Types of Irreducible Divisor Graphs of Noncommutative Domains, II

In this paper, we continue investigation of the directed and undirected irreducible divisor graph concepts $G(x)$ and Γ (x) of x∈ D^ U(D), respectively, which were introduced in [7]. Consequently, we introduce two generalizations of these concepts. The first one is the irreducible divisor simplicial complex $S(x)$ of x∈ D^ U(D) in a noncommutative atomic domain D, which simultaneously extends the commutative case that was introduced by R. Baeth and J. Hobson in [3]. The second one is the directed and undirected τ-irreducible divisor graphs Gτ(x) and Γ τ(x) of x∈ D^ U(D), respectively, in a noncommutative τ-atomic domain D with a symmetric and associate preserving relation τ on D^ U(D). Those graphs also extend the commutative case that was introduced by C. P. Mooney in [5]. Furthermore, we extend the results of [3] and [5] to give a characterization of n-unique factorization domains via those two generalizations.