The criterion of Brouwerian and closure algebras to be finitely generated
Leo Esakia, Revaz Grigolia · Bulletin of the Section of Logic · 1977
When investigating the semantics of superprintuitionistic logics and extensions of the modal system S4, as well as from purely algebraic point of view, it is important to know characteristic properties and structure of finitely generated Brouwerian (alias pseudo-Boolean) algebras [1] and closure algebras [1]. In this paper we propose a criterion for the algebras to be finitely generated. The duality between closure algebras (resp., Brouwerian lattices) and perfect Kripke models, on which this paper rests, is developed in [2]. The key definitions are recalled here for convenience. A subset A of a pre-ordered set (X,R) is a cone if x ∈ A and xRy imply y ∈ A. The following proposition will be useful.