Principal Mappings between Posets

Abstract

We introduce and study principal mappings between posets which generalize the notion of principal elements in a multiplicative lattice, in particular, the principal ideals of a commutative ring. We also consider some weaker forms of principal mappings such as meet principal, join principal, weak meet principal, and weak join principal mappings which also generalize the corresponding notions on elements in a multiplicative lattice, considered by Dilworth, Anderson and Johnson. The principal mappings between the lattices of powersets and chains are characterized. Finally, for any PID R, it is proved that a mapping F:Idl(R)→Idl(R) is a contractive principal mapping if and only if there is a fixed ideal I∈Idl(R) such that F(J)=IJ for all J∈Idl(R). This exploration also leads to some new problems on lattices and commutative rings.