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# 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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### Covering a poset by minimum number of chains and antichains

Covering a poset by minmum number of chains is given by Dilworth's theorem and covering a poset by minimum number of antichains is given by Mirsky's theorem. I was wondering what happens if we allow ...
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### Posets with cardinality bounds on upward-closed subsets

Let $(P,\leq)$ be a finite poset that contains a (global) minimal element $0$ and a (global) maximal element $1$. We say that a subset $U \subset P$ is upward closed if $x \in U$ and $y \geq x$ forces ...
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### Smallest ordinal $\mu$ not embeddable in ${\cal P}(\omega)$

The motivation for this question is the startling fact that there is an order-preserving injective map (embedding) from $\mathbb{R}$ into ${\cal P}(\omega)$. (Think Dedekind cuts.) I am wondering how &...
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### 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 ...
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### Which spectra arise from partially ordered commutative monoids?

Thomason showed how any connective spectrum arises from a symmetric monoidal category: Robert W. Thomason, Symmetric monoidal categories model all connective spectra, Theory Appl. Categ. 1 (1995), 78–...
1 vote
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### Birkhoff's representation theorem vs matroid-geometric lattice correspondence

This question is motivated by the superficial observation that Birkhoff's representation theorem and the cryptomorphism between matroids and geometric lattices are sort of similar. The former says ...
1 vote
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### Digraph without "immediately isomorphic" vertices?

Say that a digraph $(V,E)$ is reducible if there exist $x,y\in V$ with $x\ne y$ and such that for all $z\in V$, $(x,z)\in E\leftrightarrow(y,z)\in E$ and $(z,x)\in E\leftrightarrow(z,y)\in E$. It is ...
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### Matrices of combinatorial sequences that are inverse in two ways

I'm interested in pairs $A=(a_{i,j})_{i,j=0,1,\ldots}$ and $B=(b_{i,j})_{i,j=0,1,\ldots}$ of infinite matrices for which: They are uni-lower-triangular, i.e., $a_{i,i}=1$ for all $i$ and $a_{i,j}=0$ ...
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### Classification of multiplicative lattices

Question 1:Is there a classification of finite lattices which admit a multiplication making them into a finite multiplicative lattices? (see https://encyclopediaofmath.org/wiki/Multiplicative_lattice ...
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### Order ideals of positive root systems and avoiding group elements in the Weyl group

Let $X$ be the poset of positive roots of a finite root system of Dynkin type $Q$. Question 1: In Dynkin type $A_n$, is it true that the poset of order ideals of $X$ is isomorphic to the poset of [2,...
1 vote
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### Is $(\omega+1)^\omega/{\cal U}$ "unique"?

If ${\cal U}_i$ free ultrafilters on $\omega$ for $i = 1,2$ , are the ultrapowers $(\omega+1)^\omega/{\cal U}_i$ necessarily isomorphic as lattices for $i = 1,2$?
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### Which finite posets are Koszul self-dual?

Let $P$ be a finite connected poset with incidence algebra $A_P$. For the definition and results on Koszul algebras for incidence algebras, see for example here Question: Which posets have the ...
Fix a finitely generated Coxeter system $(W, S)$, and let $W_J$ denote the standard parabolic proper subgroup generated by a subset $J \subset S$. It is well known that the poset of cosets $\{xW_J\}$ ...