Representing homomorphisms of distributive lattices as restrictions of congruences of rectangular lattices
This paper is the newest contribution to the problem of representing finite distributive lattices as congruence lattices of lattices with special properties. The results are mostly of technical character and are prompting alternative proofs of known theorems (cf. [\textit{G. Grätzer} and \textit{E. Knapp}, Acta Sci. Math. 75, No. 1--2, 29--48 (2009; Zbl 1199.06029); \textit{E. T. Schmidt}, Acta Sci. Math. 77, No. 1--2, 47--52 (2011; Zbl 1249.06022)]. We can present a typical example: Every finite nontrivial distributive lattice \(D\) is isomorphic to the congruence lattice of a rectangular lattice. (Recall that a finite lattice \(M\) is called rectangular, if \(M\) is planar, semimodular, \(|M|\geq 3\), and there exists only one couple \(a, b\in M-\{0,1\}\) of double-irreducible elements such that \(a\) is the left weak corner and \(b\) is the right weak corner of \(M,\) respectively, and both elements are complementary in \(M.\))
scientific article; zbMATH DE number 6087533
scientific article