Decompositions in lattices and some representations of algebras (Q2760589)

The plan of this paper is as follows. In Chapter 1 the author investigates the properties of consistency, strongness and semimodularity, each of which may be viewed as a generalization of modularity. Chapters 2 and 3 present the decomposition theory of lattices. Here the author characterizes modularity in terms of the Kurosh-Ore replacement property. Next, he studies \(c\)-decompositions of elements in lattices (the notion of \(c\)-decomposition is introduced as a generalization of those of join and direct decompositions). He finds a common generalization of the Kurosh-Ore theorem and the Schmidt-Ore theorem for arbitrary modular lattices, solving a problem of G. Grätzer. The decomposition theory of Chapters 2 and 3 enables us to develop a structure theory for algebras. In Chapter 4 the author considers weak direct representations of a universal algebra. The existence of such representations is studied. Here some applications to algebras whose congruences permute (groups, rings, modules, quasigroups relatively complemented lattices, etc.) and to congruence distributive algebras (lattices, modular median algebras, etc.) are indicated. Chapter 5 contains a common generalization of full subdirect products and of weak direct products. This is an \(\langle{\mathcal L},\varphi\rangle\) representation of a subalgebra \(A\) of a direct product \(B\) of algebras, where \({\mathcal L}\) is an ideal in the power set of the index set of the direct product and \(\varphi\) is a binary relation on \(B\), and \(A\) is a subdirect product satisfying certain conditions involving \({\mathcal L}\) and \(\varphi\).