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Intervals of posets of a zero-divisor graph

  • John D. LaGrange
Published/Copyright: August 14, 2024
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Abstract

This article is concerned with bounded partially ordered sets P such that for every pP ∖ {1} there exists qP ∖ {0} such that 0 is the only lower bound of {p, q}. The posets P such that PQ if and only if P and Q have isomorphic zero-divisor graphs are completely characterized. In the case of finite posets, this result is generalized by proving that posets with isomorphic zero-divisor graphs form an interval under the partial order given by PQ if and only if there exists a bijective poset-homomorphism PQ. In particular, the singleton intervals correspond to the posets that are completely determined by their zero-divisor graphs. These results are obtained by exploring universal and couniversal orderings with respect to posets that have isomorphic zero-divisor graphs.

Acknowledgement

The author is grateful for the referee’s suggestions for improving the structure of the article, which benefited the readability of the work.

  1. Communicated by Anatolij Dvurečenskij

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Received: 2022-06-20
Accepted: 2023-12-08
Published Online: 2024-08-14
Published in Print: 2024-08-27

© 2024 Mathematical Institute Slovak Academy of Sciences

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