Congruence normality: The characterization of the doubling class of convex sets
Let \(D= (D_ L)\), where \(L\) ranges over the class of finite lattices and \(D_ L\) is a system of convex subsets of \(L\). A finite lattice, \(M\), is said to be constructible from \(D\) by doubling if there is a finite sequence of finite lattices \(1= L_ 0, L_ 1, \dots, L_ n =M\) and convex sets \(C_ i \subseteq L_ i\) with \(C_ i\in D_{L_ i}\), \(i<n\), such that \(L_{i+1} \cong L_ i [C_ i]\). The doubling class \(D\) is the class of all finite lattices that are constructible from \(D\) by doubling: it is denoted by \(1[D]\). The author characterizes the doubling classes of several classes as: the class \(CC\) of all connected convex sets of finite lattices, the classes \(LC\) of all lower and \(UC\) all upper pseudo-intervals, and the class \(INT\) of all the intervals, where a lower pseudo-interval in \(L\) in a (finite) lattice is a member of \(C_ L\) with \(\bigwedge_ L V\in V\) and an upper pseudo-interval is defined dually. An interval in \(L\) is a member of \(C_ L\) of the form \(u\setminus v\). The characterizations obtained are, respectively: the class \(CN\) of all congruence normal finite lattices, the classes \(CU_ \vee\), \(CU_ \wedge\) and \(CU\) of all join resp. meet resp. fully congruence uniform lattices. A comparison with a paper of W. Geyer written in the language of concept lattices is made.
scientific article; zbMATH DE number 590887