Irreducible Elements in a Lattice and Application to Hypercube

Abstract

The concept of irreducible submodule of a module over a ring is well known.  Analogously, in this paper, we define $e$-irreducibility and prove several properties that involve $e$-irreducibility, the complement in a lattice. Furthermore, we establish an isomorphism between the graph constructed on the equivalence classes of the set of pseudo-complements in a lattice and the directed hypercube of dimension $n$ (or $n$-cube).