×

Isometrical embeddings of lattices into geometric lattices. (English) Zbl 1288.06018

Embedding lattices into geometric lattices gained substantial interest since the fundamental result of Dilworth on finite lattices. The nice paper under review elaborates on Dilworth’s approach and shows that an algebraic lattices endowed with a pseudorank function and having all compact elements of finite height does admit an isometric embedding into some geometric lattice. This also gives a new proof for the result of Grätzer and Kiss in the case of finite lattices.

MSC:

06C10 Semimodular lattices, geometric lattices
06B15 Representation theory of lattices
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Burris, S.N., Sankappanavar, H.P.: A Course in Universal Algebra. Springer, Berlin (1981) · Zbl 0478.08001 · doi:10.1007/978-1-4613-8130-3
[2] Crawley, P., Dilworth, R.P.: Algebraic Theory of Lattices. Prentice-Hall, Englewood Cliffs, NJ (1973) · Zbl 0494.06001
[3] Czédli, G., Schmidt, E.T.: A cover-preserving embedding of semimodular lattices into geometric lattices. Adv. Math. 225, 2455-2463 (2010) · Zbl 1226.06004 · doi:10.1016/j.aim.2010.05.001
[4] Finkbeiner, D.T.: A semimodular imbedding of lattices. Can. J. Math. 12, 582-591 (1960) · Zbl 0098.02702 · doi:10.4153/CJM-1960-051-4
[5] Grätzer, G.: General Lattice Theory. Birkhäuser Verlag, Basel-Stuttgart (1978); 2nd edn. Birkhäuser Verlag (1998) · Zbl 0385.06014
[6] Grätzer, G., Kiss, E.W.: A construction of semimodular lattices. Order 2, 351-365 (1986) · Zbl 0586.06001 · doi:10.1007/BF00367424
[7] Oxley, J.G.: Infinite matroids. Proc. Lond. Math. Soc. 37, 259-272 (1978) · Zbl 0445.05039 · doi:10.1112/plms/s3-37.2.259
[8] Oxley, J.G.: Infinite matroids. In: White, N. (ed.) Matroid Application. Encyclopedia of Mathematics and its Applications, vol. 40, pp. 73-90. Cambridge University Press (1992) · Zbl 0766.05016
[9] Stern, M.: Semimodular Lattices: Theory and Applications. Cambridge University Press (1999) · Zbl 0957.06008
[10] Wild, M.: Cover preserving embedding of modular lattices into partition lattices. Discrete Math. 112, 207-244 (2002) · Zbl 0808.06009 · doi:10.1016/0012-365X(93)90235-L
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.