Related papers: Coset Posets of Infinite Groups
We introduce decomposition complexes of posets, which generalize order complexes. The main advantage of our construction is that decomposition complexes are closed under taking products. Other special instances of this theory include nested…
We introduce a new notion for geometric families called self-coverability and show that homothets of convex polygons are self-coverable. As a corollary, we obtain several results about coloring point sets such that any member of the family…
We introduce partially ordered sets (posets) with an additional structure given by a collection of vector subspaces of an algebra $A$. We call them algebraically equipped posets. Some particular cases of these, are generalized equipped…
We prove that every finite group $G$ can be realized as the automorphism group of a poset with $4|G|$ points. We also provide bounds for the minimum number of points of a poset with cyclic automorphism group of a given prime power order.
We determine the finite groups whose real irreducible representations have different degrees.
Various characterizations of finite convex geometries are well known. This note provides similar characterizations for possibly infinite convex geometries whose lattice of closed sets is strongly coatomic and lower continuous. Some classes…
The coefficients of the chain polynomial of a finite poset enumerate chains in the poset by their number of elements. It has been a challenging open problem to determine which posets have real-rooted chain polynomials. Two new classes of…
In this article, we investigate the existence of joins in the weak order of an infinite Coxeter group W. We give a geometric characterization of the existence of a join for a subset X in W in terms of the inversion sets of its elements and…
Given an infinite group G, we consider the finitely additive measure defined on finite unions of cosets of finite index subgroups. We show that this shares many properties with the size of subsets of a finite group, for instance we can…
Interval-closed sets of a poset are a natural superset of order ideals. We initiate the study of interval-closed sets of finite posets from enumerative and dynamical perspectives. In particular, we use the generalized toggle group to define…
Given a symmetric monoidal category $C$ with product $\sqcup$, where the neutral element for the product is an initial object, we consider the poset of $\sqcup$-complemented subobjects of a given object $X$. When this poset has finite…
We study the connectivity of proper power graphs of some family of finite groups including nilpotent groups, groups with a non-trivial partition, and symmetric and alternating groups.
In this paper, we explore combinatorial properties of the posets associated with Kohnert polynomials. In particular, we determine a sufficient condition guaranteeing when such ``Kohnert posets'' are bounded and two necessary conditions for…
We consider a category of all finite partial orderings with quotient maps as arrows and construct a Fra\"iss\'e sequence in this category. Then we use commonly known relations between partial orders and lattices to construct a sequence of…
We provide a combinatorial recipe for constructing all posets of height at most two for which the corresponding type-A Lie poset algebra is contact. In the case that such posets are connected, a discrete Morse theory argument establishes…
We study the realization problem of finite groups as the group of homotopy classes of self-homotopy equivalences of finite spaces. Let $G$ be a finite group. Using an infinite family of pairwise non weakly homotopic asymmetric spaces we…
We report on recent progress concerning the relationship that exists between the algebraic structure of a finite group and certain features of its class-size prime graph.
We develop a theory of limits of finite posets in close analogy to the recent theory of graph limits. In particular, we study representations of the limits by functions of two variables on a probability space, and connections to…
We introduce Cayley posets as posets arising naturally from pairs $S<T$ of semigroups, much in the same way that Cayley graph arises from a (semi)group and a subset. We show that Cayley posets are a common generalization of several known…
Given a Polish space X and a countable family of analytic hypergraphs on X, I consider the sigma-ideal generated by Borel sets which are anticliques in at least one hypergraph in the family. It turns out that many of the quotient posets are…