Gamma invariants for dense lattices
In this paper a subclass of dense lattices, called strongly dense lattices, is introduced and studied. Let \(L\) be a bounded lattice and \(0<\sigma \) be an ordinal. A sequence \(\mathcal A=(a_{\alpha } : \alpha < \sigma)\) of elements of \(L\) is a cofinal strictly decreasing chain (or c.d.c) in \(L\) if (0) \(a_0=1\) (1) \(a_{\alpha } \not =0\) for all \(\alpha < \sigma \) (2) \(a_{\alpha +1} < a_{\alpha }\) for all \(\alpha < \sigma \) (3) \(a_{\beta } = \inf _{\alpha <\beta } a_{\alpha }\) for all limit ordinals \(\beta <\sigma \) (4) If \(0\not = a\in L\) then there is \(\alpha < \sigma \) with \(a_{\alpha } \leq a\). The ordinal \(\sigma \) is called the length of \(\mathcal A\). The lattice \(L\) is called strongly dense when \(L\) possesses a c.d.c. The dimension of a strongly dense lattice is the minimal length of a c.d.c. in \(L\). The importance of dense lattices comes from the study of a module \(M\) through the investigation of properties of its lattice of submodules. More precisely, strongly uniform modules are precisely those modules whose submodule lattice is strongly dense. In the first part the structure of strongly dense modular lattices is studied. For those of uncountable dimension an invariant called Gamma invariant is defined, which takes values in a certain Boolean algebra. This invariant measures in some sense the failure of the lattice to be relatively complemented. It is shown that all possible invariants are actually achieved in algebraic distributive lattices and an embedding theorem for a minimal algebraic distributive lattice with a given invariant is proved. In the second part connected problems coming from module theory are studied.
scientific article; zbMATH DE number 1407115