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 joinendomorphisms 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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