This paper studies basic properties for five special types of implicative ideals (modular, pentagonal, even, rectangular and medial). The results are used to prove characterizations of modularity and distributivity.

We introduce the concept of complementary elements in ordered sets. If an ordered set S is a lattice, this concept coincides with that for lattices. The connections between distributivity and the uniqueness of complements are shown and it is also shown that modular complemented ordered sets represents \geometries" which are more general than projective planes. It was shown in 2], 4] and 6] that...