Basic retracts and counting of lattices

Abstract

In this paper, we treat nullity of a poset as the nullity of its cover graph.Using nullity, we introduce the concepts of a basic retract associated to a poset and a fundamental basic block associated to a dismantlable lattice. We prove that a lattice in which all the reducible elements are comparable is dismantlable but the converse is not true. We establish recurrence relations and count up to isomorphism all fundamental basic blocks and all basic retracts/blocks associated to the lattices (with respect to arbitrary large nullity and/or the number of reducible elements) in which all the reducible elements are comparable.