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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2404.04873 |
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Table of Contents:
- In this paper, we continue investigation of the directed and undirected irreducible divisor graph concepts $G(x)$ and $Γ(x)$ of $x\in D^{\ast} \backslash U(D)$, 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 $x\in D^{\ast} \backslash U(D)$ 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 $G_{τ}(x)$ and $Γ_{τ}(x)$ of $x\in D^{\ast} \backslash U(D)$, respectively, in a noncommutative $τ$-atomic domain $D$ with a symmetric and associate preserving relation $τ$ on $D^{\ast} \backslash U(D)$. 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.