A construction of semimodular lattices

✍ Scribed by George Grätzer; Emil W. Kiss

• Publisher: Springer Netherlands
• Year: 1985
• Tongue: English
• Weight: 871 KB
• Volume: 2
• Category: Article
• ISSN: 0167-8094
• DOI: 10.1007/bf00367424

✦ Synopsis

In this paper we prove that if -Y is a finite lattice, and r is an integral valued function on 2 satisfying some very natural conditions, then there exists a finite geometric (that is, semimodular and atomistic) lattice '/ containing 9 as a sublattice such that r is the height function of prestricted to 2. Moreover, we show that if, for all intervals [e, f] of iy' , semimodular lattices Ye f, of length at most r(f) -r(e) are given, then 'j can be chosen to contain 'je f in its interval [e, f] as a cover preserving { O)-sublattice. As applications, we obtain results of R. P. Dilworth and D. T. Finkbeiner. AMS (MOS) subject classifications (1980). 06ClO.