In this paper we study irreducible representations and symbolic Rees algebrasof monomial ideals. Then we examine edge ideals associated to vertex-weightedoriented graphs. These are digraphs having no oriented cycles of length twowith weights on the vertices. For a monomial ideal with no embedded primes weclassify the normality of its symbolic Rees algebra in terms of that of itsprimary components. If the primary components of a monomial ideal are normal,we present a simple procedure to compute its symbolic Rees algebra usingHilbert bases, and give necessary and sufficient conditions for the equalitybetween its ordinary and symbolic powers. Then we study the case of vanishingideals of finite sets of projective points. We give an effectivecharacterization of the Cohen--Macaulay vertex-weighted oriented forests. Foredge ideals of transitive weighted oriented graphs we show that Alexanderduality holds. It is shown that edge ideals of weighted acyclic tournaments areCohen--Macaulay and satisfy Alexander duality
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