A cover-preserving embedding of semimodular lattices into geometric lattices
The authors prove that for any semimodular lattice \(\mathcal L\) of finite length there exists a cover-preserving embedding into a geometric lattice \(G({\mathcal L})\) of the same length such that the number of atoms of \(G({\mathcal L})\) coincides with the number of join-irreducible elements of \(\mathcal L\). For finite semimodular lattices this has been shown by \textit{G. Grätzer} and \textit{E. W. Kiss} [Order 2, 351--365 (1986; Zbl 0586.06001)], and again by \textit{M. Wild} [Discrete Math. 112, No.~1--3, 207--244 (1993; Zbl 0808.06009)] with a short proof using matroid theory. It is pointed out that the finite geometric lattice constructed by Grätzer and Kiss differs from \(G({\mathcal L})\). Detailed historical notes are given, see also Chapter 6.2 in [\textit{M. Stern}, Semimodular lattices. Theory and applications. Cambridge: Cambridge University Press (1999; Zbl 0957.06008)].
scientific article; zbMATH DE number 5796918