ON ESSENTIALITY AND IRREDUCIBILITY IN A LATTICE

We consider a bounded lattice (L, ∧, ∨) with the smallest element 0 and the greatest element 1. In this paper, we deal with the essentiality concepts associated with a lattice. For an arbitrary element θ of L, we define a θ-e-irreducible element in L, which is an analogy to the concept of the e-irreducible submodule in a module over a ring. It is well known that e-irreducible submodules have no proper essential extension. Indeed, we prove this remains true for elements in a bounded lattice. We establish a relation between the θ-complement and θ-e-irreducible element with suitable examples. We define the notion θ-socle and prove several properties when a lattice is compactly generated. Further, we construct a generalized complement graph of a distributive lattice and relate the properties such as connectedness, diameter, and cut vertices to atoms in a lattice.