We show that all convex directed subgroups of a pl-group form distributive lattice under inclusions is Brouwer lattice. succeeded in extending some l-group results concerning rectifying and regular to the class $$ \mathcal{AO} -groups. Necessary sufficient conditions are given for an element be with unique value. In order prove this, properties lexicographic extensions -groups pl-groups investi...

Given a set S, we deene Galois connections between the lattice of fuzzy subgroups of transformations in S, the lattice of the similarities in S and the lattice of the distances in S.

We showed in [Oh] that for a simple real Lie group G with real rank at least 2, if a discrete subgroup Γ of G contains lattices in two opposite horospherical subgroups, then Γ must be a non-uniform arithmetic lattice in G, under some additional assumptions on the horospherical subgroups. Somewhat surprisingly, a similar result is true even if we only assume that Γ contains a lattice in one horo...