Cohen-Macaulay Weighted Oriented Edge Ideals and its Alexander Dual

The study of the edge ideal Formula: see text of a weighted oriented graph Formula: see text with underlying graph Formula: see text started in the context of Reed–Muller type codes. We generalize some Cohen–Macaulay constructions for Formula: see text, which Villarreal gave for edge ideals of simple graphs. Our constructions can be used to produce large classes of Cohen–Macaulay weighted oriented edge ideals. We use these constructions to classify all the Cohen–Macaulay weighted oriented edge ideals, whose underlying graph is a cycle. We also show that Formula: see text is Cohen–Macaulay if and only if Formula: see text is unmixed and Formula: see text is Cohen–Macaulay, where Formula: see text denotes the cycle of length Formula: see text. Miller generalized the concept of Alexander dual ideals of square-free monomial ideals to arbitrary monomial ideals, and in that direction, we study the Alexander dual of Formula: see text and its conditions to be Cohen–Macaulay.