UC Riverside

The goal of this paper is to continue building lattice theory by generalizing known results from commutative rings. Our focus will be results concerning strong Mori lattices as Mori lattices have been rather developed already in [7], [22], and others sources.

Our first main objective will be to obtain more results concerning quotient field lattices, which have been significantly developed in [6], [7], and [22]. Most notably, the quotient eld lattice version of the Nagata Theorem will be proven, following the approach of Chang Hwan Park and Mi Hee Park in [23]. This result had not yet been generalized to quotient field lattices. Also, some results concerning composition series will receive lattice versions, generalizing results from [23] as well as [15].

Some concepts from commutative ring theory that have not yet been applied to lattices will be introduced, such as certain star operations, particularly the w-operation. This will allow us to develop strong Mori lattices and begin to characterize them, working toward achieving lattice

versions of results from [27] and [28]. We will also introduce a lattice version of w-invertibility and obtain lattice versions of basic results. This will give a brief characterization of strong Mori lattices and lay the groundwork for further, more detailed characterizations as discussed in the final section.