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Communication Dans Un Congrès Année : 2020

Counting and Computing Join-Endomorphisms in Lattices

Résumé

Structures involving a lattice and join-endomorphisms on it are ubiquitous in computer science. We study the cardinality of the set E(L) of all join-endomorphisms of a given finite lattice L. In particular, we show that when L is Mn, the discrete order of n elements extended with top and bottom, |E(L)| = n!Ln(-1) + (n + 1) 2 where Ln(x) is the Laguerre polynomial of degree n. We also study the following problem: Given a lattice L of size n and a set S ⊆ E(L) of size m, find the greatest lower bound E(L) S. The join-endomorphism E(L) S has meaningful interpretations in epistemic logic, distributed systems, and Aumann structures. We show that this problem can be solved with worst-case time complexity in O(n + m log n) for powerset lattices, O(mn 2) for lattices of sets, and O(mn + n 3) for arbitrary lattices. The complexity is expressed in terms of the basic binary lattice operations performed by the algorithm.
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Dates et versions

hal-04352165 , version 1 (19-12-2023)

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Santiago Quintero, Sergio Ramirez, Camilo Rueda, Frank Valencia. Counting and Computing Join-Endomorphisms in Lattices. Relational and Algebraic Methods in Computer Science - 18th International Conference, RAMiCS 2020, Apr 2020, Palaiseau, France. pp.253-269, ⟨10.1007/978-3-030-43520-2_16⟩. ⟨hal-04352165⟩
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