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 J(L) of all join-endomorphisms of a given finite lattice L. We show that the cardinality of J(L) is sub-exponential, exponential and super-exponential in the size of the lattice for boolean algebras, linear-orders, and arbitrary lattices, respectively. We also study the following problem: Given a lattice L of size n and a set S ⊆ J(L) of size m, find the greatest lower bound in J(L) of S. This join-endomorphism 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(mn^(log_2 3)) 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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