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 Torrezão 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 Fraïssé limits of the countable homogeneous coloured partial orders