Regularity: Lattice-valued Cauchy spaces

A category of lattice-valued Cauchy spaces is defined, and its properties are investigated. The relationship between this category and the recent work by Jنger is presented. The notion of regularity in the Cauchy space context is the primary emphasis here. In particular, the category consisting of the regular Cauchy spaces is shown to be bireflective in the category of all lattice-valued Cauchy spaces having Cauchy-continuous maps as its morphisms. Moreover, completions are also investigated.