Alignments closed under a binary product

Abstract

This paper is concerned with closure systems defined on groups such that the product of any two closed sets is again closed. The main objective of this paper is to show that if enough “small” subsets are closed, then all subsets are closed. It is shown that this is true for groups without any elements of order 2 provided all 2-element subsets are closed. The same holds for all Abelian groups with one exception: Boolean groups with the closed sets taken to be the cosets.