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# An Arithmetical Theorem for Partially Ordered Sets

## Citation

Doyle, Worthie Lefler (1950) An Arithmetical Theorem for Partially Ordered Sets. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/TP0K-A379. https://resolver.caltech.edu/CaltechTHESIS:09162013-151810143

## Abstract

The simplest multiplicative systems in which arithmetical ideas can be defined are semigroups. For such systems irreducible (prime) elements can be introduced and conditions under which the fundamental theorem of arithmetic holds have been investigated (Clifford (3)). After identifying associates, the elements of the semigroup form a partially ordered set with respect to the ordinary division relation. This suggests the possibility of an analogous arithmetical result for abstract partially ordered sets. Although nothing corresponding to product exists in a 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 expressible as meets of proper divisors can be introduced. The assumption of the ascending chain condition then implies that each element is representable as a reduced meet of irreducibles. The central problem of this thesis is to determine conditions on the structure of the partially ordered set in order that each element have a unique such representation.

Part I contains preliminary results and introduces the principal tools of the investigation. In the second part, basic properties of the lattice of ideals and the connection between its structure and the irreducible decompositions of elements are developed. The proofs of these results are identical with the corresponding ones for the lattice case (Dilworth (2)). The last part contains those results whose proofs are peculiar to partially ordered sets and also contains the proof of the main theorem.

Item Type: Thesis (Dissertation (Ph.D.)) (Mathematics and Physics) California Institute of Technology Physics, Mathematics and Astronomy Mathematics Physics Public (worldwide access) Dilworth, Robert P. Unknown, Unknown 1 January 1950 CaltechTHESIS:09162013-151810143 https://resolver.caltech.edu/CaltechTHESIS:09162013-151810143 10.7907/TP0K-A379 No commercial reproduction, distribution, display or performance rights in this work are provided. 7957 CaltechTHESIS Benjamin Perez 16 Sep 2013 22:53 11 Apr 2023 01:07

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