Reciprocation and Pointwise Product in Vector Lattices of Functions

Abstract

If [omega] is a vector lattice of real-valued functions defined on a set containing the constant functions such that the reciprocal of each nonvanishing member of [omega] remains in [omega] , then [omega] is stable under pointwise product. We survey the literature on stability under reciprocation and pointwise product in the context of metric domains, with particular attention given to the uniformly continuous real-valued functions defined on them. For the first time, we present necessary and sufficient conditions for stability for the vector lattice of real-valued coarse maps defined on an arbitrary metric space. Membership of a function to this class means that its associated modulus of continuity is finite-valued, and need not entail continuity of the function.