Combinatorial Properties Of Finite Lattices

In lattice theory, semimodular lattices, modular lattices and distributive lattices arethree classes of important research objects. Therefore, it is important to find out sufficientconditions for a lattice being semimodular, for a semimodular lattice being modular, andfor a modular lattice being distributive. In the thesis, we establish a condition for them,by using cutset, a combinatorial concept. Several properties of semi-regular cyclicallygenerated lattices and modular cyclically generated lattices are also offered. At last, oneproperty of Orlik-Solomon algebras based on semi-regular cyclically, generated lattices isproved.The first chapter consists of notations, terminologies and the background.The second chapter proves that an atomistic lattice is geometric if it has a cutsetconsisting of nontrivial modular elements.The third chapter proves that a geometric lattice is modular if it has a cutset con-sisting of nontrivial modular elements. This result is also generalized to lattices generatedby cycles.In the fourth chapter we first prove that a geometric lattice is boolean if it has acutset consisting of nontrivial neutral elements, and then we offer several properties ofneutral cycles. At last, we prove that a modular cyclically generated lattice is distributiveif it has a cutset of special type.In the first section of the last chapter, the properties of the Mobius function onsemiregular cyclically generated lattices are researched. In the next section, it is provedthat for the Orlik-Solomon algebras based on a semi-regular cyclically generated lattice isisomorphic to the Orlik-Solomon algebras based on its reduced geometric sublattice.