Semimodular lattices and the Hall-Dilworth gluing construction
A gluing construction for lattices has been defined by \textit{M. Hall} and \textit{R. P. Dilworth} [Ann. Math. (2) 45, 450--456 (1944; Zbl 0060.06102)]. Another gluing construction was described by the author in a previous paper [Algebra Univers. 64, No. 1--2, 101--102 (2010; Zbl 1207.06007)]. Here, the lattices \(L\) and \(K\) under consideration are assumed to be semimodular of finite length and \(C\) be a maximal chain in \(L\); the result of gluing of \(L\) and \(K\) over \(C\) is denoted by \(L\cup_C K\). The main result of the paper under review is the following theorem. Under the notation as above, let \(G=L\cup_C K\). Then \(G\) is the cover-preserving join-homomorphic image of a semimodular lattice \(H\), \(\psi \colon H\rightarrow G\), such that: (1) \(H\) is the Hall-Dilworth gluing of the semimodular lattices \(P\) and \(K\) over a chain isomorphic to \(C\); (2) \(P\) is the direct power of a chain isomorphic to \(C\); (3) \(L\) is the cover-preserving join-homomorphic image of \(P\), \(\varphi \colon P\rightarrow L\); (4) the restriction of \(\psi \) to the filter \(P\) of \(H\) is \(\varphi\) and the restriction to \(K\) is the identity.
scientific article; zbMATH DE number 5881184