A lattice construction and congruence-preserving extensions
A ``chopped lattice'' is a partial lattice \(M\), such that \(M\) is a meet-semilattice with 0 and any two elements with a common upper bound have a join. Equivalently, \(M = L - \{1\}\), where \(L\) is a lattice with 0 and 1. For such a chopped lattice, ``Condition (FG)'' means: Every finitely generated ideal of \(M\) is a finite union of principal ideals. In the paper it is shown: If \(M\) satisfies (FG), then the finitely generated ideals of \(M\) form a sublattice \(\text{Idfg} (M)\) of the ideal lattice \(\text{Id} (M)\) of \(M\), and \(\text{Idfg} (M)\) is a congruence-preserving extension of \(M\) (i.e., every congruence of \(M\) has exactly one extension to \(\text{Idfg} (M)\)). Let \(A\) and \(B\) be two lattices sharing the sublattice \(A \cap B \neq \emptyset\). Then a set of sufficient conditions is given under which the amalgamation \(M(A,B) = A \cup B\) (with the natural partial ordering induced by the lattices \(A\) and \(B\)) is a chopped lattice. If \(A\), \(B\) satisfy these conditions, they are called a ``chopped pair''. Furthermore, two stronger versions of chopped pairs are introduced: ``sharp'' and ``full'' chopped pairs. The main results on sharp chopped pairs \(A\), \(B\) are the following: (i) If \(A \cap B\) satisfies the Ascending Chain Condition, then \(M(A,B)\) satisfies (FG). (ii) If \(A\), \(B\) are both modular, then \(M(A,B)\) satisfies (FG). For a full chopped pair \(A\), \(B\) it is shown: If one of the lattices \(A\), \(B\) is modular, then \(M(A,B)\) satisfies (FG). In the last section, these results are applied to the problem whether a given lattice has a proper congruence-preserving extension. The following theorem is proved: If \(L\) is a lattice with a non-trivial distributive interval, then \(L\) has a proper congruence-preserving extension.
scientific article; zbMATH DE number 755483