Resumen de MacNeille completions of modal algebras

Guram Bezhanishvili, John Harding

• For a modal algebra (B,f), there are two natural ways to extend f to an operation on the MacNeille completion of B. The resulting structures are called the lower and upper MacNeille completions of (B,f). In this paper we consider lower and upper MacNeille completions for various varieties of modal algebras. In particular, we characterize the varieties of closure algebras and diagonalizable algebras that are closed under lower and upper MacNeille completions. We also introduce the variety of Sierpinski algebras, and show that although this variety is not closed under lower or upper MacNeille completions, it follows from the axiom of choice that each Sierpinski algebra has a MacNeille completion that is also a Sierpinski algebra, and that this result implies the Boolean ultrafilter theorem.