Properties such as automaticity, growth and decidability are investigated for the class of finitely generated semigroups which have regular sets of unique normal forms. Knowledge obtained is then applied to the task of demonstrating that a class of semigroups derived from free inverse semigroups under certain closure operations is not automatic.

By blending techniques from set theory and algebraic topology we investigate the order of any homeomorphism of the nth power of the long ray or long line L having finite order, finding all possible orders when n = 1, 2, 3 or 4 in the first case and when n = 1 or 2 in the second. We also show that all finite powers of L are acyclic with respect to Alexander-Spanier cohomology.

In this paper we give a complete description of diameter-preserving linear bijections on the space of affine continuous functions on a compact convex set whose extreme points are split faces. We also give a description of such maps on function algebras considered on their maximal ideal space. We formulate and prove similar results for spaces of vector-valued functions.

A subset F of an ordered set X is a fibre of X if F intersects every maximal antichain of X. We find a lower bound on the function ƒ (D), the minimum fibre size in the distributive lattice D, in terms of the size of D. In particular, we prove that there is a constant c such that In the process we show that minimum fibre size is a monotone property for a certain class of distributive lattices. This fact depends upon being able to split every maximal antichain of this class of distributive lattices into two parts so that the lattice is the union of the upset of one part and the downset of the other.

In this paper we construct a family of variational families for a Legendrian embedding, into the 1-jet bundle of a closed manifold, that can be obtained from the zero section through Legendrian embdeddings, by discretising the action functional. We compute the second variation of a generating funciton obtained as above at a nondegenerate critical point and prove a formula relating the signature of the second variation to the Maslov index as the mesh goes to zero. We use this to prove a generlisation of the Morse inequalities thus refining a theorem of Chekanov.

We investigate certain norm and continuity conditions that provide us with ‘uniqe Hahn-Banch Theorems’ from (nc0) to (nℓ∞) and from N(nE) to N(nE″). We show that there is a unique norm-preserving extension for norm-attaining 2-homogeneous polynomials on complex c0 to ℓ∈ but there is no unique norm-preserving extension from (3c0) to (3ℓ∈).

Using a result on arithmetic progressions, we describe a method for finding the rational h–tuples ρ = (ρl,…,ρh) such that all the multiples mρ (for m coprime to a denominator of ρ) lie in a linear variety modulo Z. We give an application to hypergeometric functions.

In this paper we derive a relation between character sums and partial sums of Dirichlet series.