Abstract
In this paper we provide an embedding of finite lattices into \((\mathbb N, \mid )\), the lattice of divisibility of natural numbers. For that purpose, we explore two representations: vector clocks, a device to provide a virtual time used in distributed systems that has gained traction as a finite lattice representation, and the log-prime function that transforms natural numbers into sequences of prime exponents. Using a generalized log-prime function and its inverse we describe how to embed any finite (width, height) lattice into \((\mathbb N, \mid )\) and provide examples for such process, prior to analysing the affordances of such encoding vis-a-vis the representation of non-global, distributed time. We also discuss how this representation may help improve the affordances of using complete lattices in data analysis both for time- and non-time related data.
CPM and FJVA have been partially supported by the Department of Research and Innovation of Madrid Regional Authority, in the EMPATIA-CM research project (reference Y2018/TCS-5046), and Grant PDC2021-121071-I00 funded by MCIN/AEI/ 10.13039/501100011033 and by the European Union “NextGenerationEU/PRTR” for this work. IPC, PC and MOA have been partially supported by the Spanish Ministry of Science, Innovation, and Universities (MCIU), State Agency of Research (AEI), Junta de Andalucía (JA), Universidad de Málaga (UMA) and European Regional Development Fund (FEDER) through the projects PGC2018-095869-B-I00 and TIN2017-89023-P (MCIU/AEI/FEDER), and UMA2018-FEDERJA-001 (JA/UMA/FEDER).
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Notes
- 1.
Recall that an element x of a finite lattice L is join-irreducible if \(x = a \vee b\) implies \(x = a\) or \(x = b\). And lattice-dually for meet-irreducibles.
- 2.
In the words of Gibson, “The affordances of the environment are what it offers the animal, what it provides or furnishes [7]”.
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Valverde-Albacete, F.J., Peláez-Moreno, C., Cabrera, I.P., Cordero, P., Ojeda-Aciego, M. (2022). Encoding Non-global Time Representations into the Lattice of Divisibility. In: Ciucci, D., et al. Information Processing and Management of Uncertainty in Knowledge-Based Systems. IPMU 2022. Communications in Computer and Information Science, vol 1601. Springer, Cham. https://doi.org/10.1007/978-3-031-08971-8_11
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