One of the standard axioms for Boolean contact algebras says that if a region __x__ is in contact with the join of __y__ and __z__, then __x__ is in contact with at least one of the two regions. Our intention is to examine a stronger version of this axiom according to which if __x__ is in contact with the supremum of some family __S__ of regions, then there is a __y__ in __S__ that is in contact with __x__. We study (...) a modal possibility operator which is definable in complete algebras in the presence of the aforementioned axiom, and we prove that the class of complete algebras satisfying the axiom is closely related to the class of modal KTB-algebras. We also demonstrate that in the class of complete extensional contact algebras the axiom is equivalent to the statement: _every region is isolated_. Finally, we present an interpretation of the modal operator in the class of the so-called _resolution contact algebras_. (shrink)

Monotone modal logics have emerged in several application areas such as computer science and social choice theory. Since many of the most studied selfextensional logics have a conjunction, in this paper we study some distributive extensions obtained from a semilattice based deductive system with monotonic modal operators, and we give them neighborhood and algebraic semantics. For each logic defined our main objective is to prove completeness with respect to its characteristic class of monotonic frames.

We introduce and study a class of betweenness algebras—Boolean algebras with binary operators, closely related to ternary frames with a betweenness relation. From various axioms for betweenness, we chose those that are most common, which makes our work applicable to a wide range of betweenness structures studied in the literature. On the algebraic side, we work with two operators of possibility and of sufficiency.