In this invited talk, the author gives a couple of generalisations of facts known for Boolean algebras. At first, he develops a generalisation of Stone duality between Boolean algebras and Boolean spaces to a duality between almost Boolean algebras and sheaves of sets over Boolean spaces. Next, he gives the definition of almost Boolean rings and shows a duality between these structures and almost Boolean algebras. Finally, he presents a number of structures that generalise almost Boolean algebras and discusses their properties, among them almost distributive lattices (ADLs), pseudo-complemented almost distributive lattices and Stone ADLs. The paper gives essentially a survey of part of the author’s research of the last three and a half decades; see, e.g. [Zbl 0311.06008; Zbl 0473.06008; Zbl 0982.06011; Zbl 1059.06009].
For the entire collection see [Zbl 1245.00043].