arXiv subjects
Publications and source records attributed to Dugald Macpherson.
We prove that non-abelian definable, definably simple groups in 1-h-minimal henselian valued fields are essentially already linear algebraic groups. Here, the group is assumed to live in the home sort. We have a similar result in pure algebraically closed valued fields of positive characteristic, under the additional assumption that the definable group is a subgroup of a linear algebraic group.
We explore two constructions of oligomorphic Jordan permutation groups preserving a `limit of betweenness relations' and a `limit of $D$-relations', from \cite{bhattmacph2006jordan} and \cite{almazaydeh2021jordan} respectively. Several issues left open in \cite{almazaydeh2021jordan} are resolved. In particular it is shown that the `limit of $D$-relations' is not homogeneous in the given language, but is `homogenizable', that is, there is a homogeneous structure over a finite relational language with the same universe and the same automorphism group. The structure is NIP, but not monadically NIP, its age is not well-quasi-ordered under embeddability, and the growth rate of the sequence enumerating orbits on $k$-sets grows faster than exponentially. The automorphism group is maximal-closed in the symmetric group. Similar results are shown for the construction in \cite{bhattmacph2006jordan}.
We develop a general framework (multidimensional asymptotic classes, or m.a.c.s) for handling classes of finite first order structures with a strong uniformity condition on cardinalities of definable sets: The condition asserts that definable families given by a formula \phi(x,y) should take on a fixed number n_\phi of approximate sizes in any M in the class, with those sizes varying with M. The prototype is the class of all finite fields, where the uniformity is given by a theorem of Chatzidakis, van den Dries and Macintyre. It inspired the development of asymptotic classes of finite structures, which this new framework extends. The underlying theory of m.a.c.s is developed, including preservation under bi-interpretability, and a proof that for the m.a.c. condition to hold it suffices to consider formulas \phi(x,y) with x a single variable. Many examples of m.a.c.s are given, including 2-sorted structures (F,V) where V is a vector space over a finite field F possibly equipped with a bilinear form, and an example arising from representations of quivers of finite representation type. We also give examples and structural results for multidimensional exact classes (m.e.c.s), where the definable sets take a fixed number of precisely specified cardinalities, which again vary with M. We also develop a notion of infinite generalised measurable structure, whereby definable sets are assigned values in an ordered semiring. We show that any infinite ultraproduct of a m.a.c. is generalised measurable, that values can be taken in an ordered ring if the m.a.c. is a m.e.c., and explore model-theoretic consequences of generalised measurability. Such a structure cannot have the strict order property, and stability-theoretic properties can be read off from the measures in the semiring.
We explore the interplay between omega-categoricity and pseudofiniteness for groups, conjecturing that omega-categorical pseudofinite groups are finite-by-abelian-by-finite. We show that the conjecture reduces to nilpotent p-groups of class 2, and give a proof that several of the known examples of omega-categorical p-groups satisfy the conjecture. In particular, we show by a direct counting argument that for any odd prime p the (omega-categorical) model companion of the theory of nilpotent class 2 exponent p groups, constructed by Saracino and Wood, is not pseudofinite, and that an omega-categorical group constructed by Baudisch with supersimple rank 1 theory is not pseudofinite. We also survey some scattered literature on omega-categorical groups over 50 years.
A $k$-uniform hypergraph $M$ is set-homogeneous if it is countable (possibly finite) and whenever two finite induced subhypergraphs $U,V$ are isomorphic there is $g\in Aut(M)$ with $U^g=V$; the hypergraph $M$ is said to be homogeneous if in addition every isomorphism between finite induced subhypergraphs extends to an automorphism. We give four examples of countably infinite set-homogeneous $k$-uniform hypergraphs which are not homogeneous (two with $k=3$, one with $k=4$, and one with $k=6$). Evidence is also given that these may be the only ones, up to complementation. For example, for $k=3$ there is just one countably infinite $k$-uniform hypergraph whose automorphism group is not 2-transitive, and there is none for $k=4$. We also give an example of a finite set-homogeneous 3-uniform hypergraph which is not homogeneous.
We construct via Fraïssé amalgamation an $ω$-categorical structure whose automorphism group is an infinite oligomorphic Jordan primitive permutation group preserving a `limit of $D$-relations'. The construction is based on a semilinear order whose elements are labelled by sets carrying a $D$-relation, with strong coherence conditions governing how these $D$-sets are inter-related.
VC-dimension and VC-density are measures of combinatorial complexity of set systems. VC-dimension was first introduced in the context of statistical learning theory, and is tightly related to the sample complexity in PAC learning. VC-density is a refinement of VC-dimension. Both notions are also studied in model theory, in the context of \emph{dependent} theories. A set system that is definable by a formula of first-order logic with parameters has finite VC-dimension if and only if the formula is a dependent formula. In this paper we study the VC-dimension and the VC-density of the edge relation $Exy$ on Johnson graphs and on Hamming graphs. On a graph $G$, the set system defined by the formula $Exy$ is the vertex set of $G$ along with the collection of all \emph{open neighbourhoods} of $G$. We show that the edge relation has VC-dimension at most $4$ on Johnson graphs and at most $3$ on Hamming graphs and these bounds are optimal. We furthermore show that the VC-density of the edge relation on the class of all Johnson graphs is $2$, and on the class of all Hamming graphs the VC-density is $2$ as well. Moreover, we show that our bounds on the VC-dimension carry over to the class of all induced subgraphs of Johnson graphs, and to the class of all induced subgraphs of Hamming graphs, respectively. It also follows that the VC-dimension of the set systems of \emph{closed neighbourhoods} in Johnson graphs and Hamming graphs is bounded. Johnson graphs and Hamming graphs are well known examples of distance transitive graphs. Neither of these graph classes is nowhere dense nor is there a bound on their (local) clique-width. Our results contrast this by giving evidence of structural tameness of the graph classes.
This is a survey, intended both for group theorists and model theorists, concerning the structure of pseudofinite groups, that is, infinite models of the first order theory of finite groups. The focus is on concepts from stability theory and generalisations in the context of pseudofinite groups, and on the information this might provide for finite group theory.
We consider profinite groups as 2-sorted first order structures, with a group sort, and a second sort which acts as an index set for a uniformly definable basis of neighbourhoods of the identity. It is shown that if the basis consists of {\em all} open subgroups, then the first order theory of such a structure is NIP (that is, does not have the independence property) precisely if the group has a normal subgroup of finite index which is a direct product of finitely many compact $p$-adic analytic groups, for distinct primes $p$. In fact, the condition NIP can here be weakened to NTP${}_2$. We also show that any NIP profinite group, presented as a 2-sorted structure, has an open prosoluble normal subgroup.
Answering a question of Junker and Ziegler, we construct a countable first order structure which is not omega-categorical, but does not have any proper non-trivial reducts, in either of two senses (model-theoretic, and group-theoretic). We also construct a strongly minimal set which is not omega-categorical but has no proper non-trivial reducts in the model-theoretic sense.
We explore a notion of pseudofinite dimension, introduced by Hrushovski and Wagner, on an infinite ultraproduct of finite structures. Certain conditions on pseudofinite dimension are identified that guarantee simplicity or supersimplicity of the underlying theory, and that a drop in pseudofinite dimension is equivalent to forking. Under a suitable assumption, a measure-theoretic condition is shown to be equivalent to local stability. Many examples are explored, including vector spaces over finite fields viewed as 2-sorted finite structures, and homocyclic groups. Connections are made to products of sets in finite groups, in particular to word maps, and a generalization of Tao's algebraic regularity lemma is noted.
Let G,H be closed permutation groups on an infinite set X, with H a subgroup of G. It is shown that if G and H are orbit-equivalent, that is, have the same orbits on the collection of finite subsets of X, and G is primitive but not 2-transitive, then G=H.
We show that any pseudofinite group with NIP theory and with a finite upper bound on the length of chains of centralisers is soluble-by-finite. In particular, any NIP rosy pseudofinite group is soluble-by-finite. This generalises, and shortens the proof of, an earlier result for stable pseudofinite groups. An example is given of an NIP pseudofinite group which is not soluble-by-finite. However, if C is a class of finite groups such that all infinite ultraproducts of members of C have NIP theory, then there is a bound on the index of the soluble radical of any member of C. We also survey some ways in which model theory gives information on families of finite simple groups, particularly concerning products of images of word maps.
We give an example of an imaginary defined in certain valued fields with analytic structure which cannot be coded in the `geometric' sorts which suffice to code all imaginaries in the corresponding algebraic setting.
A famous result by Jeavons, Cohen, and Gyssens shows that every constraint satisfaction problem (CSP) where the constraints are preserved by a semi-lattice operation can be solved in polynomial time. This is one of the basic facts for the so-called universal-algebraic approach to a systematic theory of tractability and hardness in finite domain constraint satisfaction. Not surprisingly, the theorem of Jeavons et al. fails for arbitrary infinite domain CSPs. Many CSPs of practical interest, though, and in particular those CSPs that are motivated by qualitative reasoning calculi from Artificial Intelligence, can be formulated with constraint languages that are rather well-behaved from a model-theoretic point of view. In particular, the automorphism group of these constraint languages tends to be large in the sense that the number of orbits of n-subsets of the automorphism group is bounded by some function in n. In this paper we present a generalization of the theorem by Jeavons et al. to infinite domain CSPs where the number of orbits of n-subsets grows sub-exponentially in n, and prove that preservation under a semi-lattice operation for such CSPs implies polynomial-time tractability. Unlike the result of Jeavons et al., this includes many CSPs that cannot be solved by Datalog.
A directed graph is set-homogeneous if, whenever U and V are isomorphic finite subdigraphs, there is an automorphism g of the digraph with U^g=V. Here, extending work of Lachlan on finite homogeneous digraphs, we classify finite set-homogeneous digraphs, where we allow some pairs of vertices to have arcs in both directions. Under the assumption that such pairs of vertices are not allowed, we obtain initial results on countably infinite set-homogeneous digraphs, classifying those which are not 2-homogeneous.
We seek to create tools for a model-theoretic analysis of types in algebraically closed valued fields (ACVF). We give evidence to show that a notion of 'domination by stable part' plays a key role. In Part A, we develop a general theory of stably dominated types, showing they enjoy an excellent independence theory, as well as a theory of definable types and germs of definable functions. In Part B, we show that the general theory applies to ACVF. Over a sufficiently rich base, we show that every type is stably dominated over its image in the value group. For invariant types over any base, stable domination coincides with a natural notion of `orthogonality to the value group'. We also investigate other notions of independence, and show that they all agree, and are well-behaved, for stably dominated types. One of these is used to show that every type extends to an invariant type; definable types are dense. Much of this work requires the use of imaginary elements. We also show existence of prime models over reasonable bases, possibly including imaginaries.