Order-Preserving Self-Maps of Complete Lattices

We study isotone self-maps of complete lattices and their fixed point sets, which are complete lattices contained as suborders, but not necessarily as subsemilattices. We develop a representation of such maps by means of relations and show how to navigate their fixed point lattices using a modification of the standard Next closure algorithm. Our approach is inspired by early work of Shmuely [8] and Crapo [1]. We improve and substantially extend our earlier publication [4].