Abstract
In this paper, we introduce the graph G(S) of a bounded semilattice S, which is a generalization of the intersection graph of the substructures of an algebraic structure. We prove some general theorems about these graphs; as an example, we show that if S is a product of three or more chains, then G(S) is Eulerian if and only if either the length of every chain is even or all the chains are of length one. We also show that if G(S) contains a cycle, then girth(G(S)) = 3. Finally, weshowthat if (S, +, ·, 0, 1) is a dually atomic bounded distributive lattice whose set of dual atoms is nonempty, and the graph G(S) of S has no isolated vertex, then G(S) is connected with diam(G(S)) ≤ 4.
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Acknowledgments
The authors are grateful to Shahriar Shahriari and John LaGrange for looking through the paper and indicating some corrections, which improved the presentation of the paper.
Funding
The research of the first author was supported in part by a grant from the Islamic Azad University of Qazvin Branch. The second author was supported by the Department of Engineering Science at the Golpayegan University of Technology and his special thanks go to the department for making all the necessary facilities available to him for successfully conducting this research.
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Malakooti Rad, P., Nasehpour, P. On Graphs of Bounded Semilattices. Math Notes 107, 264–273 (2020). https://doi.org/10.1134/S0001434620010265
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DOI: https://doi.org/10.1134/S0001434620010265