Decidability of well quasi-order and atomicity for equivalence relations under embedding orderings

We consider the posets of equivalence relations on finite sets under the standard embedding ordering and under the consecutive embedding ordering. In the latter case, the relations are also assumed to have an underlying linear order, which governs consecutive embeddings. For each poset we ask the well quasi-order and atomicity decidability questions: Given finitely many equivalence relations $\rho_1,\dots,\rho_k$, is the downward closed set Av$(\rho_1,\dots,\rho_k)$ consisting of all equivalence relations which do not contain any of $\rho_1,\dots,\rho_k$: (a) well-quasi-ordered, meaning that it contains no infinite antichains? and (b) atomic, meaning that it is not a union of two proper downward closed subsets, or, equivalently, that it satisfies the joint embedding property

Decidability of well quasi-order and atomicity for equivalence relations under embedding orderings

We consider the posets of equivalence relations on finite sets under the standard embedding ordering and under the consecutive embedding ordering. In the latter case, the relations are also assumed to have an underlying linear order, which governs consecutive embeddings. For each poset we ask the well quasi-order and atomicity decidability questions: Given finitely many equivalence relations ρ1,...,ρk, is the downward closed set Av(ρ1,...,ρk) consisting of all equivalence relations which do not contain any of ρ1,...,ρk (a) well-quasi-ordered, meaning that it contains no infinite antichains? and (b) atomic, meaning that it is not a union of two proper downward closed subsets, or, equivalently, that it satisfies the joint embedding property?Peer reviewe

Orderings on words and permutations

Substructure orderings are ubiquitous throughout combinatorics and all of mathematics. In this thesis we consider various orderings on words, as well as the consecutive involvement ordering on permutations. Throughout there will be a focus on deciding certain order-theoretic properties, primarily the properties of being well-quasi-ordered (WQO) and of being atomic. In Chapter 1, we establish the background material required for the remainder of the thesis. This will include concepts from order theory, formal language theory, automata theory, and the theory of permutations. We also introduce various orderings on words, and the consecutive involvement ordering on permutations. In Chapter 2, we consider the prefix, suffix and factor orderings on words. For the prefix and suffix orderings, we give a characterisation of the regular languages which are WQO, and of those which are atomic. We then consider the factor ordering and show that the atomicity is decidable for finitely-based sets. We also give a new proof that WQO is decidable for finitely-based sets, which is a special case of a result of Atminas et al. In Chapters 3 and 4, we consider some general families of orderings on words. In Chapter 3 we consider orderings on words which are rational, meaning that they can be generated by transducers. We discuss the class of insertion relations introduced in a paper by the author, and introduce a generalisation. In Chapter 4, we consider three other variations of orderings on words. Throughout these chapters we prove various decidability results. In Chapter 5, we consider the consecutive involvement on permutations. We generalise our results for the factor ordering on words to show that WQO and atomicity are decidable. Through this investigation we answer some questions which have been asked (and remain open) for the involvement on permutations

On well quasi-order of graph classes under homomorphic image orderings

In this paper we consider the question of well quasi-order for classes defined by a single obstruction within the classes of all graphs, digraphs and tournaments, under the homomorphic image ordering (in both its standard and strong forms). The homomorphic image ordering was introduced by the authors in a previous paper and corresponds to the existence of a surjective homomorphism between two structures. We obtain complete characterisations in all cases except for graphs under the strong ordering, where some open questions remain.PostprintPeer reviewe

Decidability and Complexity in Weakening and Contraction Hypersequent Substructural Logics

We establish decidability for the infinitely many axiomatic extensions of the commutative Full Lambek logic with weakening FLew (i.e. IMALLW) that have a cut-free hypersequent proof calculus. Specifically: every analytic structural rule exten- sion of HFLew. Decidability for the corresponding extensions of its contraction counterpart FLec was established recently but their computational complexity was left unanswered. In the second part of this paper, we introduce just enough on length functions for well-quasi-orderings and the fast-growing complexity classes to obtain complexity upper bounds for both the weakening and contraction extensions. A specific instance of this result yields the first complexity bound for the prominent fuzzy logic MTL (monoidal t-norm based logic) providing an answer to a long- standing open problem

On the least exponential growth admitting uncountably many closed permutation classes

We show that the least exponential growth of counting functions which admits uncountably many closed permutation classes lies between 2^n and (2.33529...)^n.Comment: 13 page