a finite lattice, as long as any vertex maintain at least one upward edge and at least one downward edge, do we still always have a lattice from the ...
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A strengthening of an inequality for posets by Chan-Pak
Suppose that $P$ is a poset, $x$ and $y$ are two minimal elements of $P$, and that $e(P)$ denotes the number of linear extensions of $P$. Chan and Pak use their recent combinatorial atlas technology ...
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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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1
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Approximation of poset
Let $(X,\leq)$ be a poset, $X=\{x_1,x_2,\dots,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 \...
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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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2
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1
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513
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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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93
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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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3
votes
1
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156
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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?
- 54.4k
3
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1
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201
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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 ...
- 568
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0
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98
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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.
...
2
votes
1
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173
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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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