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 | References | Similar Articles | Additional Information

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 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 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.

**[1]**James C. Abbott,*Sets, lattices, and Boolean algebras*, Allyn and Bacon, Inc., Boston, Mass., 1969. MR**0242723****[2]**Garrett Birkhoff,*Lattice theory*, Third edition. American Mathematical Society Colloquium Publications, Vol. XXV, American Mathematical Society, Providence, R.I., 1967. MR**0227053****[3]**Garrett Birkhoff and Orrin Frink Jr.,*Representations of lattices by sets*, Trans. Amer. Math. Soc.**64**(1948), 299–316. MR**0027263**, https://doi.org/10.1090/S0002-9947-1948-0027263-2**[4]**Paul Conrad,*The lattice of all convex 𝑙-subgroups of a lattice-ordered group*, Czechoslovak Math. J.**15 (90)**(1965), 101–123 (English, with Russian summary). MR**0173716****[5]**S. Papert,*Which distributive lattices are lattices of closed sets?*, Proc. Cambridge Philos. Soc.**55**(1959), 172–176. MR**0104601**

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Additional Information

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