Increasing and decreasing operators on complete lattices

Abstract

The (isotone) map f : X → X is an increasing (decreasing) operator on the poset X if f ( x ) ⩽ f 2 ( x ) ( f 2 ( 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, f ̂ , of f . f f , 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.