ST-Distributive and ST-Modular Lattices

For two subsets S and T of a given lattice L, we define a relative distributive (modular) property over L, that underlies a large family including the usual class of distributive (modular) lattices. Our proposed class will be called ST-distributive (ST-modular) lattices. In this paper, we explore elemental properties of ST-distributivity (ST-modularity) and find examples of maximal S and T to form ST-distributive lattices for some non-distributive finite lattices of small order. We also characterize the maximal pairs of subsets (S,T), subject to certain constraints, that induce ST-distributivity in the lattice family M_n,n for all natural numbers n greater than or equal to 3. Afterwards, we present an application of ST-modular to convex sets and polytopes. This application has been the first example found and the main guiding light for our new definitions. The aforementioned definitions are closely related to distributive elements of Birkhoff-Gratzer and Stanley's SS-lattices.