Related papers: Germs in a poset

Let $G$ be an acylic directed graph. For each vertex $g \in G$, we define an involution on the independent sets of $G$. We call these involutions flips, and use them to define a new partial order on independent sets of $G$. Trim lattices…

We study the lattice of finite-index extensions of a given finitely generated subgroup $H$ of a free group $F$. This lattice is finite and we give a combinatorial characterization of its greatest element, which is the commensurator of $H$.…

A poset is called upper homogeneous, or "upho," if every principal order filter of the poset is isomorphic to the whole poset. We study (finite type $\mathbb{N}$-graded) upho lattices, with an eye towards their classification. Any upho…

We prove a version of the fundamental theorems of Morse Theory in the setting of finite spaces or partially ordered sets. By using these results we extend Forman's discrete Morse theory to more general cell complexes and derive the…

For a finite poset (partially ordered set) $U$ and a natural number $n$, let Sp$(U,n)$ denote the largest number of pairwise unrelated copies of $U$ in the powerset lattice (AKA subset lattice) of an $n$-element set. If $U$ is the singleton…

We construct an invariant of the bi-Lipschitz contact equivalence of continuous function germs definable in a polynomially bounded o-minimal structure, such as semialgebraic functions. For a germ $f,$ the invariant is given in terms of the…

The fourth listed author and Hans Parshall (\cite{IosevichParshall}) proved that if $E \subset {\mathbb F}_q^d$, $d \ge 2$, and $G$ is a connected graph on $k+1$ vertices such that the largest degree of any vertex is $m$, then if $|E| \ge C…

The formal class of a germ of diffeomorphism $\phi$ is embeddable in a flow if $\phi$ is formally conjugated to the exponential of a germ of vector field. We prove that there are complex analytic unipotent germs of diffeomorphisms at…

A poset is called upper homogeneous, or "upho," if all of its principal order filters are isomorphic to the whole poset. In previous work of the first author, it was shown that each (finite-type N-graded) upho lattice has associated to it a…

Motivated by the Hofmann--Lawson theorem, which states that every continuous lattice is inf-generated by its irreducible elements, we explore how to represent posets by extreme points with respect to a closure operator. For this purpose, we…

Motivated by quite recent research involving the relationship between the dimension of a poset and graph-theoretic properties of its cover graph, we show that for every $d\geq 1$, if $P$ is a poset and the dimension of a subposet $B$ of $P$…

An explanation is given for the initially surprising ubiquity of separating sets in normal complex surface germs. It is shown that they are quite common in higher dimensions too. The relationship between separating sets and the geometry of…

The number of embeddings of a partially ordered set $S$ in a partially ordered set $T$ is the number of subposets of $T$ isomorphic to $S$. If both, $S$ and $T$, have only one unique maximal element, we define good embeddings as those in…

We prove a Fundamental Theorem of Finite Semidistributive Lattices (FTFSDL), modelled on Birkhoff's Fundamental Theorem of Finite Distributive Lattices. Our FTFSDL is of the form "A poset L is a finite semidistributive lattice if and only…

Let $m$ and $n$ be cardinals with $3\leq m,n\leq\omega$. We show that the class of posets that can be embedded into a distributive lattice via a map preserving all existing meets and joins with cardinalities strictly less than $m$ and $n$…

We introduce the notion of wide representation of an inverse semigroup and prove that with a suitably defined topology there is a space of germs of such a representation which has the structure of an etale groupoid. This gives an elegant…

We study a certain poset on the free monoid on a countable alphabet. This poset is determined by the fact that its total extensions are precisely the standard term orders. We also investigate the poset classifying degree-compatible standard…

We study contact posets and show that every contact poset can be embedded into a Boolean poset with overlap contact relation. Contact posets and (nonadditive) contact semilattices have the superamalgamation property, Fra\"\i ss\'e limits…

Given a poset $P$ and a standard closure operator $\Gamma:\wp(P)\to\wp(P)$ we give a necessary and sufficient condition for the lattice of $\Gamma$-closed sets of $\wp(P)$ to be a frame in terms of the recursive construction of the…

We introduce several new constructions of finite posets with the number of linear extensions given by generalized continued fractions. We apply our results to the problem of the minimum number of elements needed for a poset with a given…