On regularity of sup-preserving maps: generalizing Zareckiĭ’s theorem
H. Ulrich, Tomasz Kubiak · Semigroup Forum · 2011
A sup-preserving map f between complete lattices L and M is regular if there exists a sup-preserving map g from M to L such that fgf = f . In the class of completely distributive lattices, this paper demonstrates a necessary and sufficient condition for f to be regular. When L = M is a power set, our theorem reduces to the well known Zareckiĭ’s theorem which characterizes regular elements in the semigroup of all binary relations on a set. Another application of our result is a generalization of Zareckiĭ’s theorem for quantale-valued relations.