Generalized Lattices

If, presumably, the definition of a Boolean algebra employs the notion of a partially ordered set, whereas a general definition of a topological space is based on the notion of a metric, then by considering a more general situation with quasi metrics one may obtain the notion of a bitopological space (i.e., a set endowed with two arbitrary topologies). A further development of the theory of bitopological spaces has made it possible to introduce and study an algebra of a new type that is based on a nonordinary variant of a quasi ordered set and the corresponding representation of which brings one to bitopological spaces. The author calls this a generalised Boolean algebra, which is interesting not only as the subject of independent research, but also can be used as an important tool in establishing its various relationships with other areas of mathematics and, in particular, as in related double Boolean algebras with mathematical studies.