Matroids on Complete Boolean Algebras

Abstract

The approach to a theory of non-finitary matroids, as outlined by the author in [20], is here extended to the case in which the relevant closure operators are defined on arbitrary complete Boolean algebras, rather than on the power sets of sets. As a preliminary to this study, the theory of derivatives of operators on complete Boolean algebras is developed and the notion, having interest in its own right, of an analytic closure operator is introduced . The class of B-matroidal closure operators is singled out for especial attention and it is proved that this class is closed under Whitney duality. Also investigated is the class of those closure operators which are both matroidal and topological.