On essential elements in a lattice and Goldie analogue theorem

We introduce the concept of essentiality in a lattice L with respect to an element δ ∈ L. We define notions such as δ-essential, δ-uniform elements and obtain some of their properties. Examples of lattices are given wherein essentiality can be retained with respect to an arbitrary element (specifically, there are elements in L which are δ-essential but not essential). We prove Goldie analogue results in terms of δ-uniform elements and δ-v-independent sets. Furthermore, we define a graph with respect to δ-essential element in a lattice and study its properties.