On the semigroup of closure operations (Q1825970)

Let (M,\(\leq)\) be a partially ordered set and denote by \({\mathcal T}_ M\) the full transformation semigroup on M. An element \(u\in {\mathcal T}_ M\) is a closure operation on M if it is idempotent, preserves order and, in addition, satisfies the condition that \(x\leq u(x)\) for all x. Denote by \(C_ M\) the family of all closure operations on M. In general, \(C_ M\) is not a subsemigroup of \({\mathcal T}_ M\) and when it is, \(C_ M\) is a semilattice. An element x of a complete lattice L is said to be \(\bigvee\)-irreducible if for each \(H\subseteq L\) such at \(x\leq \sup H\), then \(x\leq y\) for some \(y\in H\). It is shown that for a given semilattice S, there exists a partially ordered set M such that \(C_ M\) is isomorphic to S if and only if S is a complete lattice, under the usual ordering, in which every element is a supremum of \(\bigvee\)-irreducible elements. This, in turn, is equivalent to the proposition that there exists a complete ring of sets R such that S is isomorphic to the semilattice (R,\(\cap)\).