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A dependence system on a set \(S\) is defined by an operator \(f\), a function on the power set of \(S\) which is extensive (\(A\) is included in \(f(A)\)) and monotone (if \(A\) is included in \(B\), then \(f(A)\) is included in \(f(B)\)). In this paper, the structure of the set \(F(S)\) of all dependence systems on a given set \(S\) is studied. The partially ordered set of operators (\(f < g\) if for every set \(A\), \(f(A)\) is included in \(g(A)\)) is a bounded, complete, completely distributive, atomic, and dual atomic lattice with an involution. It is shown that every operator is a join of transitive operators (usually called closure operators, operators which are idempotent \(ff = f\)). The study of the class of transitive operators that join-generate all operators makes it possible to express \(F(n)\) (the cardinality of the set \(F(S)\) of all operators on a set \(S\) with \(n\) elements) by the Dedekind number \(D(n)\). The formula has interesting consequences for dependence system theory.