Principal Mappings between Posets

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 , it is proved that a mapping is a contractive principal mapping if and only if there is a fixed ideal such that for all . This exploration also leads to some new problems on lattices and commutative rings. 1. Introduction A multiplicative lattice [1–3] is a complete lattice together with a binary operation, called multiplication, that is associative, commutative, and distributive over arbitrary joins and has the greatest element as the multiplication identity. The complete lattice of all ideals of a commutative ring is a typical example of multiplicative lattices. If is a principal ideal of , then satisfies the following equations: for any where is the ideal quotient [4]. In terms of the order and the multiplication on the lattice , the above two equations can be rephrased as In his efforts to obtain an abstract ideal theory of commutative rings, Dilworth introduced principal elements, the analogues of principal ideals, in a multiplicative lattice. The definition of principal elements makes use of the corresponding properties of principal ideals given in (2). Based on this notion of principal elements, Dilworth successfully established Noether’s Decomposition Theorems and Krull’s Principal Ideal Theorem for multiplicative lattices. Thereafter, the principal elements have been studied extensively by many people including Anderson, Johnson, and others [5–10]. As pointed out by Anderson and Johnson [2], “principal elements are the cornerstone on which the theory of multiplicative lattices and abstract ideal theory now largely rest.” Dilworth’s original definition of principal elements is only valid for a multiplicative lattice as it makes use of the multiplication, meet and join operations in a lattice, and so it does not apply on a general lattice. It is thus natural to wonder whether it is possible to extend this notion to arbitrary lattices or even posets. Let us relook at the principal ideals of a commutative ring from the perspective of mappings. Each ideal？？ ？？of？？