A theory of partitions of partially ordered sets

A partition of a set A is a set of nonempty pairwise disjoint subsets of A whose union is A. An equivalent definition of a partition can be given using functions. In these terms, a partition of a set is the set of fibres of a surjective function. The latter definition allows us to introduce the notion of partition for finite partially ordered sets. Analyzing the category Poset of partially ordered sets, or posets, and order-preserving maps, we see that two kinds of surjections have to be taken into account. Therefore, we must deal with two different classes of partitions, namely, monotone and regular partitions of a poset. These two notions form the basis for a theory of partitions of posets. In analogy with the set-theoretic case, our first step is to obtain characterizations of monotone and regular partitions. Then, in Chapter 4, we study the collection of all monotone and regular partitions of a poset. We endow these classes with a lattice structure, obtaining the monotone and regular partition lattices of a poset. A bijection between monotone and regular partitions and certain classes of quasiorders is established. This result generalizes the usual correspondence between partitions and equivalence relations. In Chapter 5, we present the well-known Birkhoff duality for Poset. We thus investigate the duals of monotone and regular partitions via this duality. In the last chapter we discuss enumeration problems within the theory of partitions of finite posets.