Increasing and decreasing operators on complete lattices

AbstractThe (isotone) map f: X → X is an increasing (decreasing) operator on the poset X if f(x) ⩽ f2(x) (f2(x) ⩽ f(x), resp.) holds for each x ∈ X. Properties of increasing (decreasing) operators on complete lattices are studied and shown to extend and clarify those of closure (resp. anticlosure) operators. The notion of the decreasing closure, f, (the increasing anticlosure, f,) of the map f: X → X is introduced extending that of the transitive closure, f̂, of f. ff, and f are all shown to have the same set of fixed points. Our results enable us to solve some problems raised by H. Crapo. In particular, the order structure of H(X), the set of retraction operators on X is analyzed. For X a complete lattice H(X) is shown to be a complete lattice in the pointwise partial order. We conclude by claiming that it is the increasing-decreasing character of the identity maps which yields the peculiar properties of Galois connections. This is done by defining a u-v connection between the posets X and Y, where u: X → X (v: Y → Y) is an increasing (resp. decreasing) operator to be a pair f, g of maps f; X → Y, g: Y → X such that gf ⩾ u, fg ⩽ v. It is shown that the whole theory of Galois connections can be carried over to u-v connections