Pseudocomplements of closure operators on posets

Some recent results provide su,cient conditions for complete lattices of closure operators on complete lattices, ordered pointwise, to be pseudocomplemented. This paper gives results of pseudocomplementation in the more general setting of closure operators on mere posets. The following result is 0rst proved: closure operators on a meet-continuous meet-semilattice form a pseudocomplemented complete lattice. Furthermore, the following orthogonal result (actually, a slightly more general result) is proved: Closure operators on a directed-complete poset which is trans0nitely generated by maximal lower bounds from its set of completely meet-irreducible elements—any poset satisfying the ascending chain condition belongs to this class—form a pseudocomplemented complete lattice. c © 2002 Elsevier Science B.V. All rights reserved. MSC: 06A15 (06A12; 06B35; 06D15)