Efimovʼs problem and Boolean algebras

We continue the study, started by P. Koszmider, of a class of Boolean algebras, the so-called TT-algebras. We prove the following.(1)All superatomic Boolean algebras belong to this class.(2)This class is contained properly in Koppelbergʼs class of minimally generated Boolean algebras.(3)The existence of an Efimov TT-algebra (i.e., a TT-algebra whose Stone space is infinite and contains no converging sequence and no copy of βω) implies a negative answer to Scarborough–Stoneʼs problem.(4)There is an Efimov TT-algebra of countable tightness in the generic extension obtained by a finite support iteration of length ω2ω2 of Hechlerʼs poset over a model of CH.