Closure operators and generating sets (Q2777532)

The authors investigate closure operators on finite sets. For the sake of simplicity the closure of the empty set is assumed to be empty. A subset \(A\) of a subset \(B\) of some set \(M\) is called a generating set of \(B\) with respect to some closure operator \(f\) on \(M\) if \(B\subseteq f(A)\). The ideal closure operator on a semigroup \((S,\cdot)\) is the operator assigning to each subset \(C\) of \(S\) the ideal of \((S,\cdot)\) generated by \(C\). For given set-theoretical operations, all closure operators on a fixed finite set are considered such that for each subset the set of its generating sets is closed with respect to the set-theoretical operations. In order to characterize the closure operators, semigroups are determined such that each closure operator is the meet of the ideal closure operators on some semigroups.