The simplest multiplicative systems in which arithmeticalideas can be defined are semigroups. For such systems irreducible(prime) elements can be introduced and conditions underwhich the fundamental theorem of arithmetic holds have been investigated(Clifford (3)). After identifying associates, the elementsof the semigroup form a partially ordered set with respectto the ordinary division relation. This suggests the possibilityof an analogous arithmetical result for abstract partially orderedsets. Although nothing corresponding to product exists ina partially ordered set, there is a notion similar to g.c.d.This is the meet operation, defined as greatest lower bound.Thus irreducible elements, namely those elements not expressibleas meets of proper divisors can be introduced. The assumptionof the ascending chain condition then implies that each elementis representable as a reduced meet of irreducibles. The centralproblem of this thesis is to determine conditions on the structureof the partially ordered set in order that each elementhave a unique such representation.
Part I contains preliminary results and introduces the principaltools of the investigation. In the second part, basic propertiesof the lattice of ideals and the connection between itsstructure and the irreducible decompositions of elements are developed.The proofs of these results are identical with the correspondingones for the lattice case (Dilworth (2)). The lastpart contains those results whose proofs are peculiar to partiallyordered sets and also contains the proof of the main theorem.
