Forbidden configurations and subframe varieties

In a series of papers, Ball, Pultr, and Sichler studied forbidden configurations in Priestley spaces and Esakia spaces. They showed, among other things, that the class of Heyting algebras whose Esakia spaces contain no copy of a given finite configuration is a variety iff the configuration is a tree. In this short note, we show that such varieties are examples of subframe varieties—the algebraic counterparts of subframe logics introduced by Fine in the 1980s.