Nathan Bowler

An excluded minors method for infinite matroids

The notion of thin sums matroids was invented to extend the notion of representability to non-nitary matroids. A matroid is tame if every circuit-cocircuit intersection is nite. We prove that a tame matroid is a thin sums matroid over a nite eld k if and only if all its nite minors are representable over k. We expect that the method we use to prove this will make it possible to lift many theorems about nite matroids representable over a nite eld to theorems about tame thin sums matroids over these elds. We give three examples of this: various characterisations of binary tame matroids and of regular tame matroids, and unique representability of ternary tame matroids.

Matroid intersection, base packing and base covering for infinite matroids

As part of the recent developments in infinite matroid theory, there have been a number of conjectures about how standard theorems of finite matroid theory might extend to the infinite setting. These include base packing, base covering, and matroid intersection and union. We show that several of these conjectures are equivalent, so that each gives a perspective on the same central problem of infinite matroid theory. For finite matroids, these equivalences give new and simpler proofs for the finite theorems corresponding to these conjectures.This new point of view also allows us to extend, and simplify the proofs of some cases where these conjectures were known to be true.

Coproducts of Monads on Set

Coproducts of monads on Set have arisen in both the study of computational effects and universal algebra. We describe coproducts of consistent monads on Set by an initial algebra formula, and prove also the converse: if the coproduct exists, so do the required initial algebras. That formula was, in the case of ideal monads, also used by Ghani and Uustalu. We deduce that coproduct embeddings of consistent monads are injective; and that a coproduct of injective monad morphisms is injective. Two consistent monads have a coproduct iff either they have arbitrarily large common fixpoints, or one is an exception monad, possibly modified to preserve the empty set. Hence a consistent monad has a coproduct with every monad iff it is an exception monad, possibly modified to preserve the empty set. We also show other fixpoint results, including that a functor (not constant on nonempty sets) is finitary iff every sufficiently large cardinal is a fixpoint.

Matroids with an infinite circuit-cocircuit intersection

Abstract We construct some matroids that have a circuit and a cocircuit with infinite intersection. This answers a question of Bruhn, Diestel, Kriesell, Pendavingh and Wollan. It further shows that the axiom system for matroids proposed by Dress in 1986 does not axiomatize all infinite matroids. We show that one of the matroids we define is a thin sums matroid whose dual is not a thin sums matroid, answering a natural open problem in the theory of thin sums matroids.

Rudimentary Recursion, Gentle Functions and Provident Sets

This paper, a contribution to “micro set theory”, is the study promised by the first author in [M4], as improved and extended by work of the second. We use the rudimentarily recursive (set-theoretic) functions and the slightly larger collection of gentle functions to initiate the study of provident sets, which are transitive models of PROVI, a subsystem of KP whose minimal model is Jensen’s J! . PROVI supports familiar definitions, such as rank, transitive closure and ordinal addition—though not ordinal multiplication—and (shown in [M8]) Shoenfield’s unramified forcing. Providence is preserved under directed unions. An arbitrary set has a provident closure, and (shown in [M8]) the extension of a provident M by a set-generic G is the provident closure of Received May 9, 2012; accepted August 14, 2013 2010 Mathematics Subject Classification: Primary 03E30, 03D65; Secondary 03E40, 03E45

Reconstruction of infinite matroids from their 3-connected minors

We show that any infinite matroid can be reconstructed from the torsos of a tree-decomposition over its 2-separations, together with local information at the ends of the tree. We show that if the matroid is tame then this local information is simply a choice of whether circuits are permitted to use that end. The same is true if each torso is planar, with all gluing elements on a common face.

Non-reconstructible locally finite graphs

Two graphs

A counterexample to Montgomery’s conjecture on dynamic colourings of regular graphs

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A counterexample to the reconstruction conjecture for locally finite trees

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The Axiom Scheme of Acyclic Comprehension

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