In this paper, using pre-semi-open L-sets and their inequality, a new notion of PS-compactness is introduced in L-topological spaces, where L is a complete De Morgan algebra. This notion does not depend on the structure of the basis lattice L and L does not need any distributivity.

Orbits and bi-invariant subsets of binary $$G$$ -spaces are studied. The problem the distributivity a action group on space $$X$$ , which was posed in 2016 by one authors, is solved.

Maltsev families of varieties which are closed under join or Maltsev product are investigated. New Maltsev conditions for congruence semi-distributivity are given.