Pseudocomplements of closure operators on posets

Some recent results provide sufficient 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 first 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 transfinitely 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.