38 research outputs found
The variety generated by order algebras
Every ordered set can be considered as an algebra in a natural way. We investigate the variety generated by order algebras. We prove, among other things, that this variety is not finitely based and, although locally finite, it is not contained in any finitely generated variety; we describe the bottom of the lattice of its subvarieties
Promotion and Rowmotion
We present an equivariant bijection between two actions--promotion and
rowmotion--on order ideals in certain posets. This bijection simultaneously
generalizes a result of R. Stanley concerning promotion on the linear
extensions of two disjoint chains and recent work of D. Armstrong, C. Stump,
and H. Thomas on root posets and noncrossing partitions. We apply this
bijection to several classes of posets, obtaining equivariant bijections to
various known objects under rotation. We extend the same idea to give an
equivariant bijection between alternating sign matrices under rowmotion and
under B. Wieland's gyration. Finally, we define two actions with related orders
on alternating sign matrices and totally symmetric self-complementary plane
partitions.Comment: 25 pages, 22 figures; final versio
WQO is decidable for factorial languages
A language is factorial if it is closed under taking factors, i.e. contiguous subwords. Every factorial language can be described by an antidictionary, i.e. a minimal set of forbidden factors. We show that the problem of deciding whether a factorial language given by a finite antidictionary is well-quasi-ordered under the factor containment relation can be solved in polynomial time. We also discuss possible ways to extend our solution to permutations and graphs
Combinatorial Properties of Finite Models
We study countable embedding-universal and homomorphism-universal structures
and unify results related to both of these notions. We show that many universal
and ultrahomogeneous structures allow a concise description (called here a
finite presentation). Extending classical work of Rado (for the random graph),
we find a finite presentation for each of the following classes: homogeneous
undirected graphs, homogeneous tournaments and homogeneous partially ordered
sets. We also give a finite presentation of the rational Urysohn metric space
and some homogeneous directed graphs.
We survey well known structures that are finitely presented. We focus on
structures endowed with natural partial orders and prove their universality.
These partial orders include partial orders on sets of words, partial orders
formed by geometric objects, grammars, polynomials and homomorphism orders for
various combinatorial objects.
We give a new combinatorial proof of the existence of embedding-universal
objects for homomorphism-defined classes of structures. This relates countable
embedding-universal structures to homomorphism dualities (finite
homomorphism-universal structures) and Urysohn metric spaces. Our explicit
construction also allows us to show several properties of these structures.Comment: PhD thesis, unofficial version (missing apple font
On Sabidussi-Fawcett subdirect representation
AbstractSabidussi's representation theorem for symmetric graphs is generalized to fairly general concrete categories. As applications, the lists of the irreducible objects in several cases (for instance, symmetric or directed graphs with or without loops, n-partite graphs, posets) are presented
Ramsey classes and partial orders
We consider the classes of finite coloured partial orders, i.e., partial orders together with unary relations determining the colour of their points. These classes are the ages of the countable homogeneous coloured partial orders, classified by Torrez茫o de Souza and Truss in 2008. We prove that certain classes can be expanded with an order to become Ramsey classes with the ordering property. The motivation for finding such classes is the 2005 paper of Kechris, Pestov and Todorcevic, showing that these concepts are important in topological dynamics for calculating universal minimal flow of automorphism groups of homogeneous structures and finding new examples of extremely amenable groups.
We introduce the elementary skeletons to enumerate the classes of ordered shaped partial orders and show that classes are Ramsey using three main approaches. With the Blowup Lemma we use the known results about the Ramsey classes of ordered partial orders, to prove results about shaped classes. We use the Structural Product Ramsey Lemma to show that a class K is Ramsey when structures in classes known to be Ramsey determine each structure in K uniquely. Finally, we use the Two Pass Lemma when each structure in the considered class has two dimensions that can be built separately and the classes corresponding to both dimensions of the structure are known to be Ramsey. We then show that the classes of unordered reducts of the structures in the classes enumerated by elementary skeletons are the Fra茂ss茅 limits of the countable homogeneous coloured partial orders