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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Unique factorization in partially ordered sets


Author: Jorge Martinez
Journal: Proc. Amer. Math. Soc. 33 (1972), 213-220
MSC: Primary 06A10
DOI: https://doi.org/10.1090/S0002-9939-1972-0292723-5
MathSciNet review: 0292723
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Abstract: Call an ideal of a lattice L regular if it is maximal with respect to not containing some element of L. If the ideal M is maximal without $ x \in L$ we say M is a value of x. A special ideal K is a regular ideal which is maximal without some element having only one value (namely K). Our main theorem is that the following three statements are equivalent: (1) The lattice of ideals of the lattice L is completely distributive. (2) Each regular ideal is special. (3) L is distributive, and each element $ x \in L$ can be written uniquely as a finite join of pairwise incomparable, finitely join irreducible elements of L.

By carefully generalizing our notions to partially ordered sets we get a similar theorem in this context.


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DOI: https://doi.org/10.1090/S0002-9939-1972-0292723-5
Keywords: Ideal of a p.o. set, regular ideal, special ideal, strong finite join irreducibility, finite join irreducibility, freely generated, prime ideal, unique factorization p.o. set
Article copyright: © Copyright 1972 American Mathematical Society

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