Related papers: The $m$-Cover Posets and Their Applications
Given any polytope $P$ and any generic linear functional ${\bf c} $, one obtains a directed graph $G(P,{\bf c})$ from the 1-skeleton of $P$ by orienting each edge $e(u,v)$ from $u$ to $v$ for ${\bf c} (u) < {\bf c} ( v)$. For $P$ a simple…
A poset is a containment of paths in a tree (CPT) if it admits a representation by containment where each element of the poset is represented by a path in a tree and two elements are comparable in the poset if and only if the corresponding…
Gr\"atzer asked in 1971 for a characterization of sublattices of Tamari lattices (associahedra). A natural candidate was coined by McKenzie in 1972 with the notion of a bounded homomorphic image of a free lattice---in short, bounded…
We introduce a construction called realisation which transforms posets into posets. We show that realisations share several key features with upper semilattices. For example, we define local dimensions of points in a poset and show that…
We introduce the class of lattice-linear monomial ideals and use the LCM-lattice to give an explicit construction for their minimal free resolution. The class of lattice-linear ideals includes (among others) the class of monomial ideals…
We introduce a novel combinatorial structure called pointed building sets, which can be viewed as families of lattices equipped with compatibility relations. To each pointed building set $\mathsf{B}$, we associate a complete lattice…
We introduce posets of simple vertex labeled minors of graphs and a generalization to the level of polymatroids, collectively termed minor posets. We show that any minor poset is isomorphic to the face poset of a regular CW sphere, and in…
An algebraization of the notion of topology has been proposed more than seventy years ago in a classical paper by McKinsey and Tarski. However, in McKinsey and Tarski's setting the model theoretical notion of homomorphism does not…
A classical tensor product $A \,\otimes\, B$ of complete lattices $A$ and $B$, consisting of all down-sets in $A \times B$ that are join-closed in either coordinate, is isomorphic to the complete lattice $Gal(A,B)$ of Galois maps from $A$…
I present an overview of the research I have conducted for the past ten years in algebraic, bijective, enumerative, and geometric combinatorics. The two main objects I have studied are the permutahedron and the associahedron as well as the…
We show that the well known {\em homotopy complementation formula} of Bj\"orner and Walker admits several closely related generalizations on different classes of topological posets (lattices). The utility of this technique is demonstrated…
We analyze marked poset polytopes and generalize a result due to Hibi and Li, answering whether the marked chain polytope is unimodular equivalent to the marked order polytope. Both polytopes appear naturally in the representation theory of…
Each rooted plane tree $\mathsf{T}$ has an associated ornamentation lattice $\mathcal{O}(\mathsf{T})$. The ornamentation lattice of an $n$-element chain is the $n$-th Tamari lattice. We study the pop-stack operator…
The concept of a sectionally pseudocomplemented lattice was introduced by I. Chajda as an extension of relative pseudocomplementation for not necessarily distributive lattices. The typical example of such a lattice is the non-modular…
We give new interpretations of the $\nu$-Tamari lattice of Pr\'eville-Ratelle and Viennot. First, we describe it as a rotation lattice of $\nu$-trees, which uncovers the relation with known combinatorial objects such as tree-like tableaux…
The Cambrian lattices, introduced in (Reading, 2006), generalize the Tamari lattice to any choice of Coxeter element in any finite Coxeter group. They are further generalized to the m-Cambrian lattices (Stump, Thomas, Williams, 2015).…
For $P$ a poset, the dimension of $P$ is defined to be the least cardinal $\kappa$ such that $P$ is embeddable in a direct product of $\kappa$ totally ordered sets. We study the behavior of this function on finite-dimensional (not…
A poset can be regarded as a category in which there is at most one morphism between objects, and such that at most one of Hom(c,c') and Hom(c',c) is nonempty for c not equal to c'. If we keep in place the latter axiom but allow for more…
We introduce a new poset structure on Dyck paths where the covering relation is a particular case of the relation inducing the Tamari lattice. We prove that the transitive closure of this relation endows Dyck paths with a lattice structure.…
For a lattice polytope $P$, the rank of $P$ is defined by $F-(\dim P+1)$, where $F$ is the number of facets of $P$. In this paper, we study matroid polytopes with small rank. More precisely, we characterize matroid independence polytopes…