On essential elements in a lattice and Goldie analogue theorem
Tapatee Sahoo, Babushri Srinivas Kedukodi, K. P. Shum, Harikrishnan Panackal, Syam Prasad Kuncham · Asian-European Journal of Mathematics · 2021
We introduce the concept of essentiality in a lattice [Formula: see text] with respect to an element [Formula: see text]. We define notions such as [Formula: see text]-essential, [Formula: see text]-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 [Formula: see text] which are [Formula: see text]-essential but not essential). We prove Goldie analogue results in terms of [Formula: see text]-uniform elements and [Formula: see text]-∨-independent sets. Furthermore, we define a graph with respect to [Formula: see text]-essential element in a lattice and study its properties.